Pulled by one value
One extreme value drags the mean along. The median hardly moves.
The mean is below nearly everyone
The mean height of Poovizhi’s family is 160.2 cm. But four of the five members are taller than that. The youngest child is 118 cm.
Which number describes the family’s height better: the mean 160.2 cm or the median 170 cm?
What this lesson covers
The idea
A very high or very low outlier greatly changes the sum and hence the mean, while the median is hardly affected, so the mean may not represent data with outliers well.
The mean is below nearly everyone
The mean height of Poovizhi’s family is 160.2 cm. But four of the five members are taller than that. The youngest child is 118 cm.
Which number describes the family’s height better: the mean 160.2 cm or the median 170 cm?
- The mean, 160.2 cm
- The median, 170 cm
- Both describe it equally well
Move the youngest child
Drag the red dot, the fifth member’s height, to very short, to like the others and to very tall. Watch the mean and the median.
The mean follows the outlier
When the red dot was far away, it changed the sum, and so it pulled the mean after it. The median only depends on the middle, so it stayed close to 170.
Without the outlier the mean rose by more than 10 cm, but the median rose by only 1.5 cm.
A very high or very low outlier greatly changes the sum and hence the mean, while the median is hardly affected. So the mean may not represent data with outliers well.
- Poovizhi’s family | Mean | Median
- With 118 cm | 160.2 | 170
- Without 118 cm | 170.75 | 171.5
Notes
A very high or very low outlier greatly changes the sum and hence the mean, while the median is hardly affected. So the mean may not represent data with outliers well.
Check yourself
Take away the outlier 118. Find the mean of the other four heights 170, 173, 165 and 175 cm.
Answer: 170.75 cm
170 + 173 + 165 + 175 = 683 and 683 ÷ 4 = 170.75 cm.
Find the median of the same four heights, 165, 170, 173, 175 cm.
Answer: 171.5 cm
The middle two are 170 and 173, and (170 + 173) ÷ 2 = 171.5 cm.
Removing 118 changed the mean from 160.2 to 170.75, and the median from 170 to 171.5. Which measure was affected less by the outlier?
Why is the mean 160.2 cm not a good “typical height” for Poovizhi’s family?
- The mean. The mean moved by more than 10 cm. The median moved much less.
- The median — correct. Yes! It moved only 1.5 cm, while the mean moved 10.55 cm.
- Both moved equally. The mean moved 10.55 cm and the median 1.5 cm.
- The mean is always a bad way to describe data.. The mean is useful, but it can be misleading when there are outliers.
- The mean is higher than the median.. The mean 160.2 is lower than the median 170, not higher.
- One very short member pulls the mean below the heights of 4 of the 5 members. — correct. Yes! A very low outlier lowers the sum and so the mean.