Interpreting results
Using statistics
Mode, median and mean are three different answers to 'what is typical?'. With a football team aged 16 to 41, choosing the wrong one gives a misleading picture.
The four measures
• The MODE is the most common value. Two modes makes a set BIMODAL. • The MEDIAN is the middle value once the data is in order of increasing size. • The MEAN is the sum of all the values divided by how many there are. • The RANGE is the largest value minus the smallest. The first three are averages — a typical value. The range is not an average: it measures SPREAD.
Working them out
The ages of eleven football players: 16, 17, 18, 18, 19, 20, 20, 21, 21, 32, 41 Mode: 18, 20 and 21 each appear twice, so all three are modes. Median: 11 players, so the 6th value in order → 20. Mean: the ages total 243, and 243 ÷ 11 = 22.1. Range: 41 − 16 = 25.
Which average to use
• MODE — when you want the most commonly occurring value. Useful for shoe sizes a shop should stock. • MEDIAN — the middle value; half the data is above it and half below. It ignores extreme values. • MEAN — depends on EVERY value, so changing one number changes it. For the football team the median of 20 is best: five players are younger and four older, so it sits nicely in the middle. The mean of 22.1 is dragged up by the two players aged 32 and 41 — older than all but one of the squad.
Range compares spread
The range tells you how spread out a set is, not what is typical. This team's range is 25 years. Another team has a range of 14 years. The first team has the greater variation in ages — a wider spread — even though nothing has been said about which team is older on average. Bigger range, more variation.
Worked examples
The mode of 16, 17, 18, 18, 19, 20, 20, 21, 21, 32, 41
- Look for the values appearing most often.
- 18, 20 and 21 each appear twice.
Answer: 18, 20 and 21 — the data is multimodal
The median of the same ages
- There are 11 values, already in order.
- The middle one is the 6th.
Answer: 20 years
The mean of the same ages
- The ages total 243.
- 243 ÷ 11 = 22.09…
- Rounded to 1 d.p.
Answer: 22.1 years
The range of the same ages
- Largest minus smallest.
- 41 − 16 = 25.
Answer: 25 years
Which average best represents this team?
- The mean is pulled up by the 32- and 41-year-olds.
- The median has 5 below and 4 above it.
Answer: The median, 20 years
Range 25 against range 14 — which varies more?
- A bigger range means a wider spread.
Answer: The team with a range of 25
Where you see this in real life
- House prices are quoted as a median, because a handful of very expensive homes would drag the mean up.
- A shop stocks the modal shoe size most heavily, since that is the size most customers ask for.
Key words
- Mode
- the most common value
- Median
- the middle value when the data is in order
- Mean
- the total divided by how many values there are
- Range
- the largest value minus the smallest
- Bimodal
- having two modes