Mean, Median & Mode

Three Measures of Average

Averages are also called measures of central tendency — they summarise a data set with a single representative value. The three main averages are mean, median, and mode.

Key Terms

• Average — a single value representing a data set
• Outlier — an extreme value that is much larger or smaller than the rest
• Spread — how varied the data is (measured by range or IQR)
• Skewed data — data that clusters more to one side than the other

Mean (Arithmetic Average)

Mean = Sum of all values ÷ Number of values

Worked Example: Find the mean of 4, 7, 7, 9, 13:

• Sum = 4 + 7 + 7 + 9 + 13 = 40
• Count = 5 values
• Mean = 40 ÷ 5 = 8

Advantage: uses every value in the data set, so nothing is ignored. Disadvantage: heavily affected by outliers (extreme values). One very high or very low number can pull the mean far from where most values cluster.

Example: The data set 2, 4, 4, 6, 100 has a mean of 23.2 — which does not represent most of the data at all.

Median (Middle Value)

The median is the middle value when data is arranged in order. It divides the data set exactly in half.

• Odd number of values: the middle one is the median
• Even number of values: find the mean of the two middle values

Worked Example: Find the median of 3, 5, 5, 8, 9, 12 (6 values):

• Already in order. Middle two values are the 3rd and 4th: 5 and 8
• Median = (5 + 8) / 2 = 6.5

Best when: the data has outliers or is skewed, because the median is not pulled by extreme values.

Mode (Most Frequent Value)

The mode is the value that appears most often in a data set.

• A data set can have no mode (all values occur once)
• It can have one mode (unimodal)
• It can have two modes (bimodal) or more

Worked Example: For 3, 5, 5, 7, 9, 9, 9 → Mode = 9 (appears 3 times)

Best when: dealing with categorical data (e.g. most common shoe size, most popular colour).

Measures of Spread

Averages alone do not tell the full story — two data sets can have the same mean but very different spreads.

Range = Maximum − Minimum

For 12, 7, 19, 3, 15: Range = 19 - 3 = 16

The range is easy to calculate but sensitive to outliers.

Interquartile Range (IQR) = Q3 − Q1

The IQR covers the middle 50% of data:

• Q1 = lower quartile (median of the lower half)
• Q3 = upper quartile (median of the upper half)

A smaller IQR means data is tightly clustered. A larger IQR means data is more spread out. The IQR is less affected by outliers than the range.

Worked Example: Full Analysis

Data: 5, 8, 8, 10, 12, 14, 20

• Mean = (5+8+8+10+12+14+20) / 7 = 77 / 7 = 11
• Median = 4th value (middle of 7) = 10
• Mode = 8 (appears twice)
• Range = 20 - 5 = 15

Which Average to Use?

Situation	Best average
Symmetric data, no outliers	Mean
Skewed data or outliers present	Median
Categorical data	Mode
Comparing typical salaries	Median (outliers distort mean)
Most popular item	Mode

Common Mistakes

• Mistake: forgetting to sort the data before finding the median. Fix: always arrange values in ascending order first.
• Mistake: adding up then dividing by the wrong count. Fix: count the values carefully before dividing.
• Mistake: confusing range with an average — range is a measure of spread, not average.
• Mistake: stating there is no mode when values repeat. Fix: look for the most frequent value carefully.

Tips and Tricks

• A handy memory aid: Mean = add and divide, Median = middle, Mode = most often.
• If a data set has an even count, always average the two middle values for the median.
• The mean is the balance point of a data set — imagine the number line as a seesaw.
• Always check whether outliers are present before choosing an average — they can make the mean misleading.