Measures of central tendency describe the "typical" value in a set of data. Measures of dispersion describe how spread out the data is. Both appear in almost every introductory statistics exam. We will use one data set throughout.
Data: the number of hours eight students studied in a week: 2, 4, 4, 4, 5, 5, 7, 9.
The mean
The mean (arithmetic average) is the sum of all values divided by the number of values.
Sum = 2 + 4 + 4 + 4 + 5 + 5 + 7 + 9 = 40. Number of values n = 8.
Mean = 40 ÷ 8 = 5 hours.
The mean uses every value, but it is affected by extreme values (outliers). One student who studied 40 hours would pull it up a lot.
The median
The median is the middle value when the data is arranged in order. Our data is already in order. With 8 values (an even number), the median is the average of the 4th and 5th values.
4th value = 4, 5th value = 5. Median = (4 + 5) ÷ 2 = 4.5 hours.
With an odd number of values, the median is simply the middle one. The median is not affected much by outliers, which makes it useful for things like income.
The mode
The mode is the value that appears most often. In our data, 4 appears three times. Mode = 4 hours.
A data set can have no mode, one mode, or more than one mode (bimodal).
The range
The range is the largest value minus the smallest: 9 − 2 = 7 hours. It is quick to calculate but only uses two values.
Variance and standard deviation
The standard deviation measures how far, on average, the values are from the mean. A small standard deviation means values are close to the mean.
Step 1: subtract the mean (5) from each value and square the result.
| x | x − 5 | (x − 5)² |
|---|---|---|
| 2 | −3 | 9 |
| 4 | −1 | 1 |
| 4 | −1 | 1 |
| 4 | −1 | 1 |
| 5 | 0 | 0 |
| 5 | 0 | 0 |
| 7 | 2 | 4 |
| 9 | 4 | 16 |
Step 2: add the squared deviations: 9 + 1 + 1 + 1 + 0 + 0 + 4 + 16 = 32.
Step 3: divide to get the variance.
- Population variance (divide by n): 32 ÷ 8 = 4. Population standard deviation = √4 = 2.
- Sample variance (divide by n − 1): 32 ÷ 7 ≈ 4.57. Sample standard deviation ≈ 2.14.
Use n when your data is the whole population and n − 1 when it is a sample from a larger population. Check which one your lecturer uses, and always state it.
Which measure should you use?
- Use the mean for data without extreme values.
- Use the median when there are outliers or the data is skewed.
- Use the mode for categories, such as the most common department in a class.
Exam tips
- Arrange data in order before finding the median.
- Draw a table for standard deviation: it reduces arithmetic mistakes and earns method marks.
- Include units in your final answer.