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How To Effectively Implement Homogeneous Grouping

As teachers, we’re always ready to make decisions that impact our classroom academically and socially. There’s a lot to consider, from deciding how best to implement our lesson plans to what kind of activity to use. One of those decisions they also need to decide is how to break the class into groups. If you’re in the education field, you’ve likely heard about homogeneous grouping. This overview will explore more about it and the pros and cons of implementing it in a classroom setting. 

Table of Contents

• What Is Homogeneous Grouping? 

Importance Of Homogeneous Grouping

• How Does Homogeneous Grouping Work? 

Advantages Of Using Homogeneous Grouping In Classrooms

Disadvantages of homogeneous grouping in classrooms.

• Examples of Homogeneous Grouping 

Understanding Homogeneous Grouping Effects

Homogeneous grouping faqs, what is homogeneous grouping.

Homogeneous grouping means placing students into groups with similar instructional levels or abilities. It’s also sometimes referred to as ability groups. The way the students get selected to be in each group can be determined via teacher observation, feedback, or possibly feedback from last year’s teacher. Students most likely will be working on similar-type assignments because they have closely matched abilities. 

Homogenous groups can allow students to problem-solve together more strategically without feeling that someone in the group is falling behind. For instance, if you’re utilizing it during literacy centers, each group student can access the same text. Using this technique can help when students need to work together to understand more complicated concepts. However, careful planning should be considered before selecting which students will be placed in similar-type groups.

How Does Homogeneous Grouping Work?

Flexible student groups are based on learning profiles, interests, and readiness, and homogeneous groups work the same way. So, although you’re placing groups of students with similar abilities together, they can change as long as you follow the same criterion. 

In these groups, you’ll most likely find that students have similar strengths or weaknesses in the material presented. However, the teacher might find it easier to facilitate and help students with similar abilities because they’ll have fewer needs to address.

If you’re considering grouping students of similar ability levels, here are a few benefits it can provide.

Better Paced Instruction For Students

Students who are similarly placed in groups will most likely find it easier to complete the task or assignment. One common issue with heterogenous groups is when higher-performing students might tend to do all the work. As a result, others in the group might copy answers or not fully understand the assignment. Having more confidence to complete the work assigned can also enable the teacher to challenge them further and move efficiently throughout the day.

Gifted Students Are More Comfortable

Since teachers have empathy and compassion for every student, it’s common for them to focus on helping lower-performing ones. Many times, gifted students aren’t getting the challenge they need with mixed ability groups. Here’s an example. If a student were to enter a kindergarten classroom knowing already how to read, they might be placed with other students starting with letter recognition. Consequently, they might become frustrated or bored.

One of the primary reasons children misbehave is out of boredom. Students who feel adequately challenged throughout the day will not only respond better to classroom instruction, but they’ll feel more confident as time passes by.

Scale Difficulty

Homogeneous groups allow teachers to challenge their kids with increasingly difficult assignments continually. Whether you tweak some instructions or use more inquiry-based learning, you’ll discover that these grouping types can achieve your expectations and far beyond.

While there are several benefits of homogeneous groups, teachers should also consider the disadvantages. Here are some possible downsides to using this grouping type.

Lower-Performing Students Feel Isolated

Higher performing students often are “leaders” of the group who help guide others. However, what happens when the low-performing students are all placed together? It’s possible that you’ll quickly have frustrated students who don’t understand assignments, and the teacher will need to spend more time addressing those concerns.

Reduced Likelihood For Academic Advancement

Students of all ability levels need to be a high level of engagement to advance academically. Unfortunately, one common trap that’s easy to fall into is the lack of advancement because of reduced expectations. Unfortunately, many students who are part of homogeneous groups with other lower performing students have a higher chance to remain there for the rest of their K-12 education.  

Grade Level Students’ Superiority

There can be a stigma felt by students who aren’t quite as ready socially or academically as others in the class. It’s more common for these students to receive lower-quality instruction because the behavior and academic challenges take more planning . While teachers are doing everything they can, students in such groups can be more likely to “fall through the cracks.” 

Examples of Homogeneous Grouping

Educators often place students in similar type ability levels without even realizing it. They seem to have a natural ability to know what students need, even without prior assessment or data. Here are some examples of how it would look in reading or math instruction.

Literacy Centers

A teacher could use baseline data, such as district placement tests, classroom exams, or informal observations to place students into groups based on ability level.

• Higher performing students may be working on literature circles where each group member has a specific role. They are working together to read and respond to the text.
• Grade-level students may all have access to the same text, addressing literary standards. For example, they may be asked to read a section and respond to questions prepared by the teacher. (The teacher should also intervene and coach, as necessary)
• Lower performing students could have decodable text that was read previously. Ensuring students have assignments they feel successful with is critical to promoting confidence.

Math Centers

Math can tend to be quicker paced than reading, so careful placement of groups is essential here. 

• Students above grade level can be given assignments to frontload the following topics, even if it hasn’t been taught yet. They could also be given extensions tothe current chapter, such as higher-level word problems , real-world examples, and so forth.
• Students performing at grade level could be working together on independent activities with the lesson that was taught. Again, the teacher should monitor to make corrections and provide feedback.
• If you’re placing students in homogeneous groups in math, hands-on activities with previously taught material can help them close gaps. You may also want to consider technology programs that are adaptive to learning to thoroughly fill help with concepts that weren’t covered in class. 

As you see, there are advantages and possible drawbacks to using homogeneous grouping in the classroom. However, if teachers consider everything with careful planning, student achievement levels can soar. 

Homogeneous groups allow teachers to place students of similar academic abilities together. As a result, students can have more challenging material presented to them to deepen their understanding.

Children can learn just as well in homogeneous groping, but they must be presented with instructions and material that’s right for their ability level. If the assignments are too complex, they might get frustrated. On the other hand, when assignments are too easy, boredom can set in.

Homogeneous grouping in education has been around for some time. It has been researched, but there is some controversy in whether the effectiveness remains comparable to heterogeneous grouping.

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Charles is a professional educator with 14 years of classroom experience in an elementary setting. He also spends time writing educational blogs and helping teachers become more aware of reading strategies to assist students of all learning abilities.

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Maslow Before Bloom: Putting Students’ Hearts Before Their Heads

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Homogeneous task is a task that requires doing identical type of work from task's beginning to task's completion. For example writing a book can be considered as homogeneous task, because in order to complete this task you should write few hundreds of pages one by one. Another example of homogenous task is painting of fence, because this process consists of typical operations as well. In other words such task can be divided into equal portions - let's say you can write one page every day during one year or you can cover with paint one section of fence per minute.

Homogeneous task is an easy object for management, because you can easily plan such task and you can easily increase or decrease intensity of your plan. When you perform series of typical operations number of factors that can influence your performance is stable and your risks are known. To perform homogeneous tasks better you should try to set plan of maximal intensity (ex: try to complete your task twice faster) for the beginning. You should aim to work with increased intensity at the beginning of your task (but do not forget about quality of work), so you will be able to complete most of your task in the first half of time allocated for this task.

VIP Task Manager is task and time management groupware that allows you to plan your tasks, business processes and projects. To manage your tasks you need to:

• start your task planning application
• divide task into smaller tasks
• plan these tasks, prioritize and categorize them
• set order of tasks
• create task templates for equal tasks

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Student Learning Groups: Homogeneous or Heterogeneous?

Whether to form groups with students of similar or mixed ability depends on the purpose of the learning activity.

“OK, kids, we’re going to be learning in groups today. Each group needs a math checker, a presenter, a writer/editor, and an illustrator. You decide who does what. You’ll be reviewing the best ways to solve polynomial problems. Please pull out the instructions and the rubric for this assignment. As a group, your task is to create a one-page, step-by-step process that someone could follow to arrive at a solution. You have 15 minutes to complete this task according to the rubric that I have handed out. Ready, set, go!”

The teacher who gives these instructions then spends the next 15 minutes roving about the classroom, reviewing the progress of each group and asking probing questions to help the individual groups clarify their thinking.

Grouping sounds so easy. What we don’t see in the above example is how the teacher has organized students in the groups in order to achieve the best results. Some educators firmly believe that a teacher must mix the groups so that students of all levels are represented in each group (heterogeneous grouping of students), while others believe that a teacher must organize the students by ability levels (homogeneous grouping of students). Robert Marzano, Debra Pickering, and Jane Pollock explain in Classroom Instruction That Works  (first edition) that there are advantages to both methods, depending on what the teacher wants to do.

Identifying Purposes

If the purpose of the group learning activity is to help struggling students, the research shows that heterogeneous groups may help most. On the other hand, if the purpose is to encourage medium ability groups to learn at high levels, homogeneous grouping would be better.

I learned this as a teacher when one of my gifted and talented students told me in confidence that she really hated being in heterogeneous groups (she said it differently, of course) all the time because by default, the other members of the group expected her to be the leader, organize things, and do all the work.

This was a tipping point for me, because it made me realize that I wasn’t grouping students for increased learning. I was using grouping mainly as a discipline management tool, and in actuality my attempt to increase student engagement had completely backfired. By always making sure that the “smart” students and the struggling students were equally divided in the groups, I was actually limiting the student participation to the de facto leaders of the groups.

Deciding Which Is Best

Because of this epiphany, I remember vowing that I would further differentiate my teaching by also seeking ways to give the upper-level students challenging and engaging learning activities. I promised to stop using the “good kids” in the hope that some of their “goodness” would rub off on the other students. An interesting thing happened when I grouped the students by ability. New leadership structures formed, and students who had never actively participated in groups before all of a sudden demonstrated skills and creativity that I never knew they had.

Students are smart and can easily figure out what we’re really doing. Students, in our classrooms, know when they are being grouped to mainly tutor and remediate less capable students and... most of the time they resent it. We can also tick them off when we form groups solely for discipline purposes by placing the calm, obedient students in each group to separate and calm down the unruly ones. My daughter Mercedes, who falls in both categories above, said that when teachers do this to her, she doesn’t learn and it’s not fun for her or the other students. Perhaps more often than not, students are savvy enough to play along when they recognize that the grouping is nothing more than a routine way to spend the time and has no real learning purpose at all.

If given a choice, students prefer to learning in groups of their peers and friends (homogeneous groups), but they also appreciate getting to know and learn from other members of the classroom. This requires that we trust students to make good decisions and hold them accountable for following the norms of learning in groups.

According to Marzano, Pickering and Pollock, effective learning in groups must have at least the following elements:

• The work must involve every member of the group.
• Each person has a valid job to perform with a known standard of completion.
• Each member is invested in completing the task or learning goal.
• Each member is accountable individually and collectively.

Remember that the desks are not attached to the floor—we can mix things up in heterogeneous and homogeneous groups in interesting and creative ways: eye color, left- or right-handedness, preferred pizza toppings, number of siblings, music preferences, gender, nationality, hair length, shoe laces, genetic traits, learning styles, etc.

Module 11: Chi-Square Tests

Test of homogeneity, learning outcomes.

• Conduct a chi-square test of homogeneity. Interpret the conclusion in context.

We have learned the details for two chi-square tests, the goodness-of-fit test, and the test of independence. Now we focus on the third and last chi-square test that we will learn, the test for homogeneity . This test determines if two or more populations (or subgroups of a population) have the same distribution of a single categorical variable.

The test of homogeneity expands the test for a difference in two population proportions, which is the two-proportion Z-test we learned in Inference for Two Proportions . We use the two-proportion Z-test when the response variable has only two outcome categories and we are comparing two populations (or two subgroups.) We use the test of homogeneity if the response variable has two or more categories and we wish to compare two or more populations (or subgroups.)

We can answer the following research questions with a chi-square test of homogeneity:

• Does the use of steroids in collegiate athletics differ across the three NCAA divisions?
• Was the distribution of political views (liberal, moderate, conservative) different for last three presidential elections in the United States?

The null hypothesis states that the distribution of the categorical variable is the same for the populations (or subgroups). In other words, the proportion with a given response is the same in all of the populations, and this is true for all response categories. The alternative hypothesis says that the distributions differ.

Note: Homogeneous means the same in structure or composition. This test gets its name from the null hypothesis, where we claim that the distribution of the responses are the same (homogeneous) across groups.

To test our hypotheses, we select a random sample from each population and gather data on one categorical variable. As with all chi-square tests, the expected counts reflect the null hypothesis. We must determine what we expect to see in each sample if the distributions are identical. As before, the chi-square test statistic measures the amount that the observed counts in the samples deviate from the expected counts.

Steroid Use in Collegiate Sports

In 2006, the NCAA published a report called “Substance Use: NCAA Study of Substance Use of College Student-Athletes.” We use data from this report to investigate the following question: Does steroid use by student athletes differ for the three NCAA divisions?

The data comes from a random selection of teams in each NCAA division. The sampling plan was somewhat complex, but we can view the data as though it came from a random sample of athletes in each division. The surveys are anonymous to encourage truthful responses.

To see the NCAA report on substance use, click here .

Step 1: State the hypotheses.

In the test of homogeneity, the null hypothesis says that the distribution of a categorical response variable is the same in each population. In this example, the categorical response variable is steroid use (yes or no). The populations are the three NCAA divisions.

• H 0 : The proportion of athletes using steroids is the same in each of the three NCAA divisions.
• H a : The proportion of athletes using steroids is not same in each of the three NCAA divisions.

Note: These hypotheses imply that the proportion of athletes not using steroids is also the same in each of the three NCAA divisions, so we don’t need to state this explicitly. For example, if 2% of the athletes in each division are using steroids, then 98% are not.

Here is an alternative way we could state the hypotheses for a test of homogeneity.

• H 0 : For each of the three NCAA divisions, the distribution of “yes” and “no” responses to the question about steroid use is the same.
• H a : The distribution of responses is not the same.

Step 2: Collect and analyze the data.

We summarized the data from these three samples in a two-way table.

We use percentages to compare the distributions of yes and no responses in the three samples. This step is similar to our data analysis for the test of independence.

We can see that Division I and Division II schools have essentially the same percentage of athletes who admit steroid use (about 1.2%). Not surprisingly, the least competitive division, Division III, has a slightly lower percentage (about 1.0%). Do these results suggest that the proportion of athletes using steroids is the same for the three divisions? Or is the difference seen in the sample of Division III schools large enough to suggest differences in the divisions? After all, the sample sizes are very large. We know that for large samples, a small difference can be statistically significant. Of course, we have to conduct the test of homogeneity to find out.

Note: We decided not to use ribbon charts for visual comparison of the three distributions because the percentage admitting steroid use is too small in each sample to be visible.

Step 3: Assess the evidence.

We need to determine the expected values and the chi-square test statistic so that we can find the P-value.

Calculating Expected Values for a Test of Homogeneity

Expected counts always describe what we expect to see in a sample if the null hypothesis is true. In this situation, we expect the percentage using steroids to be the same for each division. What percentage do we use? We find the percentage using steroids in the combined samples. This calculation is the same as we did when finding expected counts for a test of independence, though the logic of the calculation is subtly different.

Here are the calculations for the response “yes”:

• Percentage using steroids in combined samples: 220/19,377 = 0.01135 = 1.135%

Expected count of steroid users for Division I is 1.135% of Division I sample:

• 0.01135(8,543) = 96.96

Expected count of steroid users for Division II is 1.135% of Division II sample:

• 0.01135(4,341) = 49.27

Expected count of steroid users for Division III is 1.135% of Division III sample:

• 0.01135(6,493) = 73.70

Checking Conditions

The conditions for use of the chi-square distribution are the same as we learned previously:

• A sample is randomly selected from each population.
• All of the expected counts are 5 or greater.

Since this data meets the conditions, we can proceed with calculating the χ 2 test statistic.

Calculating the Chi-Square Test Statistic

There are no changes in the way we calculate the chi-square test statistic.

[latex]{\chi }^{2}\text{}=\text{}∑\frac{{(\mathrm{observed}-\mathrm{expected})}^{2}}{\mathrm{expected}}[/latex]

We use technology to calculate the chi-square value. For this example, we show the calculation. There are six terms, one for each cell in the 3 × 2 table. (We ignore the totals, as always.)

Finding Degrees of Freedom and the P-Value

For chi-square tests based on two-way tables (both the test of independence and the test of homogeneity), the degrees of freedom are ( r − 1)( c − 1), where r is the number of rows and c is the number of columns in the two-way table (not counting row and column totals). In this case, the degrees of freedom are (3 − 1)(2 − 1) = 2.

We use the chi-square distribution with df = 2 to find the P-value. The P-value is large (0.4561), so we fail to reject the null hypothesis.

Step 4: Conclusion.

The data does not provide strong enough evidence to conclude that steroid use differs in the three NCAA divisions (P-value = 0.4561).

First Use of Anabolic Steroids by NCAA Athletes

The NCAA survey includes this question: “When, if ever, did you start using anabolic steroids?” The response options are: have never used, before junior high, junior high, high school, freshman year of college, after freshman year of college. We focused on those who admitted use of steroids and compared the distribution of their responses for the years 1997, 2001, and 2005. (These are the years that the NCAA conducted the survey. Counts are estimates from reported percentages and sample size.) Recall that the NCAA uses random sampling in its sampling design.

Use this simulation to answer the questions below.

We now know the details for the chi-square test for homogeneity. We conclude with two activities that will give you practice recognizing when to use this test.

Gender and Politics

Consider these two situations:

• A: Liberal, moderate, or conservative: Are there differences in political views of men and women in the United States? We survey a random sample of 100 U.S. men and 100 U.S. women.
• B: Do you plan to vote in the next presidential election? We ask a random sample of 100 U.S. men and 100 U.S. women. We look for differences in the proportion of men and women planning to vote.

Steroid Use for Male Athletes in NCAA Sports

We plan to compare steroid use for male athletes in NCAA baseball, basketball, and football. We design two different sampling plans.

• A: Survey distinct random samples of NCAA athletes from each sport: 500 baseball players, 400 basketball players, 900 football players.
• B. Survey a random sample of 1,800 NCAA male athletes and categorize players by sport and admitted steroid use. Responses are anonymous.

Let’s Summarize

In “Chi-Square Tests for Two-Way Tables,” we discussed two different hypothesis tests using the chi-square test statistic:

• Test of independence for a two-way table
• Test of homogeneity for a two-way table

Test of Independence for a Two-Way Table

• In the test of independence, we consider one population and two categorical variables.
• In Probability and Probability Distribution , we learned that two events are independent if P ( A | B ) = P ( A ), but we did not pay attention to variability in the sample. With the chi-square test of independence, we have a method for deciding whether our observed P ( A | B ) is “too far” from our observed P ( A ) to infer independence in the population.
• The null hypothesis says the two variables are independent (or not associated). The alternative hypothesis says the two variables are dependent (or associated).
• To test our hypotheses, we select a single random sample and gather data for two different categorical variables.
• Example: Do men and women differ in their perception of their weight? Select a random sample of adults. Ask them two questions: (1) Are you male or female? (2) Do you feel that you are overweight, underweight, or about right in weight?

Test of Homogeneity for a Two-Way Table

• In the test of homogeneity, we consider two or more populations (or two or more subgroups of a population) and a single categorical variable.
• The test of homogeneity expands on the test for a difference in two population proportions that we learned in Inference for Two Proportions by comparing the distribution of the categorical variable across multiple groups or populations.
• The null hypothesis says that the distribution of proportions for all categories is the same in each group or population. The alternative hypothesis says that the distributions differ.
• To test our hypotheses, we select a random sample from each population or subgroup independently. We gather data for one categorical variable.
• Example: Is the rate of steroid use different for different men’s collegiate sports (baseball, basketball, football, tennis, track/field)? Randomly select a sample of athletes from each sport and ask them anonymously if they use steroids.

The difference between these two tests is subtle. They differ primarily in study design. In the test of independence, we select individuals at random from a population and record data for two categorical variables. The null hypothesis says that the variables are independent. In the test of homogeneity, we select random samples from each subgroup or population separately and collect data on a single categorical variable. The null hypothesis says that the distribution of the categorical variable is the same for each subgroup or population.

Both tests use the same chi-square test statistic.

Chi-Square Test Statistic and Distribution

For all chi-square tests, the chi-square test statistic χ 2 is the same. It measures how far the observed data are from the null hypothesis by comparing observed counts and expected counts. Expected counts are the counts we expect to see if the null hypothesis is true.

The chi-square model is a family of curves that depend on degrees of freedom. For a two-way table, the degrees of freedom equals ( r − 1)( c − 1). All chi-square curves are skewed to the right with a mean equal to the degrees of freedom.

A chi-square model is a good fit for the distribution of the chi-square test statistic only if the following conditions are met:

• The sample is randomly selected.
• All expected counts are 5 or greater.

If these conditions are met, we use the chi-square distribution to find the P-value. We use the same logic that we have used in all hypothesis tests to draw a conclusion based on the P-value. If the P-value is at least as small as the significance level, we reject the null hypothesis and accept the alternative hypothesis. The P-value is the likelihood that results from random samples have a χ 2 value equal to or greater than that calculated from the data if the null hypothesis is true.

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The Trouble With Homogeneous Teams

Diversity in the workplace can increase conflict. But research also suggests that if teams lack diversity, they will be more susceptible to making flawed decisions.

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Many companies today understandably focus on workplace diversity — issues such as how to increase diversity, how to foster sensitivity to it, and how to manage a diverse workforce. But, according to MIT Sloan School professor Evan Apfelbaum, managers should also be cognizant of another, related topic: the problems associated with homogeneity. Recent research, including Apfelbaum’s own, has found, for example, that racially homogeneous groups are less rigorous in their decision-making — and make more mistakes — than diverse ones.

Apfelbaum, the W. Maurice Young (1961) Career Development Professor of Management and an associate professor of work and organization studies at the MIT Sloan School, spoke with MIT Sloan Management Review editorial director Martha E. Mangelsdorf. What follows is a condensed and edited version of their conversation.

MIT Sloan Management Review : You’re an expert in research on diversity and how it affects group decision-making. And one thing you and others have found is that diverse groups often do better in decision-making than more homogeneous ones. Can you tell us a bit about some of the important studies in that area and what they found?

Apfelbaum: Sure. A good way to think about it is that diverse groups have the potential to do better than homogeneous ones. In reality, there are a number of examples and reasons why that often doesn’t happen. But I do think there’s a unique advantage to diverse groups in certain areas.

I’ll start off by talking about cooperative decision-making scenarios, where people are trying to work together to come to some best solution. Early work from several decades ago provided the first evidence that diverse groups yielded more creative solutions, and that spurred much of the more recent research in that area.

One paper that was particularly important and useful took place in a legal setting, with jurors. In that study, a researcher now at Tufts University got access to a real jury pool and randomly assigned jurors to deliberate in six-person, all-white or racially diverse juries. The groups all considered the same fictitious case, and their deliberations were recorded on video.

In general, the diverse juries were far more rigorous in how they approached their decisions. The racially diverse juries spent a longer time deliberating. They considered a wider range of perspectives and angles with respect to the case — different things that could have happened or might have been important. And they made fewer factually inaccurate statements in their discussions.

It wasn’t the case that diverse juries were outperforming homogeneous ones primarily because, say, the black jurors were adding new information that wasn’t there in the all-white juries. It was actually white jurors on diverse juries whose behavior showed the most dramatic change. This suggested that it was something about being in the presence of a racially diverse environment that changed how people thought and discussed issues.

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Subsequent research has looked at student groups working on projects, and this work has shown similar effects. For example, one project demonstrated that when you tell students that they’re going to be having a discussion regarding a written article, they prepare more rigorously if they know that they’re going to have to debate in a more diverse group.

In that case, it might have been because the article had some race-related component to it, and the students wanted to make sure that they had a well-rehearsed way to think about that. The study also showed that when researchers asked students to write essays after the fact to reflect on what they took away from the discussion, these essays typically were more complex and of higher quality when the discussion took place in a racially diverse group.

Those are some important effects of diversity. Now, let’s take a more traditional business setting, and one that’s competitive, not cooperative. In one paper that I coauthored, we asked: What happens if you put diverse versus homogeneous groups together in a naturally occurring competitive scenario? To examine this case, we looked at trading markets.

We randomly assigned people to either racially diverse or homogeneous groups in two different studies, one in Asia and one in North America. People were brought to the lab, and we formed mini trading markets of six people. Think of it as a six-person competitive group. The participants were given real money, and there were several rounds of trading where the groups were networked through computers. The participants were making decisions about whether they would like to buy and sell assets, and their goal was to end up with the most money at the end.

The only difference between the two groups was that at the very beginning of the study, people saw who would be in their market based on who was sitting with them in the waiting room. And we randomly assigned half of these groups to be homogeneous. The beauty of this is that it wasn’t just an all-white sort of homogeneity, because we were able to do this experiment in Asia in a way where the dominant culture identity was not white. So half of the groups see a singular ethnicity, and half see a more diverse one.

Then the group members are separated and begin trading. A couple of interesting results emerge here. The first thing is that there’s a difference in accuracy, in how closely people are pricing assets to their actual value. In the homogeneous groups, there was more inaccuracy and mispricing; there was a tendency to spend more for things than they were actually worth.

The other thing that was interesting is that these mistakes were, in a sense, more contagious in homogeneous than diverse groups. That is, not only were people in homogeneous groups more likely to make pricing errors, but other people in those groups were more likely to copy those errors. People in homogeneous groups were more likely to assume that other people in the group knew what they were doing.

In diverse groups, people were less likely to trust the wisdom of other people’s purchasing choices. And the result of these two different dynamics that played out is that pricing bubbles, which are very problematic in financial markets, and for society at large, were more likely in this experimental context to form in homogeneous groups. The reason for this is that people in homogeneous groups were more likely to copy another person’s mistake — presumably assuming that the mistake had some value that they just didn’t understand. In homogeneous groups, there was this escalating effect where people would copy poor decisions.

So we were able to see very different trajectories even within the lab, and in very similar ways in two very different cultures, suggesting that there is something fundamental about working with similar versus different others that affects individuals’ decision-making. Again, this is a competitive context: People were really motivated to try to eke out as much money as they could because, at the end of the experiment, they kept the money they had made. That suggests that people in the homogeneous groups were trying to make the right decisions — but something about the group context constrained their ability to do so.

One of the ideas we had is that maybe this is just something basic about conformity. So we essentially ran a variation of what is one of the most famous social psychological experiments ever to be run in the domain of groups: psychologist Solomon Asch’s conformity paradigm from the 1950s. In that famous study, participants would sit at a table with people they thought were other study participants, and they would simply look at a picture of three lines of three different lengths.

The study participants believed that the other people at the table were also participants, but the other people were actually working for the person running the experiment. And the experimenter said, for example, “Tell me which line is the longest.” What the participants would experience is that they would hear other people answer before them and all say that the third line was the longest, when it was clearly evident that actually it was the second line that was the longest. It was such an obvious answer, but the experimenters in this case were looking at how likely participants are to yield to majority opinion, even when they know that it’s the wrong answer. In other words, how much can social pressure affect us?

Asch’s research found that people yield to what they know to be the wrong answer around roughly 30% of the time — which is a pretty large frequency. We wondered if diversity would change people’s susceptibility to this bias.

We replicated the same paradigm with a few changes; instead of lines, we used a task involving fictitious college applications, where we could establish that one candidate was a clearly stronger applicant for admission than another. What we found is that in all-white groups, the rate of conformity to the clearly wrong applicant was about 30% — which was similar to the classic research on conformity. In diverse groups, however, the frequency with which people would yield to what they know to be the wrong answer dropped significantly, to 20% in some experiments, and even lower in others.

What’s interesting about these studies is that we didn’t allow people to talk with the other people in the room. What we were looking at is not the effect of having a discussion or being persuaded by arguments. We were asking: Does simply sitting down in a room and seeing the demographic makeup of the people at the table affect people’s propensity to conform to others’ decisions?

And the answer was that it does — and that people were less likely to conform in diverse groups. Similar to our suspicions in the stock-pricing experiment, there’s almost this benefit-of-the-doubt effect that happens in homogeneous groups that we don’t see in diverse groups; people in homogeneous groups are more likely to assume that the other people in the group must know something or have picked up on something that they didn’t. In diverse groups, people are more likely to not rely on those types of assumptions and come to an independent assessment of what they think to be the case.

I wonder if homogeneous groups make people feel more comfortable, and they then work less hard cognitively. Do researchers know what the mechanisms are that cause these differences in behavior?

Apfelbaum: What I say is, diversity is not better or worse — it’s just harder. It’s harder socially, it’s harder cognitively, and it makes us work. And I think that’s a useful framework to think about why diversity can be both advantageous and complicated in the workplace and in decision-making groups.

When we find ourselves in social events, there’s a natural inclination to gravitate toward people who are similar to us. It’s easier. It’s easier to find common ground with people who have similar backgrounds to us — whether it’s in terms of culture, organizational expertise, language, or school affiliation. That’s natural. It provides us with a sense of belonging and it’s easier — and I think that’s OK. Diversity is harder for the same kind of reasons. It doesn’t allow us to rest on our laurels, and we are less concerned, in some sense, with retaining our membership in diverse groups.

I think people just end up being more independent and objective in diverse groups. And that can go well in the scenarios I just talked about, but it is also, I believe, at the root of other research that has shown that diversity can breed conflict and mistrust. Some research, for example, has shown that even a normal level of team conflict is more quickly perceived as a really serious type of conflict in the eyes of managers when the group is racially diverse as compared to homogeneous. One very recent study presented participants with an exchange between members of a team. And the researchers just changed a very, very small component of that exchange — which was whether people believed that the people involved in this team exchange, which was designed to be medium-level debate and back and forth — were a diverse or homogeneous group.

What the researchers found is that people in the role of managers were much more likely to think that the diverse teams’ level of conflict was higher, even though it was exactly the same exchange. And in turn, the managers were less likely to suggest that greater resources be provided to the diverse teams to assist them with completing future projects. In some sense, managers were saying, “We can’t invest; this is an irreparable form of conflict.” So the level of conflict seems to be perceived to be artificially higher in diverse groups than it is in homogeneous groups.

I think that when it comes to gender, race, and ethnicity, these are issues in our society that are fraught and laced with mistrust and uncertainty, so there’s a lower threshold for people to find evidence that is consistent with that and either disengage from their groups, accuse others, or devolve into unproductive forms of conflict in groups.

Interesting. It sounds like in most of the studies you’re discussing, the diversity is racial. And are there similar findings with different types of diversity, such as gender diversity?

Apfelbaum: I would say race and culture are two of the most frequent ones that have been explored. Gender has been explored, and there’s been some similar experiments there. I think that research now is really only just beginning to look at, for example, how race and gender may play out differently. There is also some research out there that has looked at cognitive diversity — for example, diversity in the way people think about problems.

Related Articles

You mentioned research about the cost of diversity and the conflicts that can happen, and how they can quickly get unproductive. tell me a little about that research..

Apfelbaum: There is a good amount of research that’s happened in the past few decades that has found in real work teams that people in diverse teams report higher degrees of conflict. They like it less. They’re less comfortable there. And there are a number of different studies that have demonstrated more interpersonal conflict in diverse teams.

Let me tell you about another finding that I think is pretty interesting. In one paper, researchers looked at a decision-making task that was cooperative. Participants in the group had to put together disparate pieces of information — clues — to make a single recommendation about the correct suspect to arrest.

What the researchers found in this task, as has been shown in previous research, is that the diverse groups tended to consider more perspectives, and ultimately were more likely than people in the homogeneous groups to narrow in on the correct suspect.

But another result that came out that was very, very interesting was that in this study the researchers also asked: How confident are you that you have identified the correct suspect? And though the diverse groups were factually more accurate than the homogeneous groups, it was actually the homogeneous groups that were more confident in their results — the exact opposite of what you would expect.

Now you could think of this as a challenge to diversity or a limitation to homogeneity. It suggests that diverse groups with the same results as homogeneous groups can come out of a meeting stating that they’re less confident that they have achieved the correct objective or have landed on a workable solution than a homogeneous group will be. And what we know about confidence in organizational settings is that it is reinforced. If you’re managing two groups and one group comes to you and says, “We are 95% sure this is going to be on time, under budget, and workable,” and the other group comes to you and says, “We’re 75% sure,” in the real world, nine times out of 10 it’s the group that says that they’re 95% sure who is going to get the opportunity to run with their project and try to deliver.

And what we’ve seen from this data is that you’re more likely to hear that 95% story from the homogeneous group — but it’s not because they are more likely to deliver better results. The homogeneous groups may just be less accurate. In homogeneous groups, there seems to be this inflated sense of confidence, in part because of the phenomena unearthed in the research that I’ve talked about earlier. Those groups may not be considering all the perspectives. And there is more of a tendency to narrowly see the issues in ways consistent with other people’s views and perhaps less comfort to disagree with others.

So the diverse groups are actually mitigating overconfidence bias in a way?

Apfelbaum: Yes.

What should managers take away from this research? What advice would you give?

Apfelbaum: The takeaway for me is that the diversity needs to be carefully managed. Managers need to mitigate the concerns about people not feeling comfortable in order to harness what can be some of these real distinct advantages of diversity.

And what about these findings about teams? Should executives be thinking, if we have a really complex decision to make, the decision-making group should be more diverse?

Apfelbaum: Certainly, when you have to make a large-level organizational change or you’re making a big decision, people are often involved from many different functional groups, so everyone can see each other’s blind spots to some degree.

I think that should sort of be the status quo in organizations. And I think that with a really inclusive culture, the lack of ease and comfort that people typically associate with diverse groups can be normalized.

If you think back to a lot of the data that we’ve just gone over, at least a good portion of it says that, well, in objective ways, homogeneity is the thing that’s producing the strange results. Think back to the overconfidence results. Think back to the amount of inaccuracy.

But how many leaders in organizations do you know who have thought, “Wow, what can we do about the problems of homogeneity? Where are the most homogeneous teams that we have in our organization, and what can we do to make sure that they are thinking really carefully and there’s some productive conflict?” I don’t know of a single program anywhere in the world that is focusing on the potential blind spots of homogeneous teams. And I think that’s just not the narrative, because, in many industries, homogeneous teams are normal in terms of their frequency.

But even if homogeneous teams are normal in the sense that they’re common, there’s reason to question how normal they are in these other ways. Instead of just looking at the management of diverse groups as a problem to be solved, it’s useful to flip that for a second and think of it from the other side: What can we do about the problematic aspects of homogeneous teams?

About the Author

Martha E. Mangelsdorf is the editorial director of MIT Sloan Management Review . She tweets at @memangelsdorf .

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1.3: Homogeneous Equations

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• Page ID 58773

• W. Keith Nicholson
• University of Calgary via Lyryx Learning

A system of equations in the variables \(x_1, x_2, \dots, x_n\) is called homogeneous if all the constant terms are zero—that is, if each equation of the system has the form

\[a_1x_1 + a_2x_2 + \dots + a_nx_n = 0 \nonumber \]

Clearly \(x_1 = 0, x_2 = 0, \dots, x_n = 0\) is a solution to such a system; it is called the trivial solution . Any solution in which at least one variable has a nonzero value is called a nontrivial solution . Our chief goal in this section is to give a useful condition for a homogeneous system to have nontrivial solutions. The following example is instructive.

Example \(\PageIndex{1}\)

Show that the following homogeneous system has nontrivial solutions.

\[ \begin{array}{rlrlrlrcr} x_1 & - & x_2 & + & 2x_3 & - & x_4 & = & 0 \\ 2x_1 & + &2x_2 & & & + & x_4 & = & 0 \\ 3x_1 & + & x_2 & + & 2x_3 & - & x_4 & = & 0 \end{array} \nonumber \]

The reduction of the augmented matrix to reduced row-echelon form is outlined below.

\[\left[ \begin{array}{rrrr|r} 1 & -1 & 2 & -1 & 0 \\ 2 & 2 & 0 & 1 & 0 \\ 3 & 1 & 2 & -1 & 0 \end{array} \right] \rightarrow \left[ \begin{array}{rrrr|r} 1 & -1 & 2 & -1 & 0 \\ 0 & 4 & -4 & 3 & 0 \\ 0 & 4 & -4 & 2 & 0 \end{array} \right] \rightarrow \left[ \begin{array}{rrrr|r} 1 & 0 & 1 & 0 & 0 \\ 0 & 1 & -1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{array} \right] \nonumber \]

The leading variables are \(x_1\), \(x_2\), and \(x_4\), so \(x_3\) is assigned as a parameter—say \(x_3 = t\). Then the general solution is \(x_1 = -t\), \(x_2 = t\), \(x_3 = t\), \(x_4 = 0\). Hence, taking \(t = 1\) (say), we get a nontrivial solution: \(x_1 = -1\), \(x_2 = 1\), \(x_3 = 1\), \(x_4 = 0\).

The existence of a nontrivial solution in Example \(\PageIndex{1}\) is ensured by the presence of a parameter in the solution. This is due to the fact that there is a nonleading variable (\(x_3\) in this case). But there must be a nonleading variable here because there are four variables and only three equations (and hence at most three leading variables). This discussion generalizes to a proof of the following fundamental theorem.

Theorem \(\PageIndex{1}\)

If a homogeneous system of linear equations has more variables than equations, then it has a nontrivial solution (in fact, infinitely many).

Proof . Suppose there are \(m\) equations in \(n\) variables where \(n > m\), and let \(R\) denote the reduced row-echelon form of the augmented matrix. If there are \(r\) leading variables, there are \(n - r\) nonleading variables, and so \(n - r\) parameters. Hence, it suffices to show that \(r < n\). But \(r \leq m\) because \(R\) has \(r\) leading 1s and \(m\) rows, and \(m < n\) by hypothesis. So \(r \leq m < n\), which gives \(r < n\).

Note that the converse of Theorem [thm:001473] is not true: if a homogeneous system has nontrivial solutions, it need not have more variables than equations (the system \(x_1 + x_2 = 0\), \(2x_1 + 2x_2 = 0\) has nontrivial solutions but \(m = 2 = n\).)

Theorem [thm:001473] is very useful in applications. The next example provides an illustration from geometry.

Example \(\PageIndex{2}\)

We call the graph of an equation \(ax^2 + bxy + cy^2 + dx + ey + f = 0\) a conic if the numbers \(a\), \(b\), and \(c\) are not all zero. Show that there is at least one conic through any five points in the plane that are not all on a line.

Let the coordinates of the five points be \((p_1, q_1)\), \((p_2, q_2)\), \((p_3, q_3)\), \((p_4, q_4)\), and \((p_5, q_5)\). The graph of \(ax^2 + bxy + cy^2 + dx + ey + f = 0\) passes through \((p_i, q_i)\) if

\[ap_i^2 + bp_iq_i + cq_i^2 + dp_i + eq_i + f = 0 \nonumber \]

This gives five equations, one for each \(i\), linear in the six variables \(a\), \(b\), \(c\), \(d\), \(e\), and \(f\). Hence, there is a nontrivial solution by Theorem [thm:001473]. If \(a = b = c = 0\), the five points all lie on the line with equation \(dx + ey + f = 0\), contrary to assumption. Hence, one of \(a\), \(b\), \(c\) is nonzero.

Linear Combinations and Basic Solutions

As for rows, two columns are regarded as equal if they have the same number of entries and corresponding entries are the same. Let \(\mathbf{x}\) and \(\mathbf{y}\) be columns with the same number of entries. As for elementary row operations, their sum \(\mathbf{x} + \mathbf{y}\) is obtained by adding corresponding entries and, if \(k\) is a number, the scalar product \(k\mathbf{x}\) is defined by multiplying each entry of \(\mathbf{x}\) by \(k\). More precisely:

\[\mbox{If } \mathbf{x} = \left[ \begin{array}{c} x_1 \\ x_2 \\ \vdots \\ x_n \end{array} \right] \mbox{and } \mathbf{y} = \left[ \begin{array}{c} y_1 \\ y_2 \\ \vdots \\ y_n \end{array} \right] \mbox{then } \mathbf{x} + \mathbf{y} = \left[ \begin{array}{c} x_1 + y_1 \\ x_2 + y_2 \\ \vdots \\ x_n + y_n \end{array} \right] \mbox{and } k\mathbf{x} = \left[ \begin{array}{c} kx_1 \\ kx_2 \\ \vdots \\ kx_n \end{array} \right]. \nonumber \]

A sum of scalar multiples of several columns is called a linear combination of these columns. For example, \(s\mathbf{x} + t\mathbf{y}\) is a linear combination of \(\mathbf{x}\) and \(\mathbf{y}\) for any choice of numbers \(s\) and \(t\).

Example \(\PageIndex{3}\)

If \(\mathbf{x} = \left[ \begin{array}{r} 3 \\ -2 \\ \end{array} \right]\) and \(\mathbf{y} = \left[ \begin{array}{r} -1 \\ 1 \\ \end{array} \right]\) then \(2\mathbf{x} + 5\mathbf{y} = \left[ \begin{array}{r} 6 \\ -4 \\ \end{array} \right] + \left[ \begin{array}{r} -5 \\ 5 \\ \end{array} \right] = \left[ \begin{array}{r} 1 \\ 1 \\ \end{array} \right]\)..

Example \(\PageIndex{4}\)

Let \(\mathbf{x} = \left[ \begin{array}{r} 1 \\ 0 \\ 1 \end{array} \right], \mathbf{y} = \left[ \begin{array}{r} 2 \\ 1 \\ 0 \end{array} \right]\) and \(\mathbf{z} = \left[ \begin{array}{r} 3 \\ 1 \\ 1 \end{array} \right]\). If \(\mathbf{v} = \left[ \begin{array}{r} 0 \\ -1 \\ 2 \end{array} \right]\) and \(\mathbf{w} = \left[ \begin{array}{r} 1 \\ 1 \\ 1 \end{array} \right]\), determine whether \(\mathbf{v}\) and \(\mathbf{w}\) are linear combinations of \(\mathbf{x}\), \(\mathbf{y}\) and \(\mathbf{z}\).

For \(\mathbf{v}\), we must determine whether numbers \(r\), \(s\), and \(t\) exist such that \(\mathbf{v} = r\mathbf{x} + s\mathbf{y} + t\mathbf{z}\), that is, whether

\[\left[ \begin{array}{r} 0 \\ -1 \\ 2 \end{array} \right] = r \left[ \begin{array}{r} 1 \\ 0 \\ 1 \end{array} \right] + s \left[ \begin{array}{r} 2 \\ 1 \\ 0 \end{array} \right] + t \left[ \begin{array}{r} 3 \\ 1 \\ 1 \end{array} \right] = \left[ \begin{array}{c} r + 2s + 3t \\ s + t \\ r + t \end{array} \right] \nonumber \]

Equating corresponding entries gives a system of linear equations \(r + 2s + 3t = 0\), \(s + t = -1\), and \(r + t = 2\) for \(r\), \(s\), and \(t\). By gaussian elimination, the solution is \(r = 2 - k\), \(s = -1 - k\), and \(t = k\) where \(k\) is a parameter. Taking \(k = 0\), we see that \(\mathbf{v} = 2\mathbf{x} - \mathbf{y}\) is a linear combination of \(\mathbf{x}\), \(\mathbf{y}\), and \(\mathbf{z}\).

Turning to \(\mathbf{w}\), we again look for \(r\), \(s\), and \(t\) such that \(\mathbf{w} = r\mathbf{x} + s\mathbf{y} + t\mathbf{z}\); that is,

\[\left[ \begin{array}{r} 1 \\ 1 \\ 1 \end{array} \right] = r \left[ \begin{array}{r} 1 \\ 0 \\ 1 \end{array} \right] + s \left[ \begin{array}{r} 2 \\ 1 \\ 0 \end{array} \right] + t \left[ \begin{array}{r} 3 \\ 1 \\ 1 \end{array} \right] = \left[ \begin{array}{c} r + 2s + 3t \\ s + t \\ r + t \end{array} \right] \nonumber \]

leading to equations \(r + 2s + 3t = 1\), \(s + t = 1\), and \(r + t = 1\) for real numbers \(r\), \(s\), and \(t\). But this time there is no solution as the reader can verify, so \(\mathbf{w}\) is not a linear combination of \(\mathbf{x}\), \(\mathbf{y}\), and \(\mathbf{z}\).

Our interest in linear combinations comes from the fact that they provide one of the best ways to describe the general solution of a homogeneous system of linear equations. When solving such a system with \(n\) variables \(x_1, x_2, \dots, x_n\), write the variables as a column 1 matrix: \(\mathbf{x} = \left[ \begin{array}{c} x_1 \\ x_2 \\ \vdots \\ x_n \end{array} \right]\). The trivial solution is denoted \(\mathbf{0} = \left[ \begin{array}{c} 0 \\ 0 \\ \vdots \\ 0 \end{array} \right]\). As an illustration, the general solution in Example [exa:001449] is \(x_1 = -t\), \(x_2 = t\), \(x_3 = t\), and \(x_4 = 0\), where \(t\) is a parameter, and we would now express this by saying that the general solution is \(\mathbf{x} = \left[ \begin{array}{r} -t \\ t \\ t \\ 0 \end{array} \right]\), where \(t\) is arbitrary.

The reason for using columns will be apparent later.

Now let \(\mathbf{x}\) and \(\mathbf{y}\) be two solutions to a homogeneous system with \(n\) variables. Then any linear combination \(s\mathbf{x} + t\mathbf{y}\) of these solutions turns out to be again a solution to the system. More generally:

\[ \mbox{ \textit{Any linear combination of solutions to a homogeneous system is again a solution.}} \nonumber \]

In fact, suppose that a typical equation in the system is \(a_1x_1 + a_2x_2 + \dots + a_nx_n = 0\), and suppose that \(\mathbf{x} = \left[ \begin{array}{c} x_1 \\ x_2 \\ \vdots \\ x_n \end{array} \right]\), \(\mathbf{y} = \left[ \begin{array}{c} y_1 \\ y_2 \\ \vdots \\ y_n \end{array} \right]\) are solutions. Then \(a_1x_1 + a_2x_2 + \dots + a_nx_n = 0\) and \(a_1y_1 + a_2y_2 + \dots + a_ny_n = 0\). Hence \(s\mathbf{x} + t\mathbf{y} = \left[ \begin{array}{c} sx_1 + ty_1 \\ sx_2 + ty_2 \\ \vdots \\ sx_n + ty_n \end{array} \right]\) is also a solution because

\[\begin{aligned} a_1(sx_1 + ty_1) &+ a_2(sx_2 + ty_2) + \dots + a_n(sx_n + ty_n) \\ &= [a_1(sx_1) + a_2(sx_2) + \dots + a_n(sx_n)] + [a_1(ty_1) + a_2(ty_2) + \dots + a_n(ty_n)] \\ &= s(a_1x_1 + a_2x_2 + \dots + a_nx_n) + t(a_1y_1 + a_2y_2 + \dots + a_ny_n) \\ &= s(0) + t(0)\\ &= 0\end{aligned} \nonumber \]

A similar argument shows that Statement [eq:homogeneousstatement] is true for linear combinations of more than two solutions.

The remarkable thing is that every solution to a homogeneous system is a linear combination of certain particular solutions and, in fact, these solutions are easily computed using the gaussian algorithm. Here is an example.

Example \(\PageIndex{5}\)

Solve the homogeneous system with coefficient matrix

\[A = \left[ \begin{array}{rrrr} 1 & -2 & 3 & -2 \\ -3 & 6 & 1 & 0 \\ -2 & 4 & 4 & -2 \\ \end{array} \right] \nonumber \]

The reduction of the augmented matrix to reduced form is

\[\left[ \begin{array}{rrrr|r} 1 & -2 & 3 & -2 & 0 \\ -3 & 6 & 1 & 0 & 0 \\ -2 & 4 & 4 & -2 & 0 \\ \end{array} \right] \rightarrow \def\arraystretch{1.5} \left[ \begin{array}{rrrr|r} 1 & -2 & 0 & -\frac{1}{5} & 0 \\ 0 & 0 & 1 & -\frac{3}{5} & 0 \\ 0 & 0 & 0 & 0 & 0 \\ \end{array} \right] \nonumber \]

so the solutions are \(x_1 = 2s + \frac{1}{5}t\), \(x_2 = s\), \(x_3 = \frac{3}{5}t\), and \(x_4 = t\) by gaussian elimination. Hence we can write the general solution \(\mathbf{x}\) in the matrix form

\[\mathbf{x} = \left[ \begin{array}{r} x_1 \\ x_2 \\ x_3 \\ x_4 \end{array} \right] = \left[ \begin{array}{c} 2s + \frac{1}{5}t \\ s \\ \frac{3}{5}t \\ t \end{array} \right] = s \left[ \begin{array}{r} 2 \\ 1 \\ 0 \\ 0 \end{array} \right] + t \left[ \begin{array}{r} \frac{1}{5} \\ 0 \\ \frac{3}{5} \\ 1 \end{array} \right] = s\mathbf{x}_1 + t\mathbf{x}_2. \nonumber \]

Here \(\mathbf{x}_1 = \left[ \begin{array}{r} 2 \\ 1 \\ 0 \\ 0 \end{array} \right]\) and \(\mathbf{x}_2 = \left[ \begin{array}{r} \frac{1}{5} \\ 0 \\ \frac{3}{5} \\ 1 \end{array} \right]\) are particular solutions determined by the gaussian algorithm.

The solutions \(\mathbf{x}_1\) and \(\mathbf{x}_2\) in Example \(\PageIndex{5}\) are denoted as follows:

Definition: \(\PageIndex{1}\) Basic Solutions

The gaussian algorithm systematically produces solutions to any homogeneous linear system, called basic solutions , one for every parameter.

Moreover, the algorithm gives a routine way to express every solution as a linear combination of basic solutions as in Example [exa:001560], where the general solution \(\mathbf{x}\) becomes

\[\mathbf{x} = s \left[ \begin{array}{r} 2 \\ 1 \\ 0 \\ 0 \end{array} \right] + t \left[ \begin{array}{r} \frac{1}{5} \\ 0 \\ \frac{3}{5} \\ 1 \end{array} \right] = s \left[ \begin{array}{r} 2 \\ 1 \\ 0 \\ 0 \end{array} \right] + \frac{1}{5}t \left[ \begin{array}{r} 1 \\ 0 \\ 3 \\ 5 \end{array} \right] \nonumber \]

Hence by introducing a new parameter \(r = t/5\) we can multiply the original basic solution \(\mathbf{x}_2\) by 5 and so eliminate fractions. For this reason:

Convention: Any nonzero scalar multiple of a basic solution will still be called a basic solution.

In the same way, the gaussian algorithm produces basic solutions to every homogeneous system, one for each parameter (there are no basic solutions if the system has only the trivial solution). Moreover every solution is given by the algorithm as a linear combination of these basic solutions (as in Example \(\PageIndex{5}\) . If \(A\) has rank \(r\), Theorem \(\PageIndex{2}\) shows that there are exactly \(n-r\) parameters, and so \(n-r\) basic solutions. This proves:

Theorem \(\PageIndex{2}\)

Let \(A\) be an \(m \times n\) matrix of rank \(r\), and consider the homogeneous system in \(n\) variables with \(A\) as coefficient matrix. Then:

• The system has exactly \(n-r\) basic solutions, one for each parameter.
• Every solution is a linear combination of these basic solutions.

Find basic solutions of the homogeneous system with coefficient matrix \(A\), and express every solution as a linear combination of the basic solutions, where

\[A = \left[ \begin{array}{rrrrr} 1 & -3 & 0 & 2 & 2 \\ -2 & 6 & 1 & 2 & -5 \\ 3 & -9 & -1 & 0 & 7 \\ -3 & 9 & 2 & 6 & -8 \end{array} \right] \nonumber \]

The reduction of the augmented matrix to reduced row-echelon form is

\[\left[ \begin{array}{rrrrr|r} 1 & -3 & 0 & 2 & 2 & 0 \\ -2 & 6 & 1 & 2 & -5 & 0 \\ 3 & -9 & -1 & 0 & 7 & 0 \\ -3 & 9 & 2 & 6 & -8 & 0 \end{array} \right] \rightarrow \left[ \begin{array}{rrrrr|r} 1 & -3 & 0 & 2 & 2 & 0 \\ 0 & 0 & 1 & 6 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} \right] \nonumber \]

so the general solution is \(x_1 = 3r - 2s - 2t\), \(x_2 = r\), \(x_3 = -6s + t\), \(x_4 = s\), and \(x_5 = t\) where \(r\), \(s\), and \(t\) are parameters. In matrix form this is

\[\mathbf{x} = \left[ \begin{array}{r} x_1 \\ x_2 \\ x_3 \\ x_4 \\ x_5 \end{array} \right] = \left[ \begin{array}{c} 3r - 2s - 2t \\ r \\ -6s + t \\ s \\ t \end{array} \right] = r \left[ \begin{array}{r} 3 \\ 1 \\ 0 \\ 0 \\ 0 \end{array} \right] + s \left[ \begin{array}{r} -2 \\ 0 \\ -6 \\ 1 \\ 0 \end{array} \right] + t \left[ \begin{array}{r} -2 \\ 0 \\ 1 \\ 0 \\ 1 \end{array} \right] \nonumber \]

Hence basic solutions are

\[\mathbf{x}_1 = \left[ \begin{array}{r} 3 \\ 1 \\ 0 \\ 0 \\ 0 \end{array} \right], \ \mathbf{x}_2 = \left[ \begin{array}{r} -2 \\ 0 \\ -6 \\ 1 \\ 0 \end{array} \right],\ \mathbf{x}_3 = \left[ \begin{array}{r} -2 \\ 0 \\ 1 \\ 0 \\ 1 \end{array} \right] \nonumber \]

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• Population Characteristics

Homogeneity and Heterogeneity

Homogeneity is the level of uniformity among sampling units within a population. Homogeneity is commonly interpreted as meaning that all the items in the sample are chosen because they have similar or identical traits (for example, people in a homogeneous sample might share the same age, location, or employment). However, the mathematical meaning of homogeneous is that a data set can be analyzed mathematically and is operating under the same rules and constraints.

The more homogenous a population, the more valid the conclusions drawn from a small sample. Lack of homogeneity, known as heterogeneity , within a population can have a major negative impact on the interpretability and validity of results obtained from a sample. When a population is heterogeneous, there is a higher likelihood that a single sample will not reflect the complexity of the population—that is, important characteristics may be misrepresented or ignored.

For this reason, assessing the heterogeneity level in a population is a key step in the sampling process, for both generalizable sampling and purposeful sampling. As a general rule, when dealing with a heterogeneous population, the population should be divided into as many groups as necessary to ensure that each subgroup is sufficiently homogeneous for the sampling purpose as defined by the audit objective and scope.

Two examples in Figure 4 illustrate the importance of understanding whether a population is homogenous or not. In the first example, the context is a hypothetical survey of employees of a government department about the adequacy of water and air quality in their work environment. Because the employees all breathe the same air and drink the same water, they form a homogenous population and, in that case, a single sample would therefore be sufficient. In the second example, the survey is about management style in the department. Because some employees are part of management and others are unionized, there are two distinct groups of people (two populations) that may have very different opinions about management style in the department. In this instance, it would therefore make more sense to take two separate samples, one from each population.

Assessing a population’s level of heterogeneity is a difficult initial step to take and is often conducted with little firm data. In some instances, auditors may not even have the information to assess the degree of heterogeneity in a population. For example, the analysis in Figure 4 would not be possible if personnel data broken down between unionized staff and management was not available.

Heterogeneity always increases both the cost and complexity of any audit, because more samples and sampling approaches could then be required to complete the audit work. Also, because it is unlikely that any population or subpopulation will be perfectly homogenous, audit teams have to judge the amount of acceptable heterogeneity based on their audit objective and scope.

Ultimately, auditors have to be comfortable that each subgroup they create is made up of reasonably similar units (in terms of materiality , risk , population characteristics, or other parameters relevant to the audit objective) that can be analyzed in the same manner.

More information on how to assess and optimize the homogeneity of populations is in Part 2 of this guide.

• Previous Population Characteristics
• Next Sampling Approaches

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11.4 Test for Homogeneity

The goodness-of-fit test can be used to decide whether a population fits a given distribution, but it will not suffice to decide whether two populations follow the same unknown distribution. A different test, called the test for homogeneity , can be used to draw a conclusion about whether two populations have the same distribution. To calculate the test statistic for a test for homogeneity, follow the same procedure as with the test of independence.

The expected value for each cell needs to be at least five for you to use this test.

Hypotheses H 0 : The distributions of the two populations are the same. H a : The distributions of the two populations are not the same.

Test Statistic Use a χ 2 χ 2 test statistic. It is computed in the same way as the test for independence.

Degrees of freedom ( df ) df = number of columns – 1

Requirements All values in the table must be greater than or equal to five.

Common Uses Comparing two populations. For example: men vs. women, before vs. after, east vs. west. The variable is categorical with more than two possible response values.

Example 11.8

Do male and female college students have the same distribution of living arrangements? Use a level of significance of 0.05. Suppose that 250 randomly selected male college students and 300 randomly selected female college students were asked about their living arrangements: dormitory, apartment, with parents, other. The results are shown in Table 11.19 . Do male and female college students have the same distribution of living arrangements?

H 0 : The distribution of living arrangements for male college students is the same as the distribution of living arrangements for female college students. H a : The distribution of living arrangements for male college students is not the same as the distribution of living arrangements for female college students. Degrees of freedom ( df ): df = number of columns – 1 = 4 – 1 = 3 Distribution for the test: χ 3 2 χ 3 2 Calculate the test statistic: χ 2 = 10.1287 (calculator or computer) Probability statement: p -value = P ( χ 2 >10.1287) = 0.0175

Using the TI-83, 83+, 84, 84+ Calculator

Compare α and the p -value: Since no α is given, assume α = 0.05. p -value = 0.0175. α > p -value. Make a decision: Since α > p -value, reject H 0 . This means that the distributions are not the same. Conclusion: At a 5 percent level of significance, from the data, there is sufficient evidence to conclude that the distributions of living arrangements for male and female college students are not the same. Notice that the conclusion is only that the distributions are not the same. We cannot use the test for homogeneity to draw any conclusions about how they differ.

Try It 11.8

Do families and singles have the same distribution of cars? Suppose that 100 randomly selected families and 200 randomly selected singles were asked what type of car they drove: sport, sedan, hatchback, truck, van/SUV. The results are shown in Table 11.20 . Do families and singles have the same distribution of cars? Test at a level of significance of 0.05.

Example 11.9

Both before and after a recent earthquake, surveys were conducted asking voters which of the three candidates they planned on voting for in the upcoming city council election. Has there been a change since the earthquake? Use a level of significance of 0.05. Table 11.21 shows the results of the survey. Has there been a change in the distribution of voter preferences since the earthquake?

H 0 : The distribution of voter preferences was the same before and after the earthquake. H a : The distribution of voter preferences was not the same before and after the earthquake. Degrees of freedom ( df ): df = number of columns – 1 = 3 – 1 = 2 Distribution for the test: χ 2 2 χ 2 2 Calculate the test statistic: χ 2 = 3.2603 (calculator or computer) Probability statement: p -value= P ( χ 2 > 3.2603) = 0.1959

Press the MATRX key and arrow over to EDIT . Press 1:[A] . Press 2 ENTER 3 ENTER . Enter the table values by row. Press ENTER after each. Press 2nd QUIT . Press STAT and arrow over to TESTS . Arrow down to C:χ2-TEST . Press ENTER . You should see Observed:[A] and Expected:[B] . Arrow down to Calculate . Press ENTER . The test statistic is 3.2603 and the p -value = 0.1959. Do the procedure a second time but arrow down to Draw instead of Calculate .

Compare α and the p -value: α = 0.05 and the p -value = 0.1959. α < p -value.

Make a decision: Since α < p -value, do not reject H o .

Conclusion: At a 5 percent level of significance, from the data, there is insufficient evidence to conclude that the distribution of voter preferences was not the same before and after the earthquake.

Try It 11.9

Ivy League schools receive many applications, but only some can be accepted. At the schools listed in Table 11.22 , two types of applications are accepted: regular and early decision.

We want to know if the number of regular applications accepted follows the same distribution as the number of early applications accepted. State the null and alternative hypotheses, the degrees of freedom and the test statistic, sketch the graph of the p -value, and draw a conclusion about the test of homogeneity.

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homogeneity_score using sklearn in Python

An entirely homogeneous clustering is one where each cluster has information that directs a place toward a similar class label. Homogeneity portrays the closeness of the clustering algorithm to this ( homogeneity_score ) perfection. 

This metric is autonomous of the outright values of the labels. A permutation of the cluster label values won’t change the score value in any way.

Syntax : sklearn.metrics.homogeneity_score(labels_true, labels_pred) The Metric is not symmetric, switching label_true with label_pred will return the completeness_score . Parameters : labels_true:< int array, shape = [n_samples] > : It accept the ground truth class labels to be used as a reference. labels_pred: < array-like of shape (n_samples,) > : It accepts the cluster labels to evaluate. Returns: homogeneity:< float > : Its return the score between 0.0 and 1.0 stands for perfectly homogeneous labeling.

Example 2: Perfectly homogeneous:

Example 3: Non-perfect labelings that further split classes into more clusters can be perfectly homogeneous:

Example 4: Include samples from different classes don’t make for homogeneous labeling:

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Homogeneous: Definition and Examples

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"Homogeneous" refers to a substance that is consistent or uniform throughout its volume . A sample taken from any part of a homogeneous substance will have the same characteristics as a sample taken from another area.

Homogeneous Examples

Air is considered a homogeneous mixture of gases. Pure salt has a homogeneous composition.

In a more general sense, a group of schoolchildren all dressed in the same uniform may be considered homogeneous.

In contrast, the term "heterogeneous" refers to a substance that has an irregular composition.

A mixture of apples and oranges is heterogeneous. A bucket of rocks contains a heterogeneous mixture of shapes, sizes, and composition. A group of different barnyard animals is heterogeneous.

A mixture of oil and water is heterogeneous because the two liquids do not mix evenly. If a sample is taken from one part of the mixture, it may not contain equal amounts of oil and water.

• What Is a Heterogeneous Mixture? Definition and Examples
• 10 Examples of Mixtures (Heterogeneous and Homogeneous)
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• Heterogeneous Definition (Science)
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Homogeneity score

This score is useful to check whether the clustering algorithm meets an important requirement: a cluster should contain only samples belonging to a single class. It's defined as:

It's bounded between 0 and 1 , with low values indicating a low homogeneity. In fact, when the knowledge of Y pred reduces the uncertainty of  Y true , H( Y true | Y pred ) becomes smaller ( h → 1 ) and viceversa. For our example, the homogeneity score can be computed as:

The   digits['target']  array contains the true labels while  Y  contains the predictions ...

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Homogeneous: definition, types, and examples.

• Post published: September 18, 2021
• Reading time: 8 mins read

Table of Contents

Homogeneous definition.

Homogeneous can be defined as “the same” or “similar.” It can be used to describe things that have similar characteristics. Homogeneous substances, for example, are substances that are homogeneous in volume and composition across their whole volume. As a result, two samples obtained from two different portions of homogeneous mixtures and substances will have the same compositions and properties.

Homogeneous Etymology

The word homogeneous is derived from two Greek words: “homo” (meaning “the same”) and “genous” (meaning “kind”). As a result, homogenous refers to individuals who are all perceived to be the same, similar, or present in the same proportion.

What is Homogeneous Mixture?

Homogenous means “of the same sort” or “similar.” It’s the ancient name for homologous in biology, which means “having matching components, similar structures, or the same anatomical locations.” Homogenous is derived from the Latin homo, which means “same,” and “genous,” which means “kind.” homogenous is a variant. Heterogeneous is the antonym of homogeneous.

A mixture is formed when two or more components combine without undergoing any chemical changes. The mechanical blending or mixing of objects like elements and compounds defines a mixture. There is no chemical bonding or chemical change in this process.

As a result, the chemical characteristics and structure of the components in a combination are preserved. Size, form, colour, height, weight, distribution, texture, temperature, radioactivity, structure, and a variety of other characteristics all stay consistent throughout the homogeneous material.

When a pigment (such as ink) is combined with water, the resultant solution is highly homogeneous, which is a fairly common example of homogeneous in our daily lives. The colour combines equally with water, and any area of the solution has the same makeup.

Mechanical techniques can be used to separate them. Centrifugation, filtration, heat, and gravity sorting are some of the methods.

That’s all there is to it when it comes to the term’s use in chemistry or biology. The term “homogenous” is used in various research areas, such as ecology, to describe a population’s homogeneity.

A group of humans raised only by asexual reproduction – with identical genes and traits — is homogeneous, for example. Scientists hypothesized that if various orientations came from the same source, the cosmos would behave similarly. Evolutionary biology is another area of biology where the term homogeneous is employed.

Homogeneous is an ancient word for homologous, which refers to anatomical components that exhibit structural similarities, such as those generated by descent from a common ancestor.

The term homogeneous has been used widely in different fields of research, such as biology, chemistry, and ecology, but it is always used to describe organisms in a mixture who have the same properties.

In chemistry, homogeneous refers to a combination in which the ingredients are uniformly distributed. However, there are no chemical connections between them at the molecular level. Air is the most typical example of a homogeneous mixture in our environment.

Homogenous vs Heterogenous

A mixture, as previously stated, is the physical coming together of components (which, in chemistry, can be elements or compounds). There are two sorts of mixtures: homogeneous and heterogeneous.

The opposite of homogeneous is heterogenous (variant: heterogeneous). It refers to the components in a combination that have distinct properties (“hetero,” which means “different”). The most obvious example of a heterogeneous combination is oil and water, which form two distinct layers that are immiscible with each other, resulting in two distinct layers.

One of the most notable characteristics of heterogeneous mixes is that the particles are not dispersed equally throughout the mixture. Analysing the combination with the naked eye reveals the heterogeneous character of the mixture. In addition, the components of all heterogeneous mixes are not uniform.

Composition is similar in homogenous mixtures and dissimilar in heterogenous. In heterogenous mixtures, various phases are seen and single phase is seen in homogenous mixtures.

Substance can be sorted from each other by physical methods such as distillation, evaporation, centrifugation, chromatography, crystallization in both types of mixtures. Variation and a smaller number of species exist in homogenous mixtures, and the reverse is seen in heterogenous mixtures.

Although the concepts and compositions of homogeneous and heterogeneous substances are vastly different, both are prone to change depending on context and composition. Let’s take the example of blood. If we look at the blood with our naked eyes, it seems to be homogeneous.

Blood, on the other hand, has a variety of components under the microscope, including red blood cells, plasma, and platelets, showing that it is heterogeneous.

Homogeneous Examples

We come across numerous examples of homogeneous mixes and entities in our daily lives. In biology, a homogeneous population is one in which all of the individuals have virtually the same genetic makeup, as a result of some types of asexual reproduction.

Asexual reproduction produces homogeneous children who are identical to each other, including their parents.

Many animals, such as goat populations, look homogenous but are not because they reproduce through sexual reproduction.

According to experts, homogeneity reduces biodiversity, and as a result, the odds of early extinction due to environmental changes are significant. Animal cloning is a frequent example of a homogeneous population.

Dolly the sheep was the first mammal to be successfully cloned from a somatic cell in an adult.

Homogeneous species are those that exhibit indistinguishable characteristics and appear to be identical. Such species appear to have a lower level of biodiversity.

The diversity and frequency of species in a particular region and period, as well as the ecosystem’s homogeneity, may be quantified using a specific fundamental unit called species richness.

Species richness refers to the number of different species found in a specific ecological community. It displays the relative abundance of species rather than the total number of species in the environment. As a result, in a homogeneous environment, species richness will be lower, as high species richness indicates variability.

This is particularly evident in endemic species, which are species that have evolved through time in a specific geographic region and aren’t found anywhere else.

Grass, trees, ants, fungus, and certain animals are all instances of homogeneous in the ecosystem. Many endemic species found nowhere else in the world may be found in New Zealand.

Homogeneous used to be a very popular term in evolutionary biology to describe physically comparable features in various species, indicating a shared evolutionary origin.

The anatomical characteristics of several animal forelimbs are depicted. A similar evolutionary ancestor is shown by the identical forelimb bone components.

Homogeneous Summary

As a result of the preceding discussion, homogeneous substances are those that are uniform in volume and composition throughout. Homogeneous mixtures in chemistry have the same size, shape, colour, texture, and many other characteristics.

A solution that does not separate from each other over time is known as a homogeneous mixture. Homogeneous species are those that are genetically similar but lack biodiversity and species richness, as defined in biology and ecology.

Similarly, various solutions are widely used in our daily lives, and the blood and DNA in our bodies are both homogeneous. Heterogeneous mixes have properties that are the polar opposite of homogeneous mixtures.

As a result, the heterogeneous mixture contains non-uniform compositions and numerous phases that cannot be distinguished by physical changes. Furthermore, they are culturally varied and affluent.

Similarly, it has been demonstrated that both homogeneous and heterogeneous mixes are prone to change depending on their environment and composition. As a result, both heterogeneous and homogeneous mixes might be seen as equally important.

Hyaline Cartilage: Definition, Function, and Examples

Homogeneous citations.

• Synthesis of Oxazolidin-2-ones from Unsaturated Amines with CO 2 by Using Homogeneous Catalysis. Chem Asian J . 2018 Sep 4;13(17):2292-2306.
• Recent Advances Utilized in the Recycling of Homogeneous Catalysis. Chem Rec . 2019 Sep;19(9):2022-2043.
• A review of thermal homogeneous catalytic deoxygenation reactions for valuable products. Heliyon . 2020 Feb 20;6(2):e03446.

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homogeneity

Definition of homogeneity

Examples of homogeneity in a sentence.

These examples are programmatically compiled from various online sources to illustrate current usage of the word 'homogeneity.' Any opinions expressed in the examples do not represent those of Merriam-Webster or its editors. Send us feedback about these examples.

Word History

see homogeneous

1625, in the meaning defined at sense 1

Dictionary Entries Near homogeneity

homogeneous

Cite this Entry

“Homogeneity.” Merriam-Webster.com Dictionary , Merriam-Webster, https://www.merriam-webster.com/dictionary/homogeneity. Accessed 1 Apr. 2024.

Medical Definition

Medical definition of homogeneity, more from merriam-webster on homogeneity.

Thesaurus: All synonyms and antonyms for homogeneity

Nglish: Translation of homogeneity for Spanish Speakers

Britannica English: Translation of homogeneity for Arabic Speakers

Britannica.com: Encyclopedia article about homogeneity

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• Is Matter Around Us Pure?
• Heterogeneous And Homogeneous Mixtures

Heterogeneous and Homogeneous Mixtures

What are Mixtures?

Mixtures are formed when two or more substances (elements or compounds) mix together without participating in a chemical change. The substances need not necessarily mix in a definite ratio to form a mixture.

Some examples of mixtures include mixtures of sand and water, mixtures of sugar and salt, and mixtures of lime juice and water. There are two primary types of mixtures, namely homogeneous mixtures and heterogeneous mixtures.

Table of Content

Recommended videos, what is a homogeneous mixture, what is a heterogeneous mixture, soft drink: homogeneous or heterogeneous mixture.

• Frequently Asked Questions – FAQs

Heterogeneous and Homogeneous Definition

These are the types of mixtures in which the components mixed are uniformly distributed throughout the mixture. In other words,  “they are uniform throughout” . We can observe only one phase of matter in a  homogeneous mixture . Key points regarding such mixtures are:

• Particles are distributed uniformly
• We can’t judge a homogeneous mixture by just seeing it
• Homogeneous mixtures are also called solutions
• Uniform composition
• Example: rainwater, vinegar, etc.

This is a type of mixture in which all the components are completely mixed and all the particles can be seen under a microscope. We can easily identify the components and more than one phase can be seen by naked eyes.

Key points regarding this type of mixture:

• Particles are distributed non-uniformly
• We can judge a heterogeneous mixture by just seeing it
• Non-uniform composition
• Example: seawater, pizza, etc.

Difference between Homogeneous and Heterogeneous Mixture

In a homogeneous mixture, all the components are uniformly distributed and in the soft drink, we find components like sweetener, carbon dioxide and water forming a single phase. Therefore, a soft drink is a homogeneous mixture.

Frequently Asked Questions – FAQs

What is a heterogeneous mixture.

A heterogeneous mixture is a mixture that is non-uniform and contains smaller component parts.

Which mixture is homogeneous?

A homogeneous mixture is a mixture throughout the solution in which the composition is uniform. The saltwater mentioned above is homogeneous due to the even distribution of the dissolved salt throughout the entire sample of saltwater.

What are heterogeneous and homogeneous mixture examples?

By combining two or more substances, a mixture is produced. A homogeneous solution tends to be identical, no matter how you sample it. Homogeneous mixtures are sources of water, saline solution, some alloys, and bitumen. Sand, oil and water, and chicken noodle soup are examples of heterogeneous mixtures.

Which best describes a heterogeneous mixture?

A heterogeneous mixture is a mixture where throughout the solution the composition is not uniform. By definition, a single-phase consists of a pure substance or a homogeneous mixture. There are two or more phases of a heterogeneous mixture.

Is air homogeneous or heterogeneous?

In air, all gases would have a uniform composition. Therefore, the air is an example of homogeneous mixture.

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