EQN-05-06

Frequency Tables & Grouped Data

Read counts from a frequency table and compute the mean by weighting each value by how often it occurs.

~5 min read•Difficulty 2/5

Each row pairs a value with its frequency (how many times it occurs). The total count is the sum of the frequencies, not the number of rows.

Multiply each value by its frequency, add those products, then divide by the total frequency.

Value : 123

Frequency : 433 (total 10)

mean = (1 \cdot 4 + 2 \cdot 3 + 3 \cdot 3) /10 = (4 + 6 + 9) /10 = 19/10 = 1.9

Find the position of the middle item using the running (cumulative) total, then read which value covers that position.

The mode is the value with the highest frequency. The range is largest value $-$ smallest value (not a frequency).

Diagram

Mean $= Σ ($ value $\times$ frequency $) \div$ total frequency.

Worked examples

Example 1 (difficulty 2/5)

Scores: 5 appears 2 times, 8 appears 3 times, 10 appears 5 times. What is the mean?

◎ Predict the answer first:

Example 2 (difficulty 1/5)

In the same table

(5 \times 2, 8 \times 3, 10 \times 5),

what is the mode?

◎ Predict the answer first:

Averaging the distinct values and ignoring the frequencies

Why: The values are the visible numbers.

Fix: Weight each value by how often it occurs before averaging.

Reporting the highest frequency as the mode

Why: The biggest count catches the eye.

Fix: The mode is the VALUE that has the highest frequency, not the frequency itself.