Limits of Comparing Distributions from Mean and Standard Deviation

Summary

The document asks how to detect when two distributions differ if only their sample means and standard deviations are available. It mentions standard tests for distribution comparison, including Jarque–Bera, Anderson–Darling, and Kolmogorov–Smirnov, but notes that such approaches require more information than the two summary statistics alone provide.

With only the mean and standard deviation, a comparison must rely on an assumption about the distribution, such as normality. Additional shape statistics, particularly skewness and kurtosis, would make the comparison more informative. The response also cautions that distinct data sets can share the same mean and standard deviation, so matching these moments does not establish that the distributions are alike. No threshold procedure or empirical example is supplied; the practical lesson is that limited summary data cannot support a general distributional similarity test without stronger assumptions.

Key ideas

• The mean and standard deviation do not uniquely determine a distribution.
• Formal distribution tests generally need observations or richer summary information.
• A normality assumption can make comparisons possible, but the conclusion depends on that assumption.
• Skewness and kurtosis provide useful information about distribution shape beyond the first two moments.

Tags

# How can I compare distributions using only mean and standard deviation?


# How can I compare distributions using only mean and standard deviation?












I only have means and standard deviations of samples of two random variables. What technique can I use to determine how similar the distributions these describe are? Assume that the values are built from very similar samples. I'm looking for a mechanism to detect when the distributions deviate from one another by some threshold. Access to historical observations is limited.

## Answer by Shane (score 13, accepted)

https://quant.stackexchange.com/a/2296

There are a number of different tests that are generally used to compare samples to different distributions, such as Jarque-Bera, Anderson-Darling, and Kolmogorov–Smirnov (see this related question).

In your case, with just the standard deviation and mean, there isn't a whole lot to say. You need to assume a distribution (e.g. normal). You would be able to tell much more if you could at least get the skewness and kurtosis.

## Answer by Carlos (score 6)

https://quant.stackexchange.com/a/2320

Be careful, remember that the mean and the standard deviation don't tell you the whole story: http://en.wikipedia.org/wiki/Anscombe%27s_quartet

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