Discover Mathematics P6: Term 1, Unit 7, Lessons 3–4
GoalSpot an outlier, see how it changes the mean, median and mode, and decide when to leave it out.
New words
outlier · a value that is far away from the rest of the dataresistant · not changed much by an outlier (the median and the mode are)recording error · a value written wrongly, for example 20 instead of 2
Explanation
An outlier is a value that lies far away from the rest of the data. Outliers happen for different reasons: a mistake when writing the number, a special day, or one case that really is unusual. An outlier pulls the mean strongly towards itself, because the mean uses every value. The median and the mode change little or not at all. So when there is an outlier, the median usually describes a typical value better. We may leave an outlier out when it is a mistake (or does not belong to the group we are studying), but then we say so. If it is a real value, we keep it and report the median together with the mean.
★Remember
Outlier = far from the rest. It pulls the mean a lot, the median a little, the mode usually not at all. Leave it out only if it is a mistake or does not belong.
Worked examples
Without 45
With 45
Mean
10
15
Median
10
10
Mode
10
10
Six pupils waited for the bus for 8, 9, 10, 10, 11 and 12 minutes. The mean, median and mode are all 10. One day the bus broke down and a pupil waited 45 minutes. Now the mean is (60 + 45) ÷ 7 = 15, but the median is still 10 and the mode is still 10. The outlier 45 changed only the mean.
Which measure is fair? Seven pupils counted the books at home: 5, 8, 8, 10, 12, 12, 85. The total is 140 and the mean is 20. But six of the seven pupils have 12 books or fewer, so 20 is not typical. The median, 10, describes the group better. The pupil with 85 books is real, so we keep it, but we report the median.
A mistake. Hazem recorded the goals of five players as 2, 3, 1, 2, 20, but the 20 should have been 2. With the mistake the mean is 28 ÷ 5 = 5.6, which is far from the truth. Corrected, the data are 2, 3, 1, 2, 2 and the mean is 10 ÷ 5 = 2. Here the outlier is an error, so we correct it.
Class activity
Spot the outlier. A grown-up writes seven numbers about family life, and one of them is far from the others (for example: the number of guests on seven days, and one day with a big party). Find the outlier, find the mean with it and without it, and say which number is more typical.
Practice
1
Which value is the outlier in 12, 14, 15, 13, 16, 60?
60, because the others are all between 12 and 16.
2
The mean of 4, 5, 5, 6 is 5. Add the outlier 25. What are the new mean and median?
The new mean is (20 + 25) ÷ 5 = 9. In order 4, 5, 5, 6, 25 the median is 5.
3
When an outlier is added, which of the mean, the median and the mode usually changes the most?
The mean.
4
When may we leave an outlier out of the data, and what must we do then?
When it is a mistake or does not belong to the group. We must say that we left it out.
5
The data 6, 7, 7, 8, 10, 10 have the mean 8 and the median 7.5. Add 36. Find the new mean and median.
The total was 48, now 84, so the mean is 84 ÷ 7 = 12. In order 6, 7, 7, 8, 10, 10, 36 the median is 8.
