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:** Σm_{i}n_{i} / N

where:

**m**The midpoint of the i_{i}:^{th}group**n**The frequency of the i_{i}:^{th}group**N:**The total sample size

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

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:** √Σn_{i}(m_{i}-μ)^{2} / (N-1)

where:

**n**The frequency of the i_{i}:^{th}group**m**The midpoint of the i_{i}:^{th}group**μ**: The mean**N:**The total sample size

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

The 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