Math · Introductory statistics · Worked example
How an outlier moves the mean, not the median
In the sleep data 4, 8, 6, 5, 3, 7, 9 and 6, suppose the 9 were mistyped as 30. How do the mean, the median and the standard deviation change?
Recompute the mean
The total rises by 21, from 48 to 69.
The median stays put
Sorted: 3, 4, 5, 6, 6, 7, 8, 30. The two middle values are still 6 and 6, so the median is 6.
The spread grows
The squared deviation of 30 from the new mean dominates the sum. The sample standard deviation rises to about 8.78, more than four times as large.
Check the outlier screen
The quartiles do not change: Q1 = 4.5 and Q3 = 7.5, so IQR = 3 and the upper fence is 7.5 + 1.5 × 3 = 12. The value 30 is far beyond it.
Result
The mean rises to 8.625 and s to about 8.78, but the median stays 6.
Your turn
The values 2, 3, 3, 4 and 18 have mean 6. What is their median, and which better describes a typical value?
Show the answer and explanation
The median is 3, and it describes a typical value better.
Four of the five values are 4 or less. The single 18 pulls the mean up to 6, above all four of them.
Keep exploring
In Statistics, change the 30 back to 9: the Summary returns to mean 6 and standard deviation 2, while the median and quartiles never moved.
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