Mathematics, three units

How to find an average from a frequency table?

Quick answer

You find the average by multiplying each data value by its frequency, adding all those products together, and dividing that total by the sum of the frequencies. Never divide by the number of rows in the table, because each row represents multiple individual data points rather than a single entry.

The topic explained

How the calculation works

When data is organized in a table, values that occur multiple times are grouped together. Determining the mean from such a grouped display requires accounting for how heavily each value weighs in the total dataset. If a test score of 80 appears 7 times, it contributes a combined value of 560 toward the grand sum and represents 7 individual tests toward the count.

Instead of adding all raw scores one by one, you multiply each distinct value by its frequency. Summing these products gives the grand total of all entries. Dividing this total by the sum of all frequencies gives the true arithmetic mean.

A step-by-step example

Suppose a test produces these scores: a mark of 70 appears 4 times, 80 appears 7 times, and 90 appears 9 times. To find the mean, first multiply each score by its frequency: 70 times 4 is 280, 80 times 7 is 560, and 90 times 9 is 810.

Next, add these products to find the grand total: 280 plus 560 plus 810 equals 1,650. Then, find the total number of students by adding the frequencies: 4 plus 7 plus 9 equals 20. Finally, divide 1,650 by 20 to get the average score, which is 82.5.

The common trap to avoid

The most frequent mistake on exams is dividing the sum of products by the number of rows instead of the total frequency. In the test score example, dividing 1,650 by 3 rows gives 550, which is impossible for a test scored out of 100. Each row holds unequal weight, so always divide by the sum of frequencies.

You can also compute the relative frequency for each row by dividing its frequency by the grand total count. For the score of 80, the relative frequency is 7 divided by 20, which is 0.35 or 35 percent. All relative frequencies in your table must always sum to 1.0 or 100 percent.

Free printable worksheet

Calculating the Mean from Frequency Tables

Designed for 11th grade students. Multiply each value by its frequency, total the products and frequencies, and find the mean.

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Calculating the Mean from Frequency TablesName:

A survey of household pets shows: 0 pets (frequency 5), 1 pet (frequency 12), 2 pets (frequency 8). Calculate the mean number of pets.
In a quiz scored out of 5: score 3 (frequency 4), score 4 (frequency 10), score 5 (frequency 6). Calculate the mean score.
Find the relative frequency of getting a score of 4 in the quiz from the previous question.
A store records daily shoe sales: 5 pairs (2 days), 10 pairs (5 days), 15 pairs (3 days). Find the average number of pairs sold per day.
A student adds three rows of values (10, 20, 30) with frequencies (1, 2, 7) and divides the total product of 310 by 3 to get 103.3. What was their error and what is the correct mean?
Grouped data shows travel times: 0 to 10 minutes has midpoint 5 (frequency 4), 10 to 20 minutes has midpoint 15 (frequency 6). Estimate the mean travel time.

Common questions

Why can you not simply average the values in the first column?

Averaging just the unique values assumes each number occurred an equal number of times. A score earned by ten students carries ten times as much weight as a score earned by one student. Multiplying by frequencies ensures every observation counts equally.

How do you find the mean if the data is grouped into intervals?

When data is grouped into intervals like 10 to 20, find the midpoint of each interval first. Multiply each midpoint by its interval frequency, add the products, and divide by the total frequency. This provides a reliable estimate of the average.

What is the difference between frequency and relative frequency?

Frequency is the raw count of how often a specific value appears. Relative frequency is that count divided by the total number of observations, expressed as a fraction, decimal, or percentage. The sum of all relative frequencies always equals one.

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