Mean, Median, and Mode

Measures of central tendency describe the "centre" of a data set. Each measure captures a different aspect of what is "typical."

Mean (x̄)

The arithmetic average. Add all values and divide by the number of values.

x̄ = Σ x/n

Median

The middle value when data is sorted. For even n, average the two middle values.

Position = n + 1/2

Mode

The most frequently occurring value. A data set can have no mode, one mode, or multiple modes.

Mean from Grouped Data

When data is presented in a frequency table with class intervals, we use the class centre (midpoint) to estimate the mean.

Estimated Mean for Grouped Data

x̄ = Σ (f × x_mid)Σ f

where f = frequency and x_mid = midpoint of each class interval.

Class Interval	Midpoint (x_mid)	Frequency (f)	f × x_mid
10 - 19	14.5	3	43.5
20 - 29	24.5	7	171.5
30 - 39	34.5	5	172.5
Total		15	387.5

Estimated mean = 387.5/15 = 25.83

Effect of Outliers

An outlier is a data value that is significantly different from the rest. Outliers have different effects on each measure:

Mean Greatly affected by outliers. A single extreme value can pull the mean significantly.

Median Resistant to outliers. It depends only on position, not extreme values.

Mode Not affected by outliers (unless the outlier itself is the most frequent).

When outliers are present, the median is generally the most reliable measure of centre.

Key Vocabulary

Mean

The arithmetic average of all values. Sum of values divided by count.

Median

The middle value in a sorted data set. Robust against outliers.

Mode

The most frequent value. A data set may be unimodal, bimodal, or have no mode.

Outlier

A data value that lies far from the rest of the data, often defined as beyond 1.5 × IQR from a quartile.

Worked Examples

1

Find the mean, median, and mode of: 3, 5, 7, 7, 9, 11.

Mean: (3+5+7+7+9+11)/6 = 42/6 = 7

Median: 6 values, so median = (3rd + 4th)/2 = (7+7)/2 = 7

Mode: 7 appears twice (most frequent). Mode = 7

2

Find the mean, median, and mode of: 3, 5, 7, 7, 9, 100.

Mean: (3+5+7+7+9+100)/6 = 131/6 ≈ 21.83

Median: (7+7)/2 = 7 (unchanged from Example 1)

Mode: 7 (unchanged)

Observation: The outlier (100) pulled the mean from 7 to 21.83, but the median and mode stayed at 7.

3

A frequency table shows: score 1 (freq 4), score 2 (freq 8), score 3 (freq 6), score 4 (freq 2). Find the mean.

Step 1: Σ(f × x) = 1(4) + 2(8) + 3(6) + 4(2) = 4 + 16 + 18 + 8 = 46

Step 2: Σf = 4 + 8 + 6 + 2 = 20

Answer: Mean = 46/20 = 2.3

Knowledge Check

Select the correct answer for each question. Click "Check Answer" to see if you are right.

Question 1

Find the mean of: 4, 8, 6, 10, 12.

6 7 10 8

Question 2

Find the median of: 15, 3, 9, 7, 12, 5, 8.

8 9 7 12

Question 3

Which measure is most affected by outliers?

Median Mean Mode Range

Question 4

Data: 2, 3, 3, 5, 5, 5, 8, 9. What is the mode?

3 3 and 5 5 5.5

Question 5

The mean of 5 numbers is 12. If a 6th number (30) is added, what is the new mean?

12 14 21 15

Measures of Central Tendency

Mean, Median, and Mode

Mean (x̄)

Median

Mode

Mean from Grouped Data

Effect of Outliers

Key Vocabulary

Mean

Median

Mode

Outlier

Worked Examples

Find the mean, median, and mode of: 3, 5, 7, 7, 9, 11.

Find the mean, median, and mode of: 3, 5, 7, 7, 9, 100.

A frequency table shows: score 1 (freq 4), score 2 (freq 8), score 3 (freq 6), score 4 (freq 2). Find the mean.

Knowledge Check

Key Concepts Summary