Choose the statistic and use the correct data order.
Core & Extended · Syllabus C9.3, E9.3
Notes and key methods
The mean is the sum of observations divided by their count. The median is the middle ordered value, or the mean of the two middle values for an even count. The mode is a most frequent value. Range is largest minus smallest.
Sort data before finding a median or range. An extreme value can strongly change the mean and range while leaving the median unchanged. A missing-value mean problem can be solved by recovering the required total first.
Check the conditions
Use the stated units and precision.
Show the method as well as the final answer.
Three worked examples
Example 1 · Mean, median, mode and range
The five observations are 3, 7, 7, 15, 23.
Find the mean.
Find the median.
Find the mode.
Find the range.
Show the reasoning
(a)
Divide the total of all observations by the number of observations.
The sum is 55; dividing by 5 gives \(11\).
Answer: \(11\).
(b)
Locate the middle ordered observation.
The third of the five ordered observations is 7.
Answer: \(7\).
(c)
Find the observation with greatest frequency.
Only 7 is repeated, so it is the mode.
Answer: \(7\).
(d)
Subtract the minimum observation from the maximum.
Largest minus smallest is \(\displaystyle 23-3=20\).
Answer: \(20\).
Example 2 · Mean, median, mode and range
The five observations are 4, 5, 5, 7, 9.
Find the mean.
Find the median.
Find the mode.
Find the range.
Show the reasoning
(a)
Divide the total of all observations by the number of observations.
The sum is 30; dividing by 5 gives \(6\).
Answer: \(6\).
(b)
Locate the middle ordered observation.
The third of the five ordered observations is 5.
Answer: \(5\).
(c)
Find the observation with greatest frequency.
Only 5 is repeated, so it is the mode.
Answer: \(5\).
(d)
Subtract the minimum observation from the maximum.
Largest minus smallest is \(\displaystyle 9-4=5\).
Answer: \(5\).
Example 3 · Mean, median, mode and range
The five observations are 5, 7, 7, 11, 15.
Find the mean.
Find the median.
Find the mode.
Find the range.
Show the reasoning
(a)
Divide the total of all observations by the number of observations.
The sum is 45; dividing by 5 gives \(9\).
Answer: \(9\).
(b)
Locate the middle ordered observation.
The third of the five ordered observations is 7.
Answer: \(7\).
(c)
Find the observation with greatest frequency.
Only 7 is repeated, so it is the mode.
Answer: \(7\).
(d)
Subtract the minimum observation from the maximum.
Largest minus smallest is \(\displaystyle 15-5=10\).
Answer: \(10\).
Common mistakes
Keep intermediate values exact until the final step.