Mean from a Frequency Table — Definition, Formula & Examples

Mean from a frequency table is the average of a data set where values are grouped with their frequencies, calculated by multiplying each value by its frequency, summing those products, and dividing by the total frequency.

Given a set of distinct data values $x_1, x_2, \ldots, x_k$ with corresponding frequencies $f_1, f_2, \ldots, f_k$ , the mean is the sum of each value times its frequency divided by the sum of all frequencies: $\bar{x} = \frac{\sum f_i x_i}{\sum f_i}$ .

Key Formula

\bar{x} = \frac{\sum f_i \, x_i}{\sum f_i}

Where:

$\bar{x}$ = the mean (average)
$x_i$ = each distinct data value
$f_i$ = the frequency (count) of each value
$\sum f_i$ = the total number of data points

How It Works

A frequency table tells you how many times each value appears in a data set. Instead of adding every individual data point, you take a shortcut: multiply each value by how often it occurs, add up all those products, then divide by the total number of data points. This gives the same result as listing out every value and finding the average the long way, but it is much faster when the same values repeat many times.

Worked Example

Problem: A teacher records quiz scores in a frequency table: Score 6 appears 2 times, Score 7 appears 5 times, Score 8 appears 4 times, Score 9 appears 3 times, and Score 10 appears 1 time. Find the mean score.

Step 1: Multiply each score by its frequency.

6 \times 2 = 12,\quad 7 \times 5 = 35,\quad 8 \times 4 = 32,\quad 9 \times 3 = 27,\quad 10 \times 1 = 10

Step 2: Add up all the products to get the total sum.

\sum f_i \, x_i = 12 + 35 + 32 + 27 + 10 = 116

Step 3: Add up all the frequencies to get the total number of data points, then divide.

\sum f_i = 2 + 5 + 4 + 3 + 1 = 15 \qquad \bar{x} = \frac{116}{15} \approx 7.73

Answer: The mean quiz score is approximately 7.73.

Why It Matters

Frequency tables appear constantly in science labs, surveys, and standardized tests. Knowing how to extract the mean quickly is essential in any middle-school or high-school statistics unit, and it builds the foundation for understanding standard deviation and other measures of spread.

Common Mistakes

Mistake: Dividing by the number of distinct values instead of the total frequency.

Correction: In the example above, there are 5 distinct scores but 15 total data points. You must divide by the total frequency (15), not the number of rows in the table (5).