Mean, Median and Mode

Grade 3

Three averages, three different ideas

An average is a single number that summarises a data set. There are three types and they don't always give the same answer — that's by design, because they measure different things.

Mean — share everything out equally. Median — the middle value when sorted. Mode — the most frequent value.

Calculating each one

EXAMPLE

Data set: 3, 7, 7, 8, 10. Find the mean, median and mode.

Mean: Add all values: 3 + 7 + 7 + 8 + 10 = 35. Divide by 5: 35 ÷ 5 = 7. Median: Already sorted. Middle value (3rd of 5) = 7. Mode: 7 appears twice — more than any other. Mode = 7. Mean = 7, Median = 7, Mode = 7

Finding the median with an even number of values

EXAMPLE

Data set: 4, 6, 9, 13. Find the median.

Four values — no single middle one. Take the two middle values (6 and 9) and find their mean: (6 + 9) ÷ 2 = 15 ÷ 2 = 7.5.

For n values sorted in order, the median position is (n + 1) ÷ 2. If n = 6, that's 3.5 — meaning average the 3rd and 4th values.

Which average to use?

The mean uses every value, so one extreme result (an outlier) pulls it up or down. The median ignores extremes — it's much better for skewed data like house prices or salaries. The mode is the only one that works for non-numerical data (e.g. favourite colour).

Exam questions often ask you to "comment on" the averages. Say which one is higher/lower and why — for example, "the mean is higher than the median because a few very large values pull it up."

Try this one

Six numbers have a mean of 9 and a range of 8.

The smallest number is 5.

(a) What is the largest number? [1]

(b) What is the total of all six numbers? [1]

Nova's hint:

Range and mean are two different statistical measures you can use to find missing information.

Ask yourself: 'How is the range calculated from the largest and smallest values, and how does the mean relate to the total?'

Remember: range = largest − smallest, and total = mean × number of values.

Try to Solve it with Nova?