Certain tests (e.g. ANOVA) require that the variances of different populations are equal. This can be determined by the following approaches:

- Comparison of graphs (esp. box plots)
- Comparison of variance, standard deviation and IQR statistics
- Statistical tests

The *F* test presented in Two Sample Hypothesis Testing of Variances can be used to determine whether the variances of two populations are equal. For three or more variables the following statistical tests for homogeneity of variances are commonly used:

- Levene’s test
- Fligner Killeen test
- Bartlett’s test

Using the terminology from Definition 1 of Basic Concepts for ANOVA, the following null and alternative hypotheses are used for all of these tests:

H_{0}: = ^{ }= ⋯ =

H_{1}: Not all variances are equal (i.e. ≠ for some *i, j*)

**Topics**:

Hello,

and thank you for such a helpful website!

I’m trying to compare the means of 4 different age groups, which are not equal (e.g. group 1=195, group 2= 499, group 3=688 and group 4=693). The aim is to compare the mean preference score that these groups assigned to 5 images to determine whether different age groups preferred different images. However the Levene test is significant for mean scores of images 2 and 4, and non significant for images 1, 3 and 5. I am just wondering why this might be? The groups are quite uneven, so I would have expected a significant Levene statistic for all of them. Perhaps I did something wrong…?

thank you for your advice!

Selena,

I am not following the scenario completely. When you say “e.g. group 1=195, …”, do you mean that group 1 has 195 elements?

Generally when you perform Levene’s test, you use all the data as input, and don’t carry out the test separately for different images (unless you are testing these images separately).

Please clarify.

Charles

Hi Charles,

thank you for your reply.

You are correct; I have 4 age groups, so age group 1 (10-20 yrs) has 195 respondents, age group 2 (21-30 yrs) has 499 respondents and so on.

I am indeed doing 5 tests separately to assess any differences in opinion among the younger/older respondents for 5 different images. So for instance, will younger respondents rate image A higher than the older respondents? but I would also like to determine if this will also be the pattern for images B, C, D and E. If I was investigating opinion for just one image, I would not have spotted this issue – but having done the 5 tests (in SPSS), I noticed that Levene’s test was significant for Images B and C, and not for A, D and E. I can’t understand why this should be the case as the groups are the same in all 5 tests. I know you do not use SPSS, but was just wondering if in your experience you might have come across something like this?

Thank you,

Selene

Selene,

Are you saying that when you use Levene’s test on images B and C together you see that the variances are significantly different? Or are you applying Levene’s test to the data in B alone and then to C alone?

Charles

Given the mean and standard deviation of a variable in panel dataset of countries, how do I determine homogeneity or heterogeneity?

Magnus,

You don’t have enough data to determine whether the variances are equal across the various countries. Generally, you need to know the raw data by country. The problem of determining whether the variances are all equal in this case is equivalent to determining whether any list of positive numbers are all equal. I don’t know of any test that does this.

Whether the variances are equal enough probably relates to what you will use this information for. For some tests as long as the largest variance is not more than 3 or 4 times the smallest variance you should be ok.

Please provide the context for which the question of homogeneity of variances is relevant.

Charles

I have seen this “rule of thumb” that largest variance should be no more than 4 times the smallest variance, but have been unable to find a citation for this – any leads? Many thanks!

AJ,

This rule of thumb is not universal. Some recommend 3 times or even 1.5 times. See the following references

http://data.library.virginia.edu/a-rule-of-thumb-for-unequal-variances/

https://www.ma.utexas.edu/users/mks/384E09/modelcheck.pdf

Charles

Thank you, this is a great start and very helpful. Best to you!

Dear sir,

How can I tell if my Homogeneity of variance assumption is met? I ran my data and calculations through SPSS and my output lists my Levene’s Test Sig = .620. My Independent Sample T-Test output reveals my Sig (two-tailed) is .050. Is there a rule to remember when the assumption is met and when it is not? Any help is appreciated.

Thank you!

Generally a significance level of .05 is used for most statistical tests, including Levene’s Test. Since p-value = .620 is much higher than .05, you can’t reject the null hypothesis that the variances are equal.

Your t test is right on the boundary with a p-value of .05. Since I typically use p-value < .05 as the measure of significance, then I would say that this test shows a non-significant result, i.e. you don't have enough evidence to reject the null hypothesis. Now although a significance level of .05 is commonly used, it is quite arbitrary. Regarding your question "Is there a rule to remember when the assumption is met and when it is not?", I would say that Levene's test shows that the homogeneity assumption is met. I don't know how you are using the t test and generally it is not used to test the homogeneity of variances assumption, and so I can't comment on this. Charles

Dear Charles

please help me how to analyze this hypothesis;

-There will be no statistically significant difference in the perceptions of EFL teachers and students with respect to teachers ( verbal and nonverbal )’ immediacy behavior in their classrooms.

since the variances of two populations are not equal. Levene’s Test for Equality of Variances indicate sig=0.000

thank you

In order for me to answer your question, you need to provide a more detailed explanation of what you are trying to test. If you are trying to perform ANOVA but the homogeneity of variances assumption is not met based on Levene’s test, then a common approach is to use Welch’s ANOVA test instead. See Welch’s Test for more details.

Charles

Would you please help me

I confused between Anova or T.test for independent sample.since I have two population( teachers- students) and I have two factors ( verbal – nonverbal).

and if I choose Anova which kind should I use.

thank you

dear Charles,

I would like to add more details to clarify my question.

-I have 50 teachers and 277 students. I used two versions of questionnaire as an instrument. one for students and the other is for teachers. each questionnaire includes two parts:

1- verbal

2- nonverbal

The purposes of the study are to explore the perceptions of EFL teachers and students concerning teachers’ verbal and nonverbal immediacy behaviors and to see if there is any difference between their perceptions.

thank you in advance

As usual, the devil is in the details. It really depends on what sort of hypotheses you want to test. E.g. if you want to see whether there is a significant difference between the teachers’ and students’ verbal scores you can conduct a t test. This has the advantage that the variances can be unequal and the sample sizes can be unequal. You can also perform a similar test comparing nonverbal scores. If you need to consider interactions, then you can use ANOVA. The problem with this is that you have said that the homogeneity of variances assumption fails.

Charles

Hello,

Could you please help me to analyze the data by using One-Way Anova as i have got these result as follows:

1. Homogeneity of Variance =.011 (It means that I need to take a look on Welch Statistic because the normal assumption has not been met)

2. Sig. in Welch’ Statistic = .06

3. Whereas, Sig. in Anova Table = .03

4. More importantly, there are * asterisk appeared in Post Hoc Tests which can show where the difference exactly is for that variable.

So, the question is “how can i conclude this result, also how to choose sig. to report when it has not been met assumption of variance”?

Hopefully, you will reply me soon and I am really looking forward to seeing your response.

Thanks very much in advance,

Earn

Earn,

1. When you say that “Homogeneity of Variance =.011”, does this mean that p = .011 for Levene’s test? If the homogeneity of variance assumption is not met then usually you would use Welch’s test.

2. Since p = .06, then using an alpha value of .05, you have a non-significant result and so you wouldn’t usually perform any post-hoc tests

3. Since the homogeneity of variance assumption is not met, I would tend to use the Welch’s result and not the ANOVA result.

4. Which post-hoc test did you use? Games-Howell would likely be a good choice. The fact that you see an asterisk, means that you are probably not using Excel, and so I don’t know how to interpret it (although in many software packages this probably indicates a significant result).

Charles

Can you tell me in layman terms, why homogeneity of variance is an important consideration in statistics and what are some methods used to determine if variances are homogenous?

Any help with simple examples is greatly appreciated.

Tiffany,

Homogeneity of variance is an important since some of the statistical tests are not valid (i.e. they don’t give accurate results) if this assumption is not met (more precisely, if the variance are too different). A good example of this is ANOVA. If the homogeneity of variance assumption is not met then you would usually use Welch’s Test instead of one-way ANOVA.

The typical tests used to determine whether the variances are sufficiently homogeneous are described in the referenced webpage (e.g. Levene’s Test).

Charles

Sir what if in a 1 way independent ANOVA, in your homogeneity of variances the variances are not equal? Should I use Welch? If so, how am i going to report it and tell that I must use the Welch in spss? thankyou

Yes, generally you would use Welch’s test. You report that you are using this test since the homogeneity of variances assumption is not satisfied to use Anova

Charles

Hi Charles,

I plan to perform a 3 (A) x 3 (B) ANOVA. However, the Levene’s test results indicate that the error variance is not equal across groups. I like to ask for your advice if I could still conduct ANOVA. If not, what tests should be done? In addition, what are the tests available for comparing group means (B1, B2 and B3) under A1, A2, and A3? Thank you.

For one-way ANOVA, the usual choice is Welch’s Test — see the webpage Dealing with Non-Homogeneous Variances.

There isn’t a similar test for two-way ANOVA. If you don’t care about the interaction of the two factors (i.e. reduce the problem to one-way ANOVA), you can simply use Welch’s Test. Otherwise you can try to make some transformations that address the problem with the variances (as described on the above webpage). If the variances are not too unequal (e.g. Levene’s test is not so unequal), then you can use two-way ANOVA but report that there is a problem.

Unfortunately, none of the solutions is that great.

Charles

Hello Charles,

I found you very helpful. Interesting to read your reply. I am having problem to do statistics for my data.

I am doing research on drying of apple in two diffferent temperature (50 C and 70 C) with 9 different pretreatments (Untreated drying at 50 C , Untreated drying at 70 C , Hot water blanching at 50 C drying at 70 C , Hot water blanching at 70 C drying at 70 C and so on…..) drying at different interval of time duration (30 min, 60 min until 240 min)

I am measuring colour measurement in each experiment : change in colour Delta E , Change in lightness (Delta L), Change in redness (Delta a).

I want to see effect of temperature, pretreatment, and drying time on colour change.

I run Anova but I face the problem with leneve’s test. It shows heterogeneity. I am not sure that I am doing right analysis. I do not have much idea about statistics.

I would be really grateful if you could help me in this matter.

Thanking you in advance

Best regards

Luna

Luna,

It is hard for me to answer your question based on the limited amount of data that you have provided. It sounds like you are measuring three types of colour-related change: colour Delta E , lightness (Delta L)and redness (Delta a). If so, then this seems like a MANOVA or multivariate regression problem. What are the scales of the three colour-oriented changes: continuous? discrete? On what groups of data did you perform Levene’s test? What was the actual p-value?

Charles

Hello Charles,

Thanks! I have factor A = Pretreatment with 9 sub level; factor B = Drying temperature with 50 C and 70 C; Factor C = Drying Time with sub level of 30 min, 60 min, 90 min, 120 min, 150 min, 240 min.

Dependent Variable = Colour change (Delta E).

The scale of Colour change is continuous ( Scale).

I just checked homogeneity of test (levene’s test ), it shows p<0,05 .

Even I did Manov , Box's test is not performed and there's no value in some of the table.

I am very bad in statistics. Please help me

Thanks

Hello,

What hypothesis are you trying to test?

Charles

Good afternoon,

I am very confused; I have been carrying out a Welch test as my data violated the assumption of homogeneity and following with the games-howell post hoc test. However my output showed that I d0 have a sig. difference between groups (0.03) but then failed to identify between which groups the difference occurred on the post hoc test . I don’t know where to go to resolve this. Would a different post hoc test show me where the differences occurred?

Kind Regards.

Michelle,

If you send me an Excel file with your data, I will try to understand what is going on better.

Charles

Dear sir,

I am in a confusion with my data i have. I want to run an one way ANOVA in SPSS for which fails the assumption of homogeneity of variances, where the p<.05(.032). i dont know to proceed with which test to follow and to report the result. please help in this regard.

Welch’s Test is usually your best alternative to ANOVA when the homogeneity of variance assumption is not met. See the following webpage for details:

Welch’s Test

Charles

Hi, I am comparing 7 groups defined under one independent variable. My problem is with the homogeneity of variance test as equal variance is one of the assumptions for the one way anova. Although my groups have equal sample size, they show unequal variance in the levene test (p-value:0.001). Can I still use Anova test or should I use Kruskall- Wallis H test. and the sample size is 20 for each group, there are 7 groups, the data is normally distributed but with unequal variance, please kindly guide me regarding choosing the correct anova and post hoc tests for this analysis. Many thanks.

With unequal variances you should probably choose the Walsh test. A post-hoc test for unequal variances is Games-Howell.

Charles

Good afternoon– I am trying to calculate the sample size required to test whether a sample variance is = 0.63 vs. H1: var < 0.63 with, say, 80% power, alpha=0.05. Could you give me some advice on how to calculate the sample size required for this test?

Thanks! Stan

This is a one sample variance test (using the chi-square distribution). You can find more information about this test at One Sample Variance Test.

You can estimate the sample size by pressing

Ctrl-mand choosing theStatistical Power and Sample Size data analysis tool. On the resulting dialog box choose theOne sample varianceandSample sizeoptions.Charles

Dear Sir,

I used Levene’s test to check homogeneity of variance for my study. The result shows that p-value of Levene’s test is 0.165, while p-value for F-test is 0.000. Can I conclude that the variances is equal?

Thank you.

A p-value of .165 indicates that there is no significant difference between the variances.

I don-t know which F-test you are referring to.

Charles

Good day sir! Pls. I am currently working on a multiple regression problem, but I don’t know how to test for the homogeneity of variances. I am using the spss package which has a built in levene’s test procedures. Is it expected of me that check for the equality of variances of the DV and all IV? When I tried comparing the means using oneway anova, I kept on getting response like: homogeneity of variances cannot be performed cos only one group has a computed variance and also because the sum of case weights is less than the number of groups. Any form of explanation or references given would be appreciated.

Thank you in anticipation of a swift response

Levene’s test is a good way to test for homogeneity of variances. If you are performing one-way ANOVA then you need to test the homogeneity of the k groups in the test. I don’t use SPSS and so I am not able to explain the error message you received.

Charles

Hi there,

I use SPSS, and have received an error that I believe is very similar to yours in wording. In my experience, it usually means that the two variables were entered incorrectly…the IV was labelled as the DV and vice versa. Try switching the group’s around and see if that doesn’t solve the problem.

Kevin

Charles,

Great info here…could you possibly do a page on the Fligner Killeen test (nonparametric assessment of homogeneity of variance with >2 groups), using concrete numbers? Every “example” I can find either just gives the chi-square result without the computation details, or just uses the abstract symbols and summation notation to explain it…really confusing and infuriating! Why can’t they just use an example with raw numbers? Thanks again!

Kevin.

I have good news for you. I just put an example using the Fligner Killeen test up on the website. You can access it at the webpage Fligner Killeen test.

Charles

dear sir

I am doing research on the effects of wet/dry and wet feeding troughs on feeding grower pigs . so i have 2 treatments wet feeding and dry feeding and 2 blocks that is males and female so i have 24 males and 24 females all pigs are of same age and genetic .Can you help me on the method of data analysis to use. I am measuring the feed in take daily and weighing pigs on entery and then once weekly for 8 weeks

It all depends on what you are trying to test, but based on what you have described it sounds like a three factor Anova. One fixed factor with two levels for feeding type, one fixed factor with two levels for gender and one repeated factor with 9 levels for week. The dependent variable is weight.

Charles

helloo

i conducted a two way anova and the levene’s test p value is 0.001, my study has a continuous dv (eating style) and two categorical iv’s

Not sure what to do as ive used this analysis to show the effects of two other eating styles?

Niki,

Sorry, but I don’t understand your question. Please explain in a little more dtail, what you have done and what you want to accomplish.

Charles

i have conducted a two way anova to see if there is a sig difference in the means of eating style scores when participants reported their stress levels (high vs low stress) and sleep quality (good vs poor). I have conducted this for both emotional and external eating style but when i conducted the analysis for restrained eating style the homogeneity assumption was violated what can i do?

Niki,

Look at the Dealing with non-heterogeneity of variances topic towards the end of the referenced webpage.

Charles

Dear sir please I need your help:

1- 40 students are divided into two groups of 20 each [the control group (i.e CG) and the experimental group(i.e. EG)] they are considered equal in their level of English language study. The CG read version A of a text with certain rhetorical organization , on the other hand the EG read version B of the same text with different rhetorical organization . They are asked to read these two versions and recall information from them so the amount of information recalled and speed of reading spent are recorded . I want to test the following hypothesis ” To what extent will the change (in the rhetorical pattern) affect the ease of information recalled and speed of reading as well? How can I use ANOVA test please when the CG have two marks of amount of information recalled and marks of speed of reading and the EG have also two marks also

It sounds like you have one independent variable Rhetorical organization (RO) and two dependent variables: Information recalled (IR) and Reading speed (RS). You can use MANOVA, or more simply Hotelling’s T-square as described on the webpage Hotelling’s T-square.

Charles

Dear sir, I consider the students (i.e. CG and EG) as dependent variables since there is no difference between them and considered equal, the RO (i.e.the two versions of texts version A and B with different rhetorical organization) as independent variable since it is varied . However the IR and SR are the responses or marks gathered and not variables .Can I use ANOVA test to examine the responses i.e. the IR (amount of information recalled) then do the same procedures to examine the RS (speed of reading) with my thanks and regards sir.

Saif,

Sorry, but I still don’t understand the question. E.g. in the statement “I consider the students (i.e. CG and EG) as dependent variables since there is no difference between them and considered equal…”, I don’t understand why this statement would make CG and EG dependent variables. I really don’t understand the other statements either.

Charles

Can one perform t-test or ANOVA using CV if the variance between group/s is not similar? If yes, how does one do it? I am a statistics illiterate

There is a version of the t-test which you can use when the variances are not similar. See the webpage two sample t-test with unequal variances.

There are also substitute tests for ANOVA when the variances are not equal. See the

Dealing with non-heterogeneity of variancestopic on the referenced webpage.You use the abbreviation “CV”. What does this stand for?

Charles

I love your website it is so useful in helping me solve statistical problems. I am a little confused about how to perform hypothesis testing when the observations are just given as one total without actually listing them separately – 125 observations (Southern States) and 132 observations (Northern States) with a sample mean of 87 and 88 respectively and a population variance of 7.0 and 6.2 respectively. Level of significance is .01 is there evidence that the workers in southern states are receiving less pay than workers in northern states?

Since you have the population variances you can use a two sample test using the normal distribution, as described in Theorem 1 of Comparing Two Means.

The null hypothesis is mean1 >= mean2 (these are population means). The test statistic is z = (m1-m2)/stdev, where m1-m2 = 88-87 = 1 (sample means) and stdev = sqrt(var) where var = v1^2/n1 + v2^2/n2 = 6.2^2/132 + 7.0^2/125. If NORMSDIST(z) > .99 then you reject the hypothesis that the the workers receive the same pay. This is a one tailed test. If you want a two-tailed test you need to replace .99 by .995.

If instead of the population variance you had the sample variances you would use Theorem 1 of Two Sample t Test instead.

Charles

Dear Sir,

I am doing the One-Way ANOVA analysis. My p-value =0,233 for Levene’s test. Since my data was not normally distributed I transformed it. Do I need to perform Levene’s test again for transformed data? (I have tried and p-value is less then 0.05)

Thank you

Alfiya.

Alfiya,

Since you will be testing the transformed data you need to make sure that the assumptions are satisfied on the transformed data. Since homogeneity of variance is an assumption for One-way ANOVA this assumption needs to be verified for the transformed data. Levene’s test is a way of checking this.

Charles

Dear Sir,

I am analyzing a field experiment on 4 maize varieties. The varieties were replicated three times in one location. Should I examine the homogeneity and normality tests?

Lucy

Lucy,

It really depends on what you are trying to test. The ANOVA tests will require homogeneity of variance and normality, but they can be quite forgiving even if these assumptions aren’t completely satisfied.

Charles

Sir,

I have conducted a experiment as two randomized complete block design ( 2 RCB) with different fertilization treatments, including control (no fertilizer application), PK, NK, NP and NPK fertilization with four replications at one location for four seasons (2 dry seasons and 2 wet seasons). The short duration rice varieties, hybrid (Pale Thwe-1) and HYV (Yadanar Toe) were tested. Can I do combine analysis for those 4 seasons’ yield data? Need to do homogeneity test? How to approach homogeneity test?

Thanks

Cho

Hello Cho,

1. You can certainly combine the the yields for the 4 seasons. You can also perform a factorial ANOVA which will let look at the fertilizer treatments, the seasons and the rice variety.

2. Generally I use Levene’s test to check homogeneity of variances. I also look at the actual values for variances to see whether they are very different or not.

Charles

sir i want to know how do i find the relationship between export growth and variation between real and nominal exchange rate based on the measure of 3 months and 6 months i have collected data but now i m confused how do i apply non-parametric test on it and which test…

Sorry, but you would need to give me more information before I can answer your question.

Charles

I want do homogeneity test for two variances please tell me to do the test

Levene’s test can be used for two variance. You can use the LEVENE function as described on the referenced webpage.

Charles

hi! i want to ask on the interpretation of Bartlet’s test on the Gen stat discover program for ANOVA, i do fail to interpret it

Sorry but I am not familiar with the Gen stat discover program for ANOVA.

Charles

sir i wanted to ask why homogeneity of variance is so important?please tell me in some points…

Sonia,

Homegeneity of variances is a requirement for many of the most used statistical tests, including ANOVA. Fortunately most such tests are pretty foriving and as long as the variances are not too unequal the tests give pretty accurate results, but when the requirement is sufficiently violated then the results of these tests can be quite unreliable.

Charles

Hi, I just wanted to ask, what happens if your levene’s test is positive so homogeneity of variance cannot be assumed in a factorial independent measures ANOVA. I know that you have to change the significance to p=0.01 instead of 0.05 (or something along those lines) but what do I do in terms of SPSS? I have run the test as normal but I don’t know how I am supposed to interpret my results considering levene’s positivity.

Please help!!! Thank you

Deborah,

What to do in case the homogeneity of variances test fails is explained towards the end of the referenced webpage under the title

“Dealing with non-heterogeneity of variances”.

Charles

Thank you:)

Sir

Will you add a real statistics function for “Bartlett’s Test” ?

Colin

Colin,

Bartlett’s Test is also called Box’s Test. This is already included in the Real Statistics Resource Pack (see multivariate statistics portion of the website).

Charles

Hi,

Can you please let me know what transformation method I should be using if both standard deviation to means and means to variances are not proportional? There is no strong correlation for both?

Thanks.

Sriya,

There is no easy answer to your question. It all depends on your data. There are an unlimited number of transformations as well (1/x, x^2, etc.). It also may turn out that a particular transformation creates more problems than it solves.

Charles

Many thanks for this… The easy to follow guide to Levene’s and Bartlett’s included in your download is just what I needed to sort out a tricky analytical problem…

Ned,

I am very pleased that the site has been useful for you. I hope that you will use it again in the future.

Charles