Introduction
Every time you look at a set of numbers (exam scores, temperatures, prices, ages) you naturally want to make sense of them. What is typical? What is the middle? What value appears most often? How spread out are the numbers? These are exactly the questions that mean, median, mode and range are designed to answer.
Together, these four measures form the foundation of statistics. They are called measures of central tendency (mean, median, mode) and measures of spread (range), and they appear in virtually every area of mathematics, science, business, and everyday life.
Key Definitions
Before diving in, here is a quick reference:
- Mean — the average. Add all values and divide by how many there are.
- Median — the middle value when data is arranged in order.
- Mode — the value that appears most often.
- Range — the difference between the highest and lowest values.
What Is Ungrouped Data?
Ungrouped data is a raw list of individual values that has not been organised into groups or frequency tables. It is the simplest form of data to work with.
Example of ungrouped data:
- Exam scores: 45, 62, 78, 55, 62, 90, 71, 62, 48, 85
All four measures — mean, median, mode and range — can be calculated directly from this list.
The Mean
What Is the Mean?
The mean is the arithmetic average of a set of values. It is the most commonly used measure of central tendency and gives a single value that represents the entire dataset.
Formula
Mean = Sum of all values ÷ Number of values
Or written formally:
x̄ = Σx / n
Where:
- x̄ = the mean
- Σx = the sum of all values
- n = the number of values
Worked Examples
Example 1 — Basic mean:
Find the mean of: 4, 7, 9, 12, 8
Step 1 — Add all values: 4 + 7 + 9 + 12 + 8 = 40
Step 2 — Divide by the number of values: 40 ÷ 5 = 8
Mean = 8
Example 2 — Mean with larger numbers:
Find the mean of: 55, 62, 78, 45, 90, 71, 48, 85, 62, 62
Step 1 — Add all values: 55 + 62 + 78 + 45 + 90 + 71 + 48 + 85 + 62 + 62 = 658
Step 2 — Divide by the number of values: 658 ÷ 10 = 65.8
Mean = 65.8
Example 3 — Finding a missing value using the mean:
The mean of five numbers is 14. Four of the numbers are 10, 18, 12, and 16. Find the fifth number.
Step 1 — Find the total sum: Mean × n = 14 × 5 = 70
Step 2 — Add the known values: 10 + 18 + 12 + 16 = 56
Step 3 — Subtract from the total: 70 − 56 = 14
The fifth number is 14
Example 4 — Mean with decimals:
Find the mean of: 3.5, 4.2, 6.8, 5.1, 2.4
Step 1 — Add all values: 3.5 + 4.2 + 6.8 + 5.1 + 2.4 = 22.0
Step 2 — Divide by 5: 22.0 ÷ 5 = 4.4
Mean = 4.4
When Is the Mean Not the Best Measure?
The mean is affected by outliers — extreme values that are much higher or lower than the rest of the data. When outliers are present, the mean can be misleading.
Example:
Salaries: £22,000 · £24,000 · £25,000 · £23,000 · £150,000
Mean = (22,000 + 24,000 + 25,000 + 23,000 + 150,000) ÷ 5
= 244,000 ÷ 5
= £48,800
The mean of £48,800 does not represent the typical salary — four out of five people earn less than £26,000. The £150,000 outlier has pulled the mean up dramatically. In cases like this, the median is a more representative measure.
The Median
What Is the Median?
The median is the middle value in a dataset when the values are arranged in ascending (or descending) order. It is not affected by outliers, which makes it a more reliable measure of central tendency when extreme values are present.
How to Find the Median
Step 1 — Arrange the data in ascending order.
Step 2 — Find the middle value.
- If there is an odd number of values, the median is the middle value.
- If there is an even number of values, the median is the mean of the two middle values.
Formula for the position of the median:
- Odd number of values: median is at position (n + 1) ÷ 2
- Even number of values: median is the mean of positions n ÷ 2 and (n ÷ 2) + 1
Worked Examples
Example 1 — Odd number of values:
Find the median of: 7, 3, 9, 1, 5
Step 1 — Arrange in order:
1, 3, 5, 7, 9
Step 2 — There are 5 values. Median position = (5 + 1) ÷ 2 = 3rd value
Median = 5
Example 2 — Even number of values:
Find the median of: 12, 7, 4, 19, 15, 8
Step 1 — Arrange in order:
4, 7, 8, 12, 15, 19
Step 2 — There are 6 values. The two middle values are the 3rd and 4th: 8 and 12
Step 3 — Find the mean of the two middle values: (8 + 12) ÷ 2 = 20 ÷ 2 = 10
Median = 10
Example 3 — Larger dataset:
Find the median of: 45, 62, 78, 55, 90, 71, 48, 85, 62, 62
Step 1 — Arrange in order:
45, 48, 55, 62, 62, 62, 71, 78, 85, 90
Step 2 — There are 10 values. The two middle values are the 5th and 6th: 62 and 62
Step 3: (62 + 62) ÷ 2 = 62
Median = 62
Example 4 — With negative numbers:
Find the median of: −5, 3, −1, 7, −3, 9, 2
Step 1 — Arrange in order:
−5, −3, −1, 2, 3, 7, 9
Step 2 — There are 7 values. Median position = (7 + 1) ÷ 2 = 4th value
Median = 2
The Mode
What Is the Mode?
The mode is the value that appears most frequently in a dataset. Unlike the mean and median, the mode can be used with non-numerical data such as colours, names, or categories.
Key Facts About the Mode
- A dataset can have one mode (unimodal)
- A dataset can have two modes (bimodal)
- A dataset can have more than two modes (multimodal)
- A dataset can have no mode if all values appear the same number of times
Worked Examples
Example 1 — One mode:
Find the mode of: 4, 7, 9, 7, 3, 7, 5, 2
Count the frequency of each value:
- 2 → once
- 3 → once
- 4 → once
- 5 → once
- 7 → three times
- 9 → once
Mode = 7
Example 2 — Two modes (bimodal):
Find the mode of: 3, 5, 7, 5, 9, 3, 8, 1
- 1 → once
- 3 → twice
- 5 → twice
- 7 → once
- 8 → once
- 9 → once
Mode = 3 and 5
Example 3 — No mode:
Find the mode of: 2, 4, 6, 8, 10
Every value appears exactly once — no value appears more than any other.
No mode
Example 4 — Mode with non-numerical data:
A class was asked to name their favourite subject. The responses were:
Maths, English, Biology, Maths, Chemistry, English, Maths, Physics, Maths
Count:
- Maths → 4 times
- English → 2 times
- Biology → once
- Chemistry → once
- Physics → once
Mode = Maths
The Range
What Is the Range?
The range measures how spread out a dataset is. It is the difference between the highest value and the lowest value. The range tells us nothing about the middle of the data — only about the extremes.
Formula
Range = Highest value − Lowest value
Worked Examples
Example 1 — Basic range:
Find the range of: 5, 12, 3, 19, 8, 14
Highest value = 19
Lowest value = 3
Range = 19 − 3 = 16
Example 2 — Range with negative numbers:
Find the range of: −8, 4, −3, 11, 0, −5, 7
Highest value = 11
Lowest value = −8
Range = 11 − (−8) = 11 + 8 = 19
Example 3 — Range with decimals:
Find the range of: 2.5, 4.8, 1.2, 6.3, 3.7
Highest value = 6.3
Lowest value = 1.2
Range = 6.3 − 1.2 = 5.1
Example 4 — Using range to compare datasets:
Class A scores: 55, 60, 62, 58, 65 → Range = 65 − 55 = 10
Class B scores: 30, 55, 72, 90, 48 → Range = 90 − 30 = 60
Class B has a much larger range — their scores are far more spread out than Class A, whose scores are clustered closely together. Class A is more consistent.
Putting It All Together — Full Worked Example
The scores of 9 students in a maths test are:
72, 55, 88, 64, 72, 91, 48, 72, 60
Find the mean, median, mode and range.
Mean:
Sum = 72 + 55 + 88 + 64 + 72 + 91 + 48 + 72 + 60 = 622
Mean = 622 ÷ 9 = 69.1 (1 d.p.)
Median:
Arrange in order: 48, 55, 60, 64, 72, 72, 72, 88, 91
9 values → median is the 5th value
Median = 72
Mode:
72 appears three times — more than any other value
Mode = 72
Range:
91 − 48 = 43
Which Measure Should You Use?
Common Mistakes to Avoid
① Forgetting to arrange data in order before finding the median — the median of an unordered list is meaningless.
② Dividing by the wrong number when finding the mean — always divide by the total number of values, not the largest value or the sum.
③ Confusing the median position with the median value — the position tells you where to look; the value at that position is the median.
④ Forgetting that a dataset can have no mode, one mode, or multiple modes.
⑤ Getting the range wrong with negative numbers.
✗ Range of −8 and 11 = 11 − 8 = 3
✓ Range of −8 and 11 = 11 − (−8) = 19
⑥ Assuming the mean must be one of the values in the dataset — it does not have to be.
Frequently Asked Questions
Q1. What is the difference between mean, median and mode?
All three are measures of central tendency — they each give a single value that represents a dataset, but they do it differently. The mean is the arithmetic average of all values. The median is the middle value when data is ordered. The mode is the most frequently occurring value. They will often give different answers for the same dataset, and choosing the right one depends on the nature of the data and whether outliers are present.
Q2. When should I use the median instead of the mean?
Use the median when the data contains outliers — extreme values that are much higher or much lower than the rest. Outliers pull the mean towards them and make it unrepresentative of the typical value. The median is not affected by outliers because it only looks at the middle value. Income and house price data almost always use the median for this reason.
Q3. Can a dataset have more than one mode?
Yes. A dataset with two modes is called bimodal. A dataset with more than two modes is called multimodal. A dataset where every value appears the same number of times has no mode. There is no limit to the number of modes a dataset can have.
Q4. Does the mean have to be one of the values in the dataset?
No. The mean is a calculated value and does not need to appear in the original dataset. For example, the mean of 3, 4, and 5 is 4 — which does appear — but the mean of 3, 4, and 6 is 4.33, which does not appear in the dataset. This is perfectly normal.
Q5. What is the difference between range and interquartile range?
The range is the difference between the highest and lowest values in the entire dataset. It is simple to calculate but is heavily influenced by outliers. The interquartile range (IQR) is the difference between the upper quartile (Q3) and the lower quartile (Q1) — it measures the spread of the middle 50% of the data and is not affected by outliers. The IQR is a more reliable measure of spread when outliers are present.
Q6. Why do we arrange data in order to find the median?
The median is defined as the middle value of an ordered dataset. If the data is not arranged in order first, you cannot correctly identify the middle value. The order is what gives the median its meaning — it splits the dataset exactly in half, with 50% of values below it and 50% above it.