How to Find Mean & Standard Deviation of Grouped Data


Often we may want to calculate the mean and standard deviation of data that is grouped in some way. For example, suppose we have the following grouped data:

While it’s not possible to calculate the exact mean and standard deviation since we don’t know the raw data values, it is possible to estimate the mean and standard deviation.

The following steps explain how to do so.

Calculate the Mean of Grouped Data

We can use the following formula to estimate the mean of grouped data:

Mean: Σmini / N

where:

  • mi: The midpoint of the ith group
  • ni: The frequency of the ith group
  • N: The total sample size

Here’s how we would apply this formula to our dataset from earlier:

Mean of grouped data formula

The mean of the dataset turns out to be 22.89

Note: The midpoint for each group can be found by taking the average of the lower and upper value in the range. For example, the midpoint for the first group is calculated as: (1+10) / 2 = 5.5.

Calculate the Standard Deviation of Grouped Data

We can use the following formula to estimate the standard deviation of grouped data:

Standard Deviation:Σni(mi-μ)2 / (N-1)

where:

  • ni: The frequency of the ith group
  • mi: The midpoint of the ith group
  • μ: The mean
  • N: The total sample size

Here’s how we would apply this formula to our dataset:

Standard deviation of grouped data exampleThe standard deviation of the dataset turns out to be 9.6377.

Additional Resources

How to Estimate the Mean and Median of Any Histogram
An Introduction to Measures of Central Tendency
An Introduction to Measures of Dispersion

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