Questions tagged [normality-assumption]

Many statistical methods assume data or a model's residuals are normally distributed. Use this tag for questions about the assumption & testing of normality, or about normality as a *property*. Use [normal-distribution] for questions about the normal distribution per se.

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361 votes
16 answers
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Is normality testing 'essentially useless'?

A former colleague once argued to me as follows: We usually apply normality tests to the results of processes that, under the null, generate random variables that are only asymptotically or ...
135 votes
3 answers
45k views

What if residuals are normally distributed, but y is not?

I've got a weird question. Assume that you have a small sample where the dependent variable that you're going to analyze with a simple linear model is highly left skewed. Thus you assume that $u$ is ...
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57 votes
3 answers
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Box-Cox like transformation for independent variables?

Is there a Box-Cox like transformation for independent variables? That is, a transformation that optimizes the $x$ variable so that the y~f(x) will make a more ...
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145 votes
6 answers
254k views

Pearson's or Spearman's correlation with non-normal data

I get this question frequently enough in my statistics consulting work, that I thought I'd post it here. I have an answer, which is posted below, but I was keen to hear what others have to say. ...
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57 votes
5 answers
96k views

Regression when the OLS residuals are not normally distributed

There are several threads on this site discussing how to determine if the OLS residuals are asymptotically normally distributed. Another way to evaluate the normality of the residuals with R code is ...
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57 votes
3 answers
243k views

ANOVA assumption normality/normal distribution of residuals

The Wikipedia page on ANOVA lists three assumptions, namely: Independence of cases – this is an assumption of the model that simplifies the statistical analysis. Normality – the distributions of the ...
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53 votes
5 answers
20k views

Interpreting QQplot - Is there any rule of thumb to decide for non-normality?

I have read enough threads on QQplots here to understand that a QQplot can be more informative than other normality tests. However, I am inexperienced with interpreting QQplots. I googled a lot; I ...
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47 votes
5 answers
103k views

What references should be cited to support using 30 as a large enough sample size?

I have read/heard many times that the sample size of at least 30 units is considered as "large sample" (normality assumptions of means usually approximately holds due to the CLT, ...). Therefore, in ...
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50 votes
5 answers
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Normality of dependent variable = normality of residuals?

This issue seems to rear its ugly head all the time, and I'm trying to decapitate it for my own understanding of statistics (and sanity!). The assumptions of general linear models (t-test, ANOVA, ...
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25 votes
5 answers
19k views

How robust is the independent samples t-test when the distributions of the samples are non-normal?

I've read that the t-test is "reasonably robust" when the distributions of the samples depart from normality. Of course, it's the sampling distribution of the differences that are important. I have ...
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31 votes
5 answers
137k views

Can I trust ANOVA results for a non-normally distributed DV?

I have analyzed an experiment with a repeated measures ANOVA. The ANOVA is a 3x2x2x2x3 with 2 between-subject factors and 3 within (N = 189). Error rate is the dependent variable. The distribution of ...
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56 votes
3 answers
39k views

Standard deviation of standard deviation

What is an estimator of standard deviation of standard deviation if normality of data can be assumed?
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19 votes
3 answers
2k views

Example of distribution where large sample size is necessary for central limit theorem

Some books state a sample size of size 30 or higher is necessary for the central limit theorem to give a good approximation for $\bar{X}$. I know this isn't enough for all distributions. I wish ...
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27 votes
4 answers
15k views

Is Shapiro–Wilk the best normality test? Why might it be better than other tests like Anderson-Darling?

I have read somewhere in the literature that the Shapiro–Wilk test is considered to be the best normality test because for a given significance level, $\alpha$, the probability of rejecting the null ...
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7 votes
2 answers
9k views

Testing normality

I have a large dataset (500000 data, V1 column include all the data). x <- read.csv("mydata.csv", header=F) hist(x) Which gives: Looking at the data, I ...
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39 votes
6 answers
133k views

Assumptions of linear models and what to do if the residuals are not normally distributed

I am a little bit confused on what the assumptions of linear regression are. So far I checked whether: all of the explanatory variables correlated linearly with the response variable. (This was the ...
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24 votes
7 answers
18k views

Appropriate normality tests for small samples

So far, I've been using the Shapiro-Wilk statistic in order to test normality assumptions in small samples. Could you please recommend another technique?
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16 votes
2 answers
18k views

QQ plot does not match histogram

I have a histogram, kernel density and a fitted normal distribution of financial log returns, which are transformed into losses (signs are changed), and a normal QQ plot of these data: The QQ plot ...
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13 votes
2 answers
23k views

If my histogram shows a bell-shaped curve, can I say my data is normally distributed?

I created a histogram for Respondent Age and managed to get a very nice bell-shaped curve, from which I concluded that the distribution is normal. Then I ran the normality test in SPSS, with n = 169. ...
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12 votes
2 answers
2k views

What is the expected distribution of residuals in a generalized linear model?

I am performing a generalized linear model, where I have to specify a family different from the normal one. What is the expected distribution of residuals? For example, should the residuals be ...
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24 votes
2 answers
46k views

How to test for differences between two group means when the data is not normally distributed?

I'll eliminate all the biological details and experiments and quote just the problem at hand and what I have done statistically. I would like to know if its right, and if not, how to proceed. If the ...
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28 votes
3 answers
56k views

What should I check for normality: raw data or residuals?

I've learnt that I must test for normality not on the raw data but their residuals. Should I calculate residuals and then do the Shapiro–Wilk's W test? Are residuals calculated as: $X_i - \...
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  • 1,767
21 votes
4 answers
14k views

Transformation to increase kurtosis and skewness of normal r.v

I'm working on an algorithm that relies on the fact that observations $Y$s are normally distributed, and I would like to test the robustness of the algorithm to this assumption empirically. To do ...
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9 votes
2 answers
10k views

Normality of residuals vs sample data; what about t-tests?

An addition to the common confusion about normality testing in inferential statistics for general linear models: I understand the assumption of normality refers to the residuals in ANOVA and ...
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34 votes
7 answers
33k views

What is normality?

In many different statistical methods there is an "assumption of normality". What is "normality" and how do I know if there is normality?
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10 votes
5 answers
15k views

What other normalizing transformations are commonly used beyond the common ones like square root, log, etc.?

In the analysis of test scores (e.g., in Education or Psychology), common analysis techniques often assume that data are normally distributed. However, perhaps more often than not, scores tend to ...
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8 votes
3 answers
14k views

Does ARIMA require normally distributed errors or normally distributed input data?

I have two questions related to time series forecasting with ARIMA: Does ARIMA require normally distributed errors or normally distributed input data ? Are there any assumptions on input time series ...
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2 votes
1 answer
2k views

How to calculate power of different normality tests such as Shapiro-Wilk, Ryan test etc [closed]

Normality test for small samples with power comparison
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53 votes
3 answers
15k views

Why do we care so much about normally distributed error terms (and homoskedasticity) in linear regression when we don't have to?

I suppose I get frustrated every time I hear someone say that non-normality of residuals and /or heteroskedasticity violates OLS assumptions. To estimate parameters in an OLS model neither of these ...
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40 votes
5 answers
18k views

Is there an explanation for why there are so many natural phenomena that follow normal distribution?

I think this is a fascinating topic and I do not fully understand it. What law of physics makes so that so many natural phenomena have normal distribution? It would seem more intuitive that they would ...
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  • 569
35 votes
6 answers
178k views

Interpretation of Shapiro-Wilk test

I'm pretty new to statistics and I need your help. I have a small sample, as follows: ...
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13 votes
1 answer
26k views

Testing large dataset for normality - how and is it reliable?

I'm examining a part of my dataset containing 46840 double values ranging from 1 to 1690 grouped in two groups. In order to analyze the differences between these groups I started by examining the ...
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8 votes
2 answers
4k views

Should the predictor variables be normally distributed for Poisson glm?

This is probably a really basic question, but it's the first time I've created a model that defines Poisson as its error family. In setting up my variables to make the model, should I be concerned ...
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8 votes
3 answers
31k views

Regression - How do I know if my residuals are normally distributed?

Performing a regression and need to find out if my residuals are normally distributed.
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5 votes
2 answers
16k views

What to do if residuals are not normally distributed?

I was wondering what to do with the following non-normal distribution of residuals of my multiple regression. Normality test of standardized residual ...
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4 votes
1 answer
3k views

Should quantitative predictors be transformed to be normally distributed?

I am always struggling with normality testing for quantitative predictors (no factors) and transforming them to normality. If I am running a GLMM and my predictors are really non-normal, should I ...
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3 votes
1 answer
313 views

How to assess normality under the OLS assumptions?

When we have a multivariate regression function, which assumption has to hold so that the OLS assumptions are not violated: ...
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39 votes
2 answers
136k views

What is the difference between the Shapiro–Wilk test of normality and the Kolmogorov–Smirnov test of normality?

What is the difference between the Shapiro–Wilk test of normality and the Kolmogorov–Smirnov test of normality? When will results from these two methods differ?
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21 votes
2 answers
3k views

Can we see shape of normal curve somewhere in nature?

I do not want to know if some phenomena in nature have normal distribution, but whether we can somewhere see shape of normal curve as we can see it for example in Galton box. See this figure from ...
6 votes
2 answers
5k views

What have to be normally distributed: groups or whole sample?

It is a general and maybe basic question, but it will help me to avoid a mistake. To test the dependence of continuous variable on nominal with parametric test (t-test, ANOVA), "the data has to be ...
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5 votes
2 answers
947 views

Is the inference from a parametric test valid when the population distribution is not normal?

This question arose from reading this post: T-test for non normal when N>50? In the response to this post, the author outlines really well that the assumption of normality with regard to the t-...
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10 votes
2 answers
271 views

Relevance of assumption of normality, ways to check and reading recommendations for non-statisticians

I started reading around the topic of modern robust methods, consulted various statistics texts and did some research on the CV forum. I ended up being rather confused regarding the relevance of the ...
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8 votes
1 answer
5k views

When is the distribution of product of two normal distributed variables near normal distribution?

It is clear the product of normal distributed variables is not normal distributed. For example, if $X \sim N( \mu_1,\sigma_1^2)$, $Y \sim N( \mu_2,\sigma_2^2)$, then $XY$ does not has the ...
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5 votes
2 answers
6k views

How to transform data to normality?

We have financial some data (500-1000 samples), which is not normally distributed (well known fact from the literature). I have some ideas to do parametric transformations of this data (using some ...
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  • 735
1 vote
2 answers
17k views

What to do when Kolmogorov-Smirnov test is significant for residuals of parametric test but skewness and kurtosis look normal?

I have conducted a parametric test in a study, n=290. I want to assess whether the residuals of this test are normally distributed. The skewness and kurtosis of the residuals are -0.017 and -0.438 ...
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52 votes
4 answers
66k views

If the t-test and the ANOVA for two groups are equivalent, why aren't their assumptions equivalent?

I'm sure I've got this completely wrapped round my head, but I just can't figure it out. The t-test compares two normal distributions using the Z distribution. That's why there's an assumption of ...
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  • 5,525
5 votes
1 answer
12k views

transformation to normality of the dependent variable in multiple regression

Is it really important to normalize dependent variables in multiple regression or are there any exceptions? My model is providing better results with more significant hypothesis when the DVs are not ...
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12 votes
2 answers
17k views

What normality assumptions are required for an unpaired t-test? And when are they met?

If we wish to conduct a paired t-test, the requirement is (if I understand correctly) that the mean difference between the matched units of measurement will be distributed normally. In paired t-test, ...
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  • 20.1k
14 votes
1 answer
40k views

R: test normality of residuals of linear model - which residuals to use

I would like to do a Shapiro Wilk's W test and Kolmogorov-Smirnov test on the residuals of a linear model to check for normality. I was just wondering what residuals should be used for this - the raw ...
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10 votes
1 answer
6k views

Linear Discriminant Analysis and non-normally distributed data

If I understand correctly, a Linear Discriminant Analysis (LDA) assumes normal distributed data, independent features, and identical covariances for every class for the optimality criterion. Since ...
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