Big picture
After collecting and compiling data, we need to summarise them. A long list of rainfall figures or crop yields is hard to understand, but a single representative value, such as the average rainfall, tells us a lot. Such values are called measures of central tendency: the mean, median and mode.
Geographers use these measures to compare places (average temperature of two cities), describe distributions (median income) and identify the most common category (modal crop). This chapter explains how to calculate each measure, step by step.
Learning goals
Explain measures of central tendency and their uses.
Calculate the mean for ungrouped and grouped data, by the direct and assumed mean methods.
Calculate the median for ungrouped and grouped data.
Calculate the mode for ungrouped and grouped data.
Compare mean, median and mode.
Calculate the range and standard deviation of a simple data set.
Core ideas
1. Measures of central tendency
A measure of central tendency is a single value that represents the centre of a data set, around which the other values cluster.
Measure | Meaning |
|---|---|
Mean | The sum of all values divided by the number of values (the arithmetic average) |
Median | The middle value when the data are arranged in order |
Mode | The value that occurs most often |
2. Mean for ungrouped data
Direct method: Mean (X̄) = ΣX ÷ N, where ΣX is the sum of values and N is the number of values.
Example. The monthly rainfall (mm) at seven stations is 34, 28, 45, 52, 30, 61 and 38.
ΣX = 34 + 28 + 45 + 52 + 30 + 61 + 38 = 288; N = 7.
Mean = 288 ÷ 7 = 41.14 mm (to two decimal places).
Assumed mean (indirect) method: used when values are large, to simplify calculation.
Choose an assumed mean (A), usually a value near the middle.
Find the deviation of each value from it: d = X − A.
Mean = A + (Σd ÷ N).
Example. For the same data, take A = 40. Deviations: −6, −12, 5, 12, −10, 21, −2. Σd = 8. Mean = 40 + 8 ÷ 7 = 40 + 1.14 = 41.14 mm, the same answer.
3. Mean for grouped data
For grouped data, we use the mid-point (m) of each class to represent it: m = (lower limit + upper limit) ÷ 2.
Direct method: Mean = Σfm ÷ N, where f is the frequency and N = Σf.
Example. Marks of 20 students:
Class | Frequency (f) | Mid-point (m) | fm |
|---|---|---|---|
30–40 | 2 | 35 | 70 |
40–50 | 4 | 45 | 180 |
50–60 | 6 | 55 | 330 |
60–70 | 5 | 65 | 325 |
70–80 | 3 | 75 | 225 |
Total | N = 20 | Σfm = 1,130 |
Mean = 1,130 ÷ 20 = 56.5 marks.
Assumed mean method: Mean = A + (Σfd ÷ N), where d = m − A.
Take A = 55. The deviations d are −20, −10, 0, 10, 20; fd are −40, −40, 0, 50, 60; Σfd = 30.
Mean = 55 + 30 ÷ 20 = 55 + 1.5 = 56.5 marks, matching the direct method.
4. Median
Ungrouped data:
Arrange the values in ascending (or descending) order.
Find the position of the median: (N + 1) ÷ 2.
If N is odd, the median is the value at that position. If N is even, it is the average of the two middle values.
Example. Rainfall values in order: 28, 30, 34, 38, 45, 52, 61. N = 7, so the median is at position (7 + 1) ÷ 2 = 4th: 38 mm.
Even example. For 12, 15, 18, 22, 25, 30 (N = 6), the middle values are the 3rd and 4th: (18 + 22) ÷ 2 = 20.
Grouped data:
Median = l + ((N/2 − cf) ÷ f) × i
where l = lower limit of the median class, N/2 = half the total frequency, cf = cumulative frequency of the class before the median class, f = frequency of the median class, and i = class width.
Example. For the marks table: N/2 = 10. Cumulative frequencies are 2, 6, 12, 17, 20. The 10th value falls in 50–60 (cumulative frequency reaches 12 there), so this is the median class.
l = 50, cf = 6, f = 6, i = 10.
Median = 50 + ((10 − 6) ÷ 6) × 10 = 50 + 6.67 = 56.67 marks.
5. Mode
Ungrouped data: the value that occurs most often.
Example. In 3, 5, 5, 7, 8, 5, 9, 7, the value 5 occurs three times, so the mode is 5. A data set may have no mode, or two modes (bimodal), or more (multimodal).
Grouped data:
Mode = l + ((f1 − f0) ÷ (2f1 − f0 − f2)) × i
where l = lower limit of the modal class (the class with the highest frequency), f1 = frequency of the modal class, f0 = frequency of the class before it, f2 = frequency of the class after it, and i = class width.
Example. In the marks table, the modal class is 50–60 (frequency 6). l = 50, f1 = 6, f0 = 4, f2 = 5, i = 10.
Mode = 50 + ((6 − 4) ÷ (12 − 4 − 5)) × 10 = 50 + (2 ÷ 3) × 10 = 50 + 6.67 = 56.67 marks.
6. Comparing mean, median and mode
Measure | Strengths | Limitations | Geographical use |
|---|---|---|---|
Mean | Uses all values; easy to calculate | Affected by extreme values | Average rainfall or temperature |
Median | Not affected by extreme values | Ignores the actual size of most values | Income distribution, where a few very rich people distort the mean |
Mode | Easy to identify; works for categories | May not exist or may be more than one | Most common crop, soil type or house type |
In a symmetrical distribution, the mean, median and mode are equal. In skewed distributions, they differ.
7. An introduction to dispersion
Two data sets can have the same mean but be spread differently. Measures of dispersion show how much values vary around the centre.
Range = largest value − smallest value. For the rainfall data, range = 61 − 28 = 33 mm. It is simple but depends only on two extreme values.
Standard deviation (σ) = square root of the mean of squared deviations from the mean: σ = √(Σ(X − X̄)² ÷ N).
Example. For 2, 4, 4, 4, 5, 5, 7, 9: the mean is 40 ÷ 8 = 5. Squared deviations: 9, 1, 1, 1, 0, 0, 4, 16; their sum is 32. σ = √(32 ÷ 8) = √4 = 2.
A smaller standard deviation means values are clustered closely around the mean, for example, a place with reliable rainfall.
A reliable approach to questions
Identify whether the data are ungrouped or grouped.
For the grouped mean, find mid-points first; use the assumed mean method for large numbers.
For the grouped median, find N/2 and the median class from cumulative frequency, then apply the formula.
For the grouped mode, find the modal class and its neighbours' frequencies, then apply the formula.
Check that the direct and assumed mean methods give the same answer.
Connections that matter
Data sources and compilation explains how frequency distributions are built.
Graphical representation shows these distributions on graphs.
Mathematics: statistics uses the same formulas.
Common mistakes
Using class limits instead of mid-points for the grouped mean.
Forgetting to arrange data in order before finding the median.
Taking cf of the median class itself in the median formula. Use the cf of the class before it.
Choosing the modal class as the one with the largest class limit. It is the class with the highest frequency.
Revision checklist
I can explain mean, median and mode and their uses.
I can calculate the mean by direct and assumed mean methods.
I can calculate the median for ungrouped and grouped data.
I can calculate the mode for ungrouped and grouped data.
I can calculate the range and standard deviation.