Statistics and Distributions

How to find the mean of grouped data?

Quick answer

To find the mean of grouped data, calculate the midpoint of each class interval by adding the lower and upper limits and dividing by two. Multiply each midpoint by its class frequency to get fx. Add these products together, then divide this total sum by the total frequency. The resulting quotient is your estimated mean.

The topic explained

What is grouped data and why we estimate its mean

In 11th grade statistics, raw data is often collected into class intervals, such as 10 to under 20 or 20 to under 30. When data is grouped this way, individual values are lost. We know how many data points fall into each interval, but not their exact numbers.

Because the true values are hidden, we cannot compute an exact arithmetic mean. Instead, we assume that the values in each group balance out around the middle. That is why any mean calculated from a grouped frequency table is called an estimated mean.

Step-by-step formula for finding the mean

First, find the midpoint, denoted by x, for each class interval. Add the lower boundary and upper boundary, then divide by 2. For an interval of 10 to 20, the midpoint is (10 + 20) / 2 = 15.

Second, multiply each midpoint x by its group frequency f to obtain fx. This represents the estimated total for that class.

Third, sum all the fx values to get sigma fx. Then sum all the frequencies f to find the total count n. Finally, divide sigma fx by total f. The formula is: Mean = (sum of f * x) / (sum of f).

A worked example with commute times

Suppose a survey records commute times for 20 students: 0 to under 10 minutes (frequency 4), 10 to under 20 minutes (frequency 10), and 20 to under 30 minutes (frequency 6).

The midpoints are 5, 15, and 25. Multiply each midpoint by its frequency: 5 * 4 = 20; 15 * 10 = 150; 25 * 6 = 150. Sum these products: 20 + 150 + 150 = 320. Sum the frequencies: 4 + 10 + 6 = 20. Divide 320 by 20 to find an estimated mean commute time of 16 minutes.

Free printable worksheet

Finding the Mean of Grouped Data Worksheet

For 11th grade statistics students. Complete each table to find the midpoint, the fx values, and the estimated mean. Round decimals to one decimal place when necessary.

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Finding the Mean of Grouped Data WorksheetName:

Find the midpoint for each interval: (a) 0 to under 10, (b) 20 to under 40, (c) 15 to under 25.
A table shows two intervals: 0 to under 10 (frequency = 6) and 10 to under 20 (frequency = 4). Calculate the estimated mean.
Quiz scores are grouped: 10 to 19 (midpoint 14.5, f = 3), 20 to 29 (midpoint 24.5, f = 5), 30 to 39 (midpoint 34.5, f = 2). Find the sum of f and the sum of fx.
Using the previous quiz scores, calculate the estimated mean score.
Weekly study hours: 0-4 hours (f = 2), 5-9 hours (f = 7), 10-14 hours (f = 8), 15-19 hours (f = 3). Find the estimated mean. Use midpoints 2, 7, 12, 17.
Why can two datasets with the exact same grouped frequency table have different actual means?

Common questions

Why is the mean from a grouped frequency table only an estimate?

Grouped data places observations into bins rather than listing each specific number. Calculating the mean requires picking a single representative value for each bin, usually the midpoint. Because the actual numbers may not sit exactly at the midpoint, the resulting mean is an estimate.

What is the difference between frequency and relative frequency?

Frequency is the raw count of how many data points fall into a given interval. Relative frequency is the proportion or percentage of the whole dataset in that interval, calculated by dividing the interval frequency by the total number of data points.

What should I do if the class intervals have unequal widths?

The calculation method stays the same. Find the specific midpoint of each interval by averaging its own upper and lower boundaries, multiply by that interval frequency, sum the products, and divide by the total frequency.

What common mistake should I avoid when calculating grouped mean?

A frequent error is dividing the sum of fx by the number of intervals instead of the sum of the frequencies. Always divide by the total count of observations, n.

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