Hello Amna,

The first question is, for what purpose are you evaluating whether this set of scores fits (adequately) to a normal distribution? Usually, for parametric statistical tests, the issue of concern is whether model residuals (difference between actual scores and scores estimated from the model) are plausibly characterized as normally distributed.

As you're consulting several different bits of information, it shouldn't be surprising that they might not all be completely congruent with one another.

1. For moment indicators like skewness and kurtosis, take the estimate, divide it by its standard error, and if the ratio is more than 3 or so (we can get more precise relative to any target type I risk level you might prefer), that would suggest that the scores in question do not conform well to a normal distribution. Your skewness appears to be fine (|-.075 / .121| ~ 0.62); the kurtosis (a measure of distribution flatness/peakedness and tail heavyness) is too low (|-.856 / .242 ~ 3.54). So, that suggests misfit (to the normal distribution).

2. Visual inspection of a histogram, or other display (e.g., quantile-quantile plot, or especially deviations from a Q-Q plot) is an acquired skill, since most statistical software invokes auto scaling of one or both of the axes (and, for histograms, the bin width; too few bins makes it harder to be certain of distributional form), so you have to be cautious.

3. Though you don't mention them, other goodness of fit tests (e.g., Kolmogorov-Smirnov, Anderson-Darling, or Shapiro-Wilk) may not completely agree, but they are often close in their appraisal of degree of misfit.

Good luck with your work.

Dear Amna,

I highly recommend you to use Kolmogorov-Smirnov or Shapiro-Wilk instead of histogram. Use first one if your sample size is bigger than 30 and use second test if your sample size is under 30.

But you should take into account which your hyopthesis really matters here.

In the end, if your data is not normally distributed, base on your hypothesis you can use different methods to make them normally distributed.

In case suggestions by David Morse and Mohammad Ali Koushesh Vatan do not work, you can either use some transformations (such as log transformation or Box Cox transformation) and check if the new data become normal. If still not normally distributed, then you may consider using non-parametric tests.

David Morse wrote:

The first question is, for what purpose are you evaluating whether this set of scores fits (adequately) to a normal distribution?

I'll add a few more specific questions of my own.

If normality tests reveal your variable is not normally distributed, you might try transforming the variable to conform to the normal distribution. You could check out Raymaekers and Rousseeuw (2021) for relevant insights. That being suggested, non-normal distributions visually displayed through histograms could be a good sign for using a nonparametric test.

Raymaekers, J., & Rousseeuw, P. J. (2021). Transforming variables to central normality. Machine Learning. https://doi.org/10.1007/s10994-021-05960-5

Article Transforming variables to central normality

Good luck,

You can use software like EASYFIT to find the best distribution fit to our data, then find the suitable transformation for that sort of distribution (or using Cox-Box formula).

David Morse & Bruce Weaver Thanks for your suggestions.

As far as the purpose of my study is concerned we have developed a model of 'information privacy concerns' which contains IV's, Dv's, MV's and Mediating variables so that we want to apply parametric test to test the model. As i read data normality is basic issue while applying parametric test. My sample size is 405.

i test outliers which identified 1 case.

I am using SPSS.

As David Morse noted in his response, the normality assumption for OLS models applies to the errors, not to the dependent variable itself, or to any other variables. Here is a slightly modified version of something I posted recently in another thread. Perhaps you'll find it helpful.

Regarding point 4 above, with n = 405, unless your outcome variable has a particularly ugly distribution, the sampling distributions of the parameter estimates should be approximately normal.

HTH.

A histogram is a terrible, terrible way to judge the shape of a distribution! I've attached four histograms of the same data, only differing by the width of the bars and the start point of the first bar. With so many aspects of the appearance of the histogram dependent on arbitrary choices of bar width and start position, the degree to which the shape of the histogram also reflects the distribution of the data is unknown.

Try a normal quantile plot. They are good for detecting unusual behaviour in the extremes of the data, which is always interesting.

Two things.

1. As @Ronan says look up quantile-quantile plots.

2. If you have specific parameters that you wish to examine, these can dictate how (and whether) you transform the data. If you interests are more qualitative, like X is bigger than Y, this is less true, but you should still think what model to you want to examine, and then there are lots of estimation methods that vary in how affected they are by different distributions (and as many note above, distributions of residuals).

Further to my graph showing four different, awful histograms, here is a normal quantile plot of the same data. It's far clearer than any or all of them.

Thanks Ronán Michael Conroy. Would you mind sharing your (Stata) code for the graphs you have uploaded? I am curious to see what happens if you add the kdensity option to your 4 histogram commands.

https://www.stata.com/support/faqs/graphics/gph/graphdocs/histogram-of-continuous-variable-with-frequencies-and-overlaid-kernel-density-estimate/

Thanks,

Bruce

The kdensity trace is, of course, the same in all cases. The slight differences in appearance are due to the range of the vertical scale. That slight hummock you can see in the density is explainable in terms of the two groups of participants : babies delivered normally and babies that had to be delivered by emergency section because they were failing to progress. The distribution of the latter group is particularly interesting, as shown in the strip plots

histogram babyweightkilos, percent kdensity name(asis, replace)

histogram babyweightkilos, bin(11) percent kdensity name(bin11, replace)

histogram babyweightkilos, start(2) percent kdensity name(start2, replace)

histogram babyweightkilos, width(.5) start(2) percent kdensity name(start2w5, replace)

gr combine asis bin11 start2 start2w5, title("Four histograms of the same thing" "With kernal density estimate")

stripplot babyweightkilos, over(modeofdelivery) jitter(3) ///

box(bfcolor(eltblue) blcolor(black) barw(0.1)) ///

iqr vertical aspect(1) ///

mcolor(blue) boffset(.1) ///

xscale(range(-0.2 1.2)) ///

title("Stripplot of birth weights" "showing normal delivery" "and emergency section groups")

Thanks Ronán Michael Conroy. I suspected that the kdensity traces would be the same on all plots.

I thought you were using the lbw.dta file that comes with Stata, but when I tried your code with that dataset, the results were different. If you are using a publicly available dataset, which one is it?

Cheers,

Bruce

The data are from a dataset I use a lot in teaching. It's a case-control study of predictors of emergency cesarean section because the baby was getting stuck. You can't say "drat, the baby's stuck" in front of the everyone, so you have to say "cephalopelvic disproportion" so it sounds like you have everything under control.

It's a fun dataset, with interesting features like a case where you have a significant t-test and a non=significant Wilcoxon Mann-Whitney and they are both right. The uploaded dataset just has the two variables I used in the examples.

There is some lovely theoretical stuff in this thread but I am unclear about the questioner's real world objective. Is it comparison (two things at a time or global?), 'is it regression? If you convert data the answer should have some meaning under conversion, or at least when your convert back.

If you are concerned about the question of normal maybe consider nonparametric approaches. However t tests, for instance, are adequate for most samples of size 30 or more and almost all samples for sample sizes of 80 or more.

Regression itself does not depend on normality, but the t test of the regression does. If your data is non-normal you can run a Spearman's rank correlation to see if there is at least an ordinal correlation which includes testing procedures.

Lewis Maccarter wrote :

However t tests, for instance, are adequate for most samples of size 30 or more and almost all samples for sample sizes of 80 or more.

I would be very interested in your reference for this. I'm writing something at the moment, and trying to find the origin of recommendations such as these, which is surprisingly difficult.

The following mostly refer to skew, but careful reading will suggest that in most cases any sample size sufficient to correct skew will correct Kurtosis.:

Ratcliffe JF. 1968. The effect on the t distribution of non-normality in the sampled population. Appl. Stat. 17:42–48.

Lumley, Thomas, Paula Diehr, Scott Emerson, and Lu Chen 2002. The importance of the normality assumption in large public health data sets, Annual Review of Public Health 23:151-169

Lovric, M. (Ed) 2010. International Encyclopedia of Statistical Science. New York: Springer, New York. London.

However, Likert scale is unsuitable for the t test and Wilcoxon rank sum tests are to be preferred for small samples especially if skew is extreme or variances are unequal and sample sizes are unequal:

Blair, R. C., and Higgins, J. J. 1980. A comparison of power of Wlicoxon's rank- sum statistic to that of Student's t statistic under various nonnormal distributions. Journal of Educational Statistics, 5, 309-335.

Bridge, P. K., and Sawilowsky, S. S. 1999.. Increasing physician's awareness of the impact of statistical tests on research outcomes: Investigating the comparative power of the Wilcoxon Rank-Sum test and independent samples t test to violations from normality. Journal of Clinical Epidemiology, 52, 229-235.

Nanna, M. J., and Sawilowsky , S. S. 1998. Analysis of Likert scale data in disability and medical rehabilitation evaluation. Psychometric Methods, 3, 55-67.

These references are from MacCarter (2021), which also touches on all this and updated sample size estimation (no more 'small', 'medium', and 'large'). In general if sample size is sufficient to have 80% power and alpha even as ridiculously large as 5% (1/20), the samples will in my experience be >30.

L.E. MacCarter 2021. RESEARCH DECISIONS AND ESTIMATION WITH CONFIDENCE AND POWER, Up To Date Statistics Reference and Text From Classical and Recent Literature. ISBN-13: 978-1532721076, Library of Congress Control Number: 2021913804 (available Amazon in US and International; reviews appreciated).

Watch this video for Non-parametric tests, tricks and conditions to use them: https://youtu.be/Xu6-kU8s-ls

Ronán Michael Conroy, I have long speculated that the so-called rule of 30 that many people espouse arose due to confusion about the way older textbooks described small-sample and large-sample inference. Let's take the one-sample test for a mean as the context. In those old books, readers were typically advised to compute tobs = (Xbar-mu0)/SE, and to refer tobs to a table of critical t-values if df was below some cut-off, typically around 30. If df was greater than the cut-off, readers were advised to compare tobs to the critical value of z from a table of the standard normal distribution. Somehow, this got twisted into the current belief held by many that one must have n=30 or more to carry out a t-test. But notice that this is directly contradicted by the advice given in the older textbooks. All of the small-sample examples in which readers were directed to a table of critical t-values had n < 30.

Again, this is speculative, but I think that may be one source of the belief that one must have n = 30+ to use a t-test.

I also want to comment on this statement by Lewis:

Regression itself does not depend on normality, but the t test of the regression does.

Aye. But normality of what? See my first post in the thread (27-Jun-2021) for my thoughts on that! ;-)

Cheers,

Bruce

Why do you want your data to be normally distributed? What have they done to you to deserve that? And why (other than being away from the others) do you think case 405 is not a valid part of your population of interest? And Lewis Maccarter , you talk about "correcting" skew. Why is a particular skewness value in need of correcting? Are you making some kind of argument that some distributions are amoral? ... Okay, I am going on, but that point is a distribution is neither good nor bad, but it may or may not be ideally suited for your needs. And things like high kurtosis in the error term is addressed by many robust estimates and things like the transformations discussed in Preprint A Robust Alternative to Pearson's Correlation for Testing As...

And I see discussion of the n=30 old notion. Beyond just being outdated (in part because more alternatives exist now), it was only about Type 1 error, but people generalized that. The responses when Tukey (1960) asked people what they would expect showed this.

Daniel Wright: Thanks for reading my comments. I suggest you will also enjoy reading the papers I cited. As for the word "correcting", it may have been an awkward choice of words. I am not intimidated by non-normal distributions, but I attempt to be moderate when trying to assist and inform the many researchers who are overly concerned due to the fact that most texts drastically need updating.

Lewis Maccarter , I figured you weren't, much more getting at the questioner not giving a reason why they were doing that. Obviously it is not for things like a regression, since the distributional assumptions aren't about the original variables (as pointed out by several). Thanks for the references.

I think Daniel Wright has a really important point here : why are we trying to make our data fit a pre-determined model when we should be trying to formulate a model that fits the data generation process.

I like to compare biostats to forensic science. Like a crime scene, everything in the data is where it is because of the processes that acted upon it. Our job is to understand the processes and to tell the story.

You have to transform your data to attain normality assumptions.

Teshita Uke Chikako wrote: You have to transform your data to attain normality assumptions.

Teshita, did you read any of the earlier replies before posting that? Please review Ronán Michael Conroy's Aug 20 post, for example. It appears just before your post.