When time is short and you just want a rough estimate of the standard deviation, turn to the range rule to quickly estimate the standard deviation value.

The standard deviation is approximately equal to the range of the data divided by 4. That’s it, simple.

Find the largest value, the maximum and subtract the smallest value, the minimum, to find the range. Then divide the range by four.

## Example

Say you have the following measurements:

9, 13, 14, 17, 18, 21, 22, 25, 29, and 35

Which has a mean of 20.3, sample standard deviation of 7.8 and population standard deviation of 7.4.

The maximum value is 35 and the minimum value is 9.

35 – 9 = 26.

26 divided by 4 is 6.5.

About the best you can say is that 6.5 is in the ball part of 7.4 and 7.8 – it’s a rough estimate at best.

## How it works

Consider the normal distribution for a moment. 95% of the data is within plus or minus two standard deviations.

For a relatively small sample, the range is likely to based on data from within this 95% range. The range of the two standard deviations is the full range of the data from the distribution, so a relatively small sample will most likely contain data inside this range. Thus the range rule provides a rough estimate, often a bit smaller than the actual standard deviation.

Even for distributions that are not normally distributed the bulk of the data will be within the plus or minus two standard deviations, so it should provide a rough estimate of most any distribution.

A quick trick to help you make sense of data.

Related:

Variance (article)

Root Sum Squared Tolerance Analysis Method (article)

Point and Interval Estimates (article)

amajid says

1-why it is divided by 4?

2-is it necessary the data to be unimodal?

Fred Schenkelberg says

Why divide by 4? – just a rough estimate of the standard deviation given for a sample the data is most likely from within the plus/minus two standard deviations of the mean. There is probably a theory or study to support it, yet I’m not sure.

Unimodel? no, I don’t think so – you could test this by creating various distributions and compare the standard deviations to the estimated value using this method.

Cheers,

Fred

mark says

How is this affected by the number of observations? How good is the estimate if there are only 2 observations?

Fred Schenkelberg says

Good Question Mark

I don’t know exactly, as I tend to avoid using the range rule as I can typically use my computer, smartphone, tablet, or even a calculator to estimate the variance or standard deviation directly. A quick study of using random numbers from a distribution, and estimate with two, three, etc values and compare many iterations to the actual value may reveal a meaningful answer to your question.

cheers,

Fred

Jay Wijaya says

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1097734/

Swati says

Dy dividing 4 ultimate result is 2sd or 3sd

Fred Schenkelberg says

Hi Swati, the assumption is the range of the dataset is most likley representing the plus/minus 2 standard deviation. This may or may not be true, yet if faced with a bit of data and wanting a quick, rough, estimate, this works. cheers, Fred