Levene 's test - wikipedia Levene 's test In statistics, Levene 's test is an inferential statistic used to assess the equality of variances for a variable calculated for two or more groups. Some common statistical procedures assume that variances of the populations from which different samples are drawn are equal. Levene 's test assesses this assumption. It tests the null hypothesis that the population variances are equal ( called homogeneity of variance or homoscedasticity ). If the resulting p - value of Levene 's test is less than some significance level ( typically 0.05 ), the obtained differences in sample variances are unlikely to have occurred based on random sampling from a population with equal variances. Thus, the null hypothesis of equal variances is rejected and it is concluded that there is a difference between the variances in the population. Some of the procedures typically assuming homoscedasticity, for which one can use Levene 's tests, include analysis of variance and t - tests. Levene 's test is often used before a comparison of means. When Levene 's test shows significance, one should switch to more generalized tests that is free from homoscedasticity assumptions ( sometimes even non-parametric tests ). Levene 's test may also be used as a main test for answering a stand - alone question of whether two sub-samples in a given population have equal or different variances. Contents 1 Definition 2 Comparison with the Brown -- Forsythe test 3 See also 4 References 5 External links Definition The test statistic, W ( displaystyle W ), is defined as follows : W = ( N − k ) ( k − 1 ) ∑ i = 1 k N i ( Z i ⋅ − Z ⋅ ⋅ ) 2 ∑ i = 1 k ∑ j = 1 N i ( Z i j − Z i ⋅ ) 2, ( displaystyle W = ( \frac ( ( N-k ) ) ( ( k - 1 ) ) ) ( \frac ( sum _ ( i = 1 ) ^ ( k ) N_ ( i ) ( Z_ ( i cdot ) - Z_ ( cdot cdot ) ) ^ ( 2 ) ) ( sum _ ( i = 1 ) ^ ( k ) sum _ ( j = 1 ) ^ ( N_ ( i ) ) ( Z_ ( ij ) - Z_ ( i cdot ) ) ^ ( 2 ) ) ), ) where k ( displaystyle k ) is the number of different groups to which the sampled cases belong, N i ( displaystyle N_ ( i ) ) is the number of cases in the i ( displaystyle i ) th group, N ( displaystyle N ) is the total number of cases in all groups, Y i j ( displaystyle Y_ ( ij ) ) is the value of the measured variable for the j ( displaystyle j ) th case from the i ( displaystyle i ) th group, Z i j = ( Y i j − Y  ̄ i ⋅, Y  ̄ i ⋅ is a mean of the i - th group, Y i j − Y ~ i ⋅, Y ~ i ⋅ is a median of the i - th group. ( displaystyle Z_ ( ij ) = ( \begin ( cases ) Y_ ( ij ) - ( \bar ( Y ) ) _ ( i cdot ), & ( \bar ( Y ) ) _ ( i cdot ) ( \text ( is a mean of the ) ) i ( \text ( - th group ) ), Y_ ( ij ) - ( \tilde ( Y ) ) _ ( i cdot ), & ( \tilde ( Y ) ) _ ( i cdot ) ( \text ( is a median of the ) ) i ( \text ( - th group ) ). end ( cases ) ) ) ( Both definitions are in use though the second one is, strictly speaking, the Brown -- Forsythe test -- see below for comparison. ) Z i ⋅ = 1 N i ∑ j = 1 N i Z i j ( displaystyle Z_ ( i cdot ) = ( \frac ( 1 ) ( N_ ( i ) ) ) sum _ ( j = 1 ) ^ ( N_ ( i ) ) Z_ ( ij ) ) is the mean of the Z i j ( displaystyle Z_ ( ij ) ) for group i ( displaystyle i ), Z ⋅ ⋅ = 1 N ∑ i = 1 k ∑ j = 1 N i Z i j ( displaystyle Z_ ( cdot cdot ) = ( \frac ( 1 ) ( N ) ) sum _ ( i = 1 ) ^ ( k ) sum _ ( j = 1 ) ^ ( N_ ( i ) ) Z_ ( ij ) ) is the mean of all Z i j ( displaystyle Z_ ( ij ) ). The test statistic W ( displaystyle W ) is approximately F - distributed with k − 1 ( displaystyle k - 1 ) and N − k ( displaystyle N-k ) degrees of freedom, and hence is the significance of the outcome w ( displaystyle w ) of W ( displaystyle W ) tested against F ( α, k − 1, N − k ) ( displaystyle F ( alpha, k - 1, N-k ) ) where F ( displaystyle F ) is a quantile of the F - distribution, with k − 1 ( displaystyle k - 1 ) and N − k ( displaystyle N-k ) degrees of freedom, and α ( displaystyle alpha ) is the chosen level of significance ( usually 0.05 or 0.01 ). Comparison with the Brown -- Forsythe test The Brown -- Forsythe test uses the median instead of the mean in computing the spread within each group ( Y  ̄ ( displaystyle ( \bar ( Y ) ) ) vs. Y ~ ( displaystyle ( \tilde ( Y ) ) ), above ). Although the optimal choice depends on the underlying distribution, the definition based on the median is recommended as the choice that provides good robustness against many types of non-normal data while retaining good statistical power. If one has knowledge of the underlying distribution of the data, this may indicate using one of the other choices. Brown and Forsythe performed Monte Carlo studies that indicated that using the trimmed mean performed best when the underlying data followed a Cauchy distribution ( a heavy - tailed distribution ) and the median performed best when the underlying data followed a chi - squared distribution with four degrees of freedom ( a heavily skewed distribution ). Using the mean provided the best power for symmetric, moderate - tailed, distributions. See also Bartlett 's test F - test of equality of variances References Jump up ^ Levene, Howard ( 1960 ). `` Robust tests for equality of variances ''. In Ingram Olkin ; Harold Hotelling ; et al. Contributions to Probability and Statistics : Essays in Honor of Harold Hotelling. Stanford University Press. pp. 278 -- 292. External links Parametric and nonparametric Levene 's test in SPSS http://www.itl.nist.gov/div898/handbook/eda/section3/eda35a.htm Retrieved from `` https://en.wikipedia.org/w/index.php?title=Levene%27s_test&oldid=779693625 '' Categories : Analysis of variance Statistical tests Talk Contents About Wikipedia Deutsch Español Euskara Magyar Nederlands 日本 語 Polski Українська Edit links This page was last edited on 10 May 2017, at 12 : 20 ( UTC ). Text is available under the Creative Commons Attribution - ShareAlike License ; additional terms may apply. By using this site, you agree to the Terms of Use and Privacy Policy. Wikipedia ® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization. About Wikipedia

Question: purpose of significant test of levene’s test for equality of variances
Answer: It tests the null hypothesis that the population variances are equal