Visual Checks for Normality and Independent, Identically Distributed Data

Summary

The document suggests simple graphical diagnostics for assessing whether observations look normally distributed and whether their distribution remains stable over time. For normality, it proposes comparing a histogram of the observations with an overlaid normal density. A mismatch in shape can reveal features such as skewness or excess kurtosis. For a rough check of identical distributions, it recommends dividing the series into periods and comparing their empirical frequency distributions; a shift between periods indicates that the distribution may have changed.

These plots are accessible exploratory checks, not formal tests. A histogram can depend on bin choices and may hide deviations, while similar period distributions do not establish independence. The examples are described qualitatively, without a dataset, test statistic, or significance assessment. The document mentions Jarque–Bera and Durbin–Watson tests as the question’s context, but its proposed visual comparisons do not replace a systematic assessment of normality, serial dependence, or the assumptions needed for an IID model.

Key ideas

• Overlaying a normal density on a histogram provides a visual check for departures from normality.
• Comparing frequency distributions across time periods can reveal distribution shifts.
• A visible change across periods is evidence against identically distributed observations.
• Visual diagnostics are exploratory and do not establish independence or replace formal testing.

Tags

# a simpler test for normality given skewness, kurtosis and autocorrelation and size of time series


# a simpler test for normality given skewness, kurtosis and autocorrelation and size of time series












I typically do a JB (Jarque Bera) test and DW (Durbin Watson) tests for check for normality given skewness, kurtosis and autocorrelation of the data. However this requires a CHI distribution table lookup and some calculation. I was wondering if there is a simple less accurate test that I can do on the data to check if it is normal or not?

Secondly How do I numerically test for I.I.D ?

## Answer by Ram Ahluwalia (score 0, accepted)

https://quant.stackexchange.com/a/3106

There are simple visual tests for normality and i.i.d.

To test for nomality, simply look at the distribution of the observations and overlay the normal density function as in the following chart:

The blue line represents the normal distribution. The black bars are from the histogram representing the empirical distribution. Clearly the data is leptokurtotic is skewed to the right.

To test whether the data is i.i.d., simply partition the data set into different periods and plot the frequency distribution:

Notice the distribution has shifted from Period 1 to Period 2 -- therefore this data is not i.i.d.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.