Understand · explore · practise
Calculate and interpret three ways to summarise the centre of data. Move an extreme value, compare the measures and practise with original worked questions.
Before you startAdding values, dividing and ordering numbers.
01 / Three meanings of average
Mean: share the total equally. Median: find the middle rank. Mode: find the most frequent value.
These measures answer different questions; they need not agree.
A single centre condenses information. Always retain the variable, units and context: an average journey time and an average number of journeys are different quantities.
Ordered values: 2, 4, 4, 5, 6.
Mean = 21 ÷ 5 = 4.2. The third value is 4, and 4 occurs twice.
The green line marks the mean; blue dots are observations. Only the largest observation changes here.
02 / Mean as equal sharing
Total = 3 + 5 + 7 + 9 = 24
Count all four days, including a zero day if one is observed.
Mean = 24/4 = 6 enquiries per day
Equal sharing replaces the four unequal counts by four sixes without changing the total.
For n observations, x̄ = Σx/n. The sigma symbol means add all the listed values. A mean can be non-integer even when the variable is a count.
Find the mean of 0, 2, 3 and 4 deliveries. Explain why a non-integer answer is allowed.
The sum is nine and there are four observations.
Mean = 9/4 = 2.25 deliveries per period. Individual counts are integers, but the equal-share summary need not be an observed possible count.
Six journeys have a mean duration of 18 minutes. What is their total duration?
Reverse division by six.
Total = 6×18 = 108 minutes.
03 / Order before finding the middle
For an odd number of numerical values, use the single middle observation. For an even number, average the two middle values. The position (n + 1)/2 is a useful way to locate the middle; a half-integer position means halfway between adjacent observations.
Find the median of 12, 3, 8, 5 and 6.
Sort all five first.
Ordered: 3, 5, 6, 8, 12. The third value is 6.
Find the median of 2, 4, 9, 11, 15 and 18.
The middle observations are third and fourth.
Median = (9 + 11)/2 = 10. It need not be one of the recorded values.
A learner takes the average of the smallest and largest observations to find the median. Why is this wrong?
That calculation uses endpoints rather than middle ranks.
It gives the midrange, a different measure. For 1, 2, 3, 4, 20, the median is 3 but the endpoint average is 10.5.
04 / Look for the highest frequency
The mode is the most frequent observed value or category. Two equally most frequent values give a bimodal set. If every value occurs once, the usual school convention is to report no mode.
For unordered categories, a mode can be meaningful when a mean or numerical median is not. Numerical category codes do not make their average meaningful.
Find the modes of 2, 2, 3, 5, 5, 7.
Compare frequencies.
2 and 5 each occur twice, more than the other values. The data are bimodal.
Travel methods are walk, bus, walk, train, bus, walk. Give an appropriate central summary.
The observations are categories, not quantities.
The modal travel method is walk, occurring three times. Averaging arbitrary codes for walk/bus/train would not give a meaningful travel method.
05 / Change an extreme value
Pause, replay or seek freely. The notes explain the same idea and stay in view.
In the model, the fifth value grows but stays above the other four. The middle rank remains 4 and the most frequent value remains 4. The mean increases because the total increases.
This illustrates resistance to an extreme value. It does not mean the median can never change: changing enough observations or moving a value across middle ranks can alter it.
For 2, 4, 4, 5 and 30, find the mean, median and mode.
Total 45; the list is already ordered.
Mean = 45/5 = 9; median = 4; mode = 4.
Only the largest observation in a five-value dataset rises by 10. How much does the mean rise?
The total rises by 10 while the count remains five.
It rises by 10/5 = 2. This statement does not require knowing the other values.
06 / Justify the choice in context
The mean uses every magnitude and is useful when totals and equal sharing matter. The median often better describes a typical value in a strongly skewed dataset. The mode identifies the most common category or available size. Report spread too when differences between observations matter.
Most waiting times are around 5 minutes but one equipment failure causes a 90-minute wait. Which centre might best describe a typical customer’s experience, and what would be lost by reporting it alone?
Distinguish typical experience from the total burden of waiting.
The median may describe the typical wait better because the exceptionally long wait has less influence on its rank. Alone it hides the severe delay; report suitable spread or context too. The mean remains useful for total waiting time per customer.
Why might a shop examine the mode of recorded shoe sizes rather than order only the mean size?
The mean can lie between sizes and does not identify demand peaks.
The mode identifies the most common recorded size. It helps describe demand, though the full frequency distribution is needed for stock quantities across all sizes.
07 / Keep count, order and context
For 1, 3, 3, 5, 8, give all three centres and explain their difference.
The total is 20.
Mean 4, median 3, mode 3. The larger observations raise the total and hence the mean, while 3 is both the middle-ranked and most frequent value.
Section 1 of 7 · Three meanings of average