Basic Statistics — Mean, Median, Mode

Statistics helps us summarise data. This lesson covers the three measures of central tendency — mean, median, and mode — and how to choose between them.

Why Statistics?

When we have a large set of numbers, we want a single value that represents the whole set. We use measures of central tendency.

The Mean (Average)

Mean

\bar{x} = \frac{\text{sum of all values}}{\text{number of values}}

Add all values, then divide by how many there are.

Data: 3, 7, 8, 5, 7

$\bar{x} = \frac{3+7+8+5+7}{5} = \frac{30}{5} = 6$

The Median

The median is the middle value when data is ordered from smallest to largest.

Odd number of values: middle value
Even number of values: mean of the two middle values

Data: 2, 4, 7, 11, 15 → Median = 7

Data: 3, 5, 8, 12 → Median = $\frac{5+8}{2} = 6.5$

The Mode

The mode is the value that appears most often.

Data: 4, 7, 4, 2, 7, 4 → Mode = 4 (appears 3 times)

Note A data set can have no mode, one mode, or multiple modes (bimodal/multimodal).

Choosing the Right Measure

Situation	Best measure
Symmetric, no outliers	Mean
Outliers present	Median
Most popular item	Mode

Test scores: 55, 62, 78, 80, 81, 83, 95. Find mean, median, mode.

Mean = $(55+62+78+80+81+83+95)/7 = 534/7 \approx 76.3$

Median = 80 (4th value of 7)

Mode = none (all values appear once)

Real world: sports league table

A football team's goals scored over 5 games: 2, 0, 3, 5, 1. Mean $= \frac{11}{5} = 2.2$ goals per game. Ordered: 0, 1, 2, 3, 5 → Median = 2. The median is more representative here because the outlier (5) inflates the mean.

Real world: weather data

Daily high temperatures (°C) for a week: 14, 16, 15, 13, 18, 20, 16. Mean $= \frac{112}{7} = 16°\text{C}$ . Mode = 16°C (appears twice). Meteorologists report the mean as the "average temperature" for the week.

Real world: most popular shoe size

A shoe shop's sales for one week: sizes 6, 7, 8, 8, 9, 8, 10, 7, 8. Mode = 8 (sold 4 times). The mode tells the manager which size to order most of — the mean (7.9) is not actually a shoe size at all.

Tip If data contains outliers (very high or very low values), the median is usually a fairer summary than the mean.

Practice Problems

Find the mean of: 14, 18, 22, 10, 16.

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(14+18+22+10+16)/5 = 80/5 = 16.

Find the median of: 3, 9, 1, 7, 5, 2.

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Ordered: 1,2,3,5,7,9. Median = (3+5)/2 = 4.

What is the mode of: 5, 3, 7, 5, 9, 3, 5?

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5 (appears 3 times).