The mean, median, mode and range, and using them to compare two sets of data.
Key ideas
Mean
Add up all the values.
Divide by how many values there are.
Mean = totalnumber of values
Median
Put the values in order and count them.
Split them into two equal halves. The median is the value left in the middle.
If no value is left on its own, the median is halfway between the two either side of the split.
Mode
The value that appears most often.
There can be more than one mode, or none at all.
Range
Range = highest − lowest.
It measures how spread out the data is. A smaller range means more consistent.
Comparing two sets of data
Compare an average (mean or median) and the range, and always quote the numbers:
"On average Class A scored higher, because their mean (14.2) is higher than Class B's (12.8)."
"Class A's scores were more consistent, because their range (6) is lower than Class B's (11)."
The mean
Statistics
Finding the mean
Find the mean of these numbers:
4, 7, 9, 3, 7
(2 marks)
Worked solutionIn plain English
1Add them up.4 + 7 + 9 + 3 + 7 = 30✓1
In plain EnglishFind the total of all the values.
2Divide by how many.Mean = 305 = 6✓2
In plain EnglishThere are 5 values, so divide the total by 5.
Marking points
M1Total = 30
M2Mean = 6
Statistics
A mean that isn't a whole number
Four friends scored these points in a game. Find the mean score.
12, 15, 9, 14
(2 marks)
Worked solutionIn plain English
1Add them up.12 + 15 + 9 + 14 = 50✓1
In plain EnglishAdd up all the scores.
2Divide by how many.Mean = 504 = 12.5✓2
In plain EnglishThe mean doesn't have to be one of the values, or even a whole number.
Marking points
M1Total = 50
M2Mean = 12.5
The mode
Statistics
Finding the mode
Find the mode of these shoe sizes:
3, 5, 2, 5, 8, 3, 5
(1 mark)
Worked solutionIn plain English
1Which appears most often?2 once, 3 twice, 5 three times, 8 onceMode = 5✓1
In plain EnglishCount how many times each value appears. 5 appears three times, more than any other.
⚠ A common mistake: giving the answer 3 because 5 appears 3 times. The mode is the value (5), not how many times it appears.
Marking points
M1Mode = 5
The median
Statistics
The median: an odd number of values
Find the median of:
8, 3, 11, 6, 9
(2 marks)
Worked solutionIn plain English
1Put them in order and count them.Data: 8, 3, 11, 6, 9Ordered (n = 5): 3, 6, 8, 9, 11✓1
In plain EnglishWrite the values from smallest to largest and count them. There are 5. Tick each one off as you use it so none go missing.
2Split into two halves.
3, 68median9, 11
Median = 8✓2
In plain EnglishWith 5 values there are 2 in each half, and 8 is left on its own in the middle. Circle it: that's the median.
Marking points
M1In order
M2Median = 8
Statistics
The median: an even number of values
Find the median of:
14, 9, 12, 20, 7, 16
(2 marks)
Worked solutionIn plain English
1Put them in order and count them.Data: 14, 9, 12, 20, 7, 16Ordered (n = 6): 7, 9, 12, 14, 16, 20✓1
In plain EnglishWrite the values from smallest to largest and count them. There are 6.
2Split into two halves.
7, 9, 12median14, 16, 20
Median = 12 + 142 = 262 = 13✓2
In plain English6 values make two halves of 3. No value sits on its own, so the median falls between 12 and 14, shown by an arrow. It is halfway between them.
In plain EnglishThe range is the gap between the biggest and smallest values.
Marking points
M1Range = 19
Comparing data sets
Statistics
Comparing two classes
Two classes sat the same test.
Mean
Range
Class A
14.2
6
Class B
12.8
11
Make two valid comparisons between the classes.
(2 marks)
Worked solutionIn plain English
1Compare the averages.On average Class A did better, because their mean (14.2) is higher than Class B's (12.8).✓1
In plain EnglishSay which is higher on average, and quote both numbers.
2Compare the spread.Class A's marks were more consistent, because their range (6) is lower than Class B's (11).✓2
In plain EnglishA lower range means the marks are less spread out, so more consistent.
⚠ Marks are lost for "Class A's mean is higher" on its own. Say what it means (did better on average) and quote the numbers.
Marking points
M1Average compared
M2Spread compared
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Your turn
Calculator allowed. For the median, always put the numbers in order first. When comparing, choose the statement that says what the numbers mean and quotes them. Stuck? Tap Need a hint?