Averages and range
Three averages and one spread
An average is one value that stands for the whole set.
The mean is the total divided by how many values there are. The median is the middle value once the list is in order. The mode is the value that appears most often. The range = largest − smallest measures how spread out the data are.
- Data: 7, 3, 9, 4, 7
- Mean: 7 + 3 + 9 + 4 + 7 = 30, and 30 ÷ 5 = 6
- Median: in order 3, 4, 7, 7, 9, so the middle (3rd) value is 7
- Mode: 7 appears twice, so the mode is 7
- Range: 9 − 3 = 6
With an even number of values the median is halfway between the middle two: for 2, 5, 8, 10 it is (5 + 8) ÷ 2 = 6.5.
Working backwards from the mean
If you know the mean and how many values there are, you know the total: total = mean × number of values.
That finds a missing value.
- The mean of 4 numbers is 9. Three of them are 6, 12 and 5.
- Total of all 4 = 4 × 9 = 36
- Total of the three = 6 + 12 + 5 = 23
- Missing number = 36 − 23 = 13
The same idea combines two groups: 10 students with mean 60 and 15 students with mean 70 have a total of 600 + 1050 = 1650, so the mean of all 25 is 1650 ÷ 25 = 66, not 65.
The slips that cost marks
Finding the median of a list that is not in order is the most common slip: always write the list in order first.
When you compare two sets of data, give an average and the range, and say what each means in context: "Team A scored more on average (higher mean), and Team B was more consistent (smaller range)."
One very large or very small value pulls the mean a long way but hardly moves the median, so the median is the better average when there is an extreme value. Use the mode for categories such as favourite colour.
See an extreme value move the mean
Open the Statistics Calculator and enter 12, 15, 9, 22, 15, 18, 95. Write down the mean and the median. Now delete 95 and look again. Which average changed more, and why?
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This question is in Brainlag Plus