Averages and measures of spread.
Calculate and interpret mean, median, mode and range, then estimate means from grouped data.
Choose and calculate averages and range · use frequencies and midpoints · identify the modal class
Choose a measure that answers the question
If several students report different journey times, there is no single time shared by everyone. An average gives one summary of a typical value, and different averages summarise the list in different ways.
The mean shares the total equally, the median finds the middle position, and the mode identifies the most common value. Range instead measures spread between extremes. Understand what each summary represents before choosing which one best answers the question.
An average describes a typical value. A measure of spread describes how variable the data are. They answer different questions.
| Measure | Calculation | Useful when… |
|---|---|---|
| Mean | Sum of values ÷ number of values | You want to use every numerical observation. |
| Median | Middle value after ordering | Extreme values could distort the mean. |
| Mode | Most frequent value or category | You want the most common outcome. |
| Range | Largest value − smallest value | You want a quick measure of spread. |
Ordered data: 2, 3, 3, 4, 8
The mean and range change with the extreme value. Here the middle value and most frequent value stay at 3.
Pause and explainWhy is range not another average?
It describes how far apart the largest and smallest values are, rather than a typical central value.
Order the data before finding the median
To find a middle value, first put the observations in order. Without ordering, the item sitting in the middle of the written list has no special statistical meaning.
For an odd number of observations, one middle value leaves equal counts on either side. For an even number, two values straddle the centre, so average those two. Keep positions separate from values: the third position might contain a value of 20, not 3.
Find the mean, median, mode and range of 2, 3, 3, 4, 8.
Median = 3; mode = 3
Range = 8 − 2 = 6
Find the median of 9, 2, 7, 4, 4, 10.
Order first: 2, 4, 4, 7, 9, 10. There are six observations, so average the third and fourth values.
Pause and explainIn six ordered observations, which positions determine the median?
The third and fourth. Their average gives a value halfway between the two central observations.
Weight each value by its frequency
A frequency table is a shortened data list. A value of 2 with frequency 7 means the original list contains seven separate copies of 2.
Multiplying value by frequency recovers that row’s contribution to the total. Adding frequencies recovers the number of observations. For the median, use cumulative counts to locate the middle observation positions rather than averaging the row labels without weights.
| Goals x | Frequency f | fx | Cumulative frequency |
|---|---|---|---|
| 0 | 2 | 0 | 2 |
| 1 | 5 | 5 | 7 |
| 2 | 7 | 14 | 14 |
| 3 | 4 | 12 | 18 |
| 4 | 2 | 8 | 20 |
| Total | 20 | 39 |
Find the mean and median number of goals.
The 10th and 11th observations both fall in the x = 2 row, so the median is 2. The mode is 2 (frequency 7), and the range is 4 − 0 = 4.
Pause and explainWhy can the mean not usually be found by averaging the distinct values alone?
Values occurring many times must contribute more weight. Ignoring frequency treats unequal counts as equal.
Estimate the mean of grouped data
A grouped table tells us how many observations lie in an interval, but does not reveal their individual values. An exact total cannot usually be reconstructed from that information alone.
Use each class midpoint as a representative value and multiply it by the class frequency. This assumes a reasonable balance of observations within each class, making the mean an estimate. Keep that qualification in the answer rather than presenting the midpoint total as the original exact data.
The individual values inside a class are unknown. Represent them by its midpoint, then weight by frequency.
Estimated mean = Σ(f × midpoint)Σf
| Class | f | Midpoint m | fm |
|---|---|---|---|
| 0 ≤ t < 10 | 4 | 5 | 20 |
| 10 ≤ t < 20 | 8 | 15 | 120 |
| 20 ≤ t < 30 | 6 | 25 | 150 |
| 30 ≤ t < 40 | 2 | 35 | 70 |
| Total | 20 | 360 |
Estimate the mean journey time and identify the modal class.
The modal class is 10 ≤ t < 20, with the highest frequency, 8. We cannot identify the exact mode or exact mean from these grouped totals.
Grouped discrete data: 0–2 books (f = 4), 3–5 (f = 6), 6–8 (f = 2). Estimate the mean.
The midpoints are 1, 4 and 7.
Pause and explainWhy is a grouped mean an estimate even when the arithmetic is exact?
The midpoint values stand in for unknown individual observations, so the reconstructed total is approximate.
Your turn. Give exact values or label estimates.
Choose a method for the data format: individual list, discrete frequency table or grouped intervals. State whether the answer is exact or estimated.
Order data before finding a median, weight values by frequencies for a mean, and give the whole modal interval when grouped data only identify a modal class. Check the final units for mean and range.
For 1, 4, 4, 6, 10, find the mean, median, mode and range.
Reveal worked answer
Find the median of 8, 1, 5, 3.
Reveal worked answer
Values 1, 2, 3 have frequencies 2, 5, 3. Find the mean.
Reveal worked answer
Classes 0 ≤ x < 10 and 10 ≤ x < 20 have frequencies 3 and 7. Estimate the mean.
Reveal worked answer
In the journey table, state the modal class.
Reveal worked answer
A list of salaries contains one exceptionally high salary. Which average may better represent a typical worker, and why?