How to Find the Variance of Grouped Data (With Example)

Often we may want to calculate the variance of a grouped frequency distribution.

For example, suppose we have the following grouped frequency distribution:

While it’s not possible to calculate the exact variance since we don’t know the raw data values, it is possible to estimate the variance using the following formula:

Variance: Σn_i(m_i-μ)² / (N-1)

where:

• n_i: The frequency of the i^th group
• m_i: The midpoint of the i^th group
• μ: The mean
• N: The total sample size

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.

The following example shows how to use this formula in practice.

Example: Calculate the Variance of Grouped Data

Suppose we have the following grouped data:

Here’s how we would use the formula mentioned earlier to calculate the variance of this grouped data:

We would then calculate the variance as:

• Variance: Σn_i(m_i-μ)² / (N-1)
• Variance: (604.82 + 382.28 + 68.12 + 477.04 + 511.21) / (23-1)
• Variance: 92.885

The variance of the dataset turns out to be 92.885.

Additional Resources

The following tutorials explain how to calculate other metrics for grouped data:

How to Find Mean & Standard Deviation of Grouped Data
How to Calculate Percentile Rank for Grouped Data
How to Find the Median of Grouped Data
How to Find the Mode of Grouped Data