The mean is the total divided by the number of values, the median is the middle value once the data is in order, the mode is the most common value, and the range is the largest value minus the smallest. With an even number of values, the median is halfway between the two middle values. Outliers affect the mean far more than the median.
Averages and spread
- Mean = total of values ÷ number of values
- Median = middle value in order; position (n + 1) ÷ 2
- Mode = most common value
- Range = largest − smallest
- Total = mean × number of values
Mean, median and mode
The mean adds all the values and divides by how many there are. The median is the middle value when the data is written in order. The mode is the value that appears most often; there can be more than one mode, or none. For 4, 7, 7, 9, 13 the mean is 40 ÷ 5 = 8, the median is 7 and the mode is 7.
The median with an even number of values
Always put the data in order first. With n values, the median is at position (n + 1) ÷ 2. If n is even, this falls between two values, so take the halfway point. For 3, 5, 8, 10, 12, 14 the median is halfway between 8 and 10, which is 9.
The range
The range measures spread: the largest value minus the smallest. A small range means the data is consistent; a large range means it is spread out. The range is a single number, not a pair of values.
Working backwards with the mean
Total = mean × number of values. If five numbers have a mean of 12, their total is 60. If four of them are 10, 15, 9 and 14, which total 48, the fifth number is 12. Combining two groups works the same way: find each total, add them, and divide by the total number of values.
Choosing the best average
The mean uses every value but is pulled by outliers, extreme values far from the rest. The median is not affected by outliers, so it is better for data such as house prices or salaries. The mode suits qualitative data, where a mean or median cannot be found.
Worked examples
Example 1. Find the mean, median, mode and range of 4, 7, 7, 9, 13.
- Mean: 40 ÷ 5 = 8.
- In order already; the middle value is 7. The mode is 7.
- Range: 13 − 4 = 9.
Answer: Mean 8, median 7, mode 7, range 9.
Example 2. Find the median of 12, 3, 10, 5, 14, 8.
- Order: 3, 5, 8, 10, 12, 14.
- Six values, so the median is halfway between the 3rd and 4th: 8 and 10.
Answer: 9.
Example 3. Five numbers have a mean of 12. Four of them are 10, 15, 9 and 14. Find the fifth number.
- Total = 5 × 12 = 60.
- The four known numbers total 48.
Answer: The fifth number is 12.
Example 4. Class A has 20 students with a mean score of 64. Class B has 30 students with a mean of 70. Find the mean for all 50 students.
- Totals: 20 × 64 = 1280 and 30 × 70 = 2100.
- (1280 + 2100) ÷ 50 = 3380 ÷ 50.
Answer: 67.6.
Example 5. Five salaries are £21 000, £23 000, £24 000, £26 000 and £95 000. Which average best represents them? Explain.
- The mean is £37 800, pulled up by the outlier of £95 000.
- The median is £24 000, which is typical of most of the salaries.
Answer: The median, because it is not affected by the outlier.
Common mistakes
- Not ordering the data before finding the median. Put the values in order first.
- Giving the median position as the median. For 6 values, the median is the 3.5th value, not 3.5.
- Writing the range as two numbers. The range is largest − smallest, a single value.
- Averaging the means of different-sized groups. Combine the totals, then divide by the total number of values.
- Using the mean when there are outliers. The median is usually more representative.
How averages from raw data are examined
The mean, median, mode and range are tested early on Foundation papers and reappear at Higher whenever two data sets are compared. Questions ask you to:
- find the mean, median, mode and range of a list
- find a missing value from a given mean
- combine the means of two groups
- choose and justify the most suitable average
The averages from raw data worksheet includes answers, and the data and statistics engine produces more lists.
Next, find averages from tables in frequency tables, and compare data sets in comparing distributions.
Frequently asked questions
What is the median of an even number of values?
Order the data and find the value halfway between the two middle numbers.
Which average is affected by outliers?
The mean. The median and mode are much less affected.
How do you find a missing value from the mean?
Multiply the mean by the number of values to get the total, then subtract the known values.