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Measures of central tendency: Class 11 Statistics
The mean, median and mode for raw data and for frequency distributions, with the formulas, how to choose between them and worked numericals.
Key points
Statistics is the science of collecting, organising, presenting, analysing and interpreting data. A measure of central tendency, or an average, is a single value that represents the centre of a set of data. The three common averages are the arithmetic mean, the median and the mode.
The arithmetic mean is the sum of all the values divided by the number of values. It uses every value, so it is affected by extremely large or small values (outliers). For a frequency distribution with values x and frequencies f, the mean is the sum of the products fx divided by the sum of the frequencies.
The median is the middle value when the data are arranged in order. If there is an odd number of values, the median is the middle one. If there is an even number, the median is the mean of the two middle values. The median is not affected by outliers, so it is a better measure for skewed data such as incomes.
The mode is the value that occurs most often. A set of data can have one mode, more than one mode or no mode. It is the only average that can be used with data that are not numbers, such as the most popular colour.
The weighted mean is used when the values have different importance. It is the sum of each value multiplied by its weight, divided by the sum of the weights. Marks in a course with different weights for assignments and the final exam are an example.
Choosing an average: use the mean for data without extreme values, the median when there are outliers or a skewed distribution, and the mode for the most common value or for non-numerical categories.
Study tip: Write the data in order first. It makes the median and the mode easy to find and helps you to catch errors.
Definitions and flashcards
Say the meaning first, then open the card to check.
Statistics
The science of collecting, organising, presenting, analysing and interpreting data.
Average
A single value that represents the centre of a set of data.
Arithmetic mean
The sum of the values divided by the number of values.
Median
The middle value of the data arranged in order.
Mode
The value that occurs most often.
Outlier
A value that is much larger or smaller than the others.
Frequency
The number of times a value occurs.
Weighted mean
A mean in which each value is multiplied by a weight showing its importance.
Formula sheet
| Name | Formula | What it means |
|---|---|---|
| Mean of raw data | x̄ = Σx ÷ n | Sum of the values divided by the number of values. |
| Mean of a frequency distribution | x̄ = Σfx ÷ Σf | Sum of value times frequency divided by the total frequency. |
| Median (odd number of values) | Value at position (n + 1) ÷ 2 | The middle value. |
| Median (even number of values) | Mean of the values at positions n ÷ 2 and (n ÷ 2) + 1 | The mean of the two middle values. |
| Weighted mean | x̄w = Σwx ÷ Σw | w is the weight of each value. |
Short questions with model answers
These are the kind of questions asked for 2, 3 and 5 marks. Try each one before opening the answer.
Define an average. Name three averages. (2 marks)
An average is a single value that represents the centre of a set of data. The arithmetic mean, the median and the mode are three averages.
Write two merits of the arithmetic mean. (2 marks)
It is easy to calculate and understand, and it uses all the values in the data.
Why is the median better than the mean for incomes? (3 marks)
A few very high incomes raise the mean and make it unrepresentative. The median is the middle value and is not affected by extreme values, so it gives a better picture of a typical income.
What is the mode? Can a set of data have more than one mode? (2 marks)
The mode is the value that occurs most often. A set of data can have more than one mode, for example data with two values that occur equally often are bimodal.
When is a weighted mean used? (2 marks)
It is used when the values do not have equal importance, for example when different parts of a course carry different marks.
Numericals with worked steps
Find the mean of 4, 8, 6, 5 and 7.
- Sum = 4 + 8 + 6 + 5 + 7 = 30.
- Number of values = 5.
- Mean = 30 ÷ 5.
Answer: 6
Find the median of 12, 15, 17 and 20.
- The values are already in order and there are 4 of them, an even number.
- The two middle values are 15 and 17.
- Median = (15 + 17) ÷ 2.
Answer: 16
Find the mode of 3, 5, 5, 7, 5 and 9.
- Count how often each value occurs: 5 occurs three times and every other value occurs once.
Answer: 5
Find the mean of the frequency distribution: values 2, 4 and 6 with frequencies 3, 5 and 2.
- Find fx for each value: 2 × 3 = 6, 4 × 5 = 20 and 6 × 2 = 12.
- Σfx = 6 + 20 + 12 = 38.
- Σf = 3 + 5 + 2 = 10.
- Mean = 38 ÷ 10.
Answer: 3.8
A student scores 70 in an assignment (weight 2) and 80 in the final exam (weight 3). Find the weighted mean.
- Σwx = (2 × 70) + (3 × 80) = 140 + 240 = 380.
- Σw = 2 + 3 = 5.
- Weighted mean = 380 ÷ 5.
Answer: 76
Common mistakes
- Taking the middle value without arranging the data in order first.
- Dividing Σfx by the number of different values instead of Σf.
- Forgetting that the median of an even number of values is the mean of the two middle values.
- Using the mean for data with extreme values when the median is more suitable.