1. KS3 Mathematics D2 Processing data
The aim of this unit is to teach pupils to:
Calculate statistics from data, using ICT as appropriate, finding the mode, mean, median and range
Material in this unit is linked the Framework’s supplement of examples pp256 –261.
D2 Processing data

2. D2 Processing data Contents D2.1 Finding the mode
D2.2 Calculating the mean
D2.3 Finding the median
D2.4 Finding the range
D2.5 Calculating statistics

3. Finding the mode
The mode or modal value in a set of data is the data value that appears the most often.
For example, the number of goals scored by the local football team in the last ten games is: 2, 1, 0, 3, 1. 2, 1, 0, 3, 1.
The modal score is 2.
Is it possible to have more than one modal value?
Yes
Is it possible to have no modal value?
Yes

4. What was the modal score?
Finding the mode
A dice was thrown ten times. These are the results:
What was the modal score?
3 is the modal score because it appears most often.

5. Finding the mode
The mode is the only average that can be used for categorical or non-numerical data.
For example, 30 pupils are asked how they usually travel to school. The results are shown in a frequency table.
Method of travel
Frequency
Bicycle 6
On foot 8
Car 2
Bus
Train 3
What is the modal method of travel? 8
Most children travel by foot.
Travelling on foot is therefore the modal method of travel.
Stress that the modal method will have the highest frequency.

6. Finding the mode from a bar chart
This bar chart shows the scores in a science test:
Number of pupils
Mark out of ten
This bar chart can be edited by double clicking on it in Normal view.
What was the modal score?
6 is the modal score because it has the highest bar.

7. Finding the mode from a pie chart
This pie chart shows the favourite food of a sample of people:
What was the modal food type?
Ask pupils to tell you how many people were surveyed (200).
Ask pupils to tell you what percentage of people liked each type of food.
The biggest sector of the pie chart is for chocolate, so this is the modal food type.

8. Finding the mode from a frequency table
This frequency table shows the frequency of different length words in a given paragraph of text.
Word length
Frequency 1 3 2 16 12 4 5 7 6 11 8 9 10 16 16
What was the modal word length?
We need to look for the word lengths that occur most frequently.
2 and 4 are the modal word lengths because they both appeared 16 times.
For this data there are two modal word lengths: 2 and 4.

9. Finding the modal class for continuous data
This grouped frequency table shows the times 50 girls and 50 boys took to complete one lap around a race track.
Frequency
Time (minutes:seconds)
Boys
Girls
2:00 ≤ 2:15 3 1
2:15 ≤ 2:30 7 6
2:30 ≤ 2:45 11 10
2:45 ≤ 3:00 13 9
3:15 ≤ 3:30 8 12
3:30 ≤ 3:45
3:45 ≤ 4:00 2
What is the modal class for the girls?
What is the modal class for the boys?
What is the modal class for the pupils regardless of whether they are a boy or a girl?
Establish that the modal class for the girls is 3:15  3:30 and modal class for the boys is 2:45  3:00.
To find the modal class for both boys and girls we must add the two frequencies in each row together. The modal class is 2:45  3:00.

10. D2 Processing data Contents D2.1 Finding the mode
D2.2 Calculating the mean
D2.3 Finding the median
D2.4 Finding the range
D2.5 Calculating statistics

11. The mean The mean is the most commonly used average.
To calculate the mean of a set of values we add together the values and divide by the total number of values.
Mean =
Sum of values
Number of values
For example, the mean of 3, 6, 7, 9 and 9 is 5 34 5 =
= 6.8

12. The mean

13. Problems involving the mean
A pupil scores 78%, 75% and 82% in three tests. What must she score in the fourth test to get an overall mean of 80%?
To get a mean of 80% the four marks must add up to
4 × 80% = 320%
The three marks that the pupils has so far add up to
78% + 75% + 82% = 235%
Encourage pupils to check the answer by adding 78, 75, 82 and 85 and dividing by four.
The mark needed in the fourth test is
320% – 235% =
85%

14. Calculating the mean from a frequency table
The following frequency table shows the scores obtained when a dice is thrown 50 times.
What is the mean score?
Score
Frequency 1 2 3 4 5 6 8 11 9 7
Total 50
Score × Frequency 8 22 18 36 45 42
171
Explain that to find the mean score we need to find the total score altogether. A 1 was scored 8 times and so we can find the score obtained by throwing 1s by multiplying 8 × 1. A 2 was scored 11 times and so we can find the score obtained by throwing 2s by finding 11 × 2. Conclude that we need to find the score × the frequency for each score. Show how this can be done by revealing the yellow row in the table.
We can then record the total number of throws and the total score in the blue column. The mean is found by dividing these totals.
171 50
The mean score =
= 3.42

15. Calculating the mean using a spreadsheet
When processing large amounts of data it is often helpful to use a spreadsheet to help us calculate the mean.
For example, 500 households were asked how many children under the age of 16 lived in the home. The results were collected in a spreadsheet.
Discuss how a spreadsheet can be used to find the mean for larger amounts of data.

16. Calculating the mean using a spreadsheet
The total number of households is found by entering =SUM(B2:J2) in cell K2 as follows:
Pressing enter shows the number of households.

17. Calculating the mean using a spreadsheet
The total number of children in households with no children is found by entering =B1*B2 into cell B3.
The total number of children in households with one child is found by entering =C1*C2 into cell C3.

18. Calculating the mean using a spreadsheet
This can be repeated along the row to find the total number of children in each type of household.
To find the total number of children altogether enter =SUM(B3:J3) in cell K3.

19. Calculating the mean using a spreadsheet
The mean number of children in each household is now found by dividing the number in cell K3 by the number in cell K2.
Mean number of children =
1061
500 =
Explain that it is a good idea to create a cell containing the formula =K3/K2. This cell will calculate the mean number of children. The advantage of entering this as a formula is that if any of the values in the table are changed, the mean will recalculated automatically.

20. Using an assumed mean
We can calculate the mean of a set of data using an assumed mean.
For example, suppose the heights of ten year 8 pupils are as follows:
148 cm, cm, cm, cm, cm,
142 cm, cm, cm, cm, cm.
To find the mean of these values using an assumed mean we start by making a guess at what the mean might be. This is the assumed mean.
Point out that the assumed mean is going to be subtracted from each of the values in the set of data. It makes sense, therefore, to choose a value that is easy to subtract, such as a value that has been rounded to the nearest 10.
Point out, that it doesn’t really matter what this value is as long as it is easy to subtract and the resulting differences are easy to add. The aim is to make the values smaller so that we can calculate the mean mentally.
For this set of values we can use an assumed mean of 150 cm.

21. Using an assumed mean
Subtract the assumed mean from each of the data values.
148 cm, cm, cm, cm, cm,
142 cm, cm, cm, cm, cm.
– 150 cm
–2 cm, cm, –5 cm, cm, cm,
–8 cm, cm, cm, cm, –12 cm.
Next, find the mean of the new set of values.
Point out that when we add together lots of numbers where some are positive and some are negative, it is often easier to add the positive numbers together and then add the negative numbers together. That way, we only have a single subtraction to do at the end of the calculation.
If we add together the positive values we get 38 and if we add the negative values together we get –27. The new set of values therefore have a sum of 11.
38 – 27 10 11 10 =
= 1.1

22. mean of the differences
Using an assumed mean
To find the actual mean of the heights, add this value to the assumed mean.
= 151.1
assumed mean
mean of the differences
actual mean
So, the actual mean of the heights is cm.
This method is often used to find the mean of numbers that are large or written to a large number of decimal places.
We can use this method to find the mean mentally, using jottings.

23. Using an assumed mean
In summary, to find the mean of a set of values using an assumed mean follow these steps:
1) Assume the mean of the values.
2) Subtract this assumed mean from each of the values.
3) Find the mean of the new values.
Add this mean to the assumed mean to find the actual
mean.
Actual mean = Assumed mean +
Sum of the differences
Total number of values

24. D2 Processing data Contents D2.1 Finding the mode
D2.2 Calculating the mean
D2.3 Finding the median
D2.4 Finding the range
D2.5 Calculating statistics

25. Finding the median
The median is the middle value of a set of numbers arranged in order.
For example, find the median of
10, 7, 9,
12, 8, 6,
Write the values in order: 6, 7, 7, 8, 9,
10,
12.
The median is the middle value.

26. Finding the median
When there is an even number of values, there will be two values in the middle.
In this case, we have to find the mean of the two middle values.
For example,
Find the median of 56, 42, 47, 51, 65 and 43.
The values in order are:
42,
43,
47,
51,
56,
65.
There are two middle values, 47 and 51.

27. Finding the median
To find the number that is half-way between 47 and 51 we can add the two numbers together and divide by 2. 2 = 98 2
= 49
Alternatively, find the difference between 47 and 51 and add half this difference to the lower number.
51 – 47 = 4
½ of 4 = 2
Pupils may wish to use other mental methods to find a number half-way between two others, such as imagining the numbers on a number line, particularly if the numbers are close together.
= 49
The median of 42, 43, 47, 51, 56 and 65 is 49.

28. Find the median

29. What is the rogue value in the following data set:
Rogue values
The median is often used when there is a rogue value – that is, a value that is much smaller or larger than the rest.
What is the rogue value in the following data set:
192, 183, 201, 177, 193, 197, 4, 186, 179?
The median of this data set is:
4, 177, 179, 183, 186, 192, 193, 197, 201.
The median of the data set is not affected by the rogue value, 4.
Point out that rogue values are not typical of the rest of the data. If we found the mean of data containing a rogue value then the mean would be unrepresentative of the data set.
The mean of the data set is 168. This is not representative of the set because it is lower than almost all the data values.

30. Mean or median?
Would it be better to use the median or the mean to represent the following data sets?
34.2, 36.8, 29.7, 356, 42.5, 37.1?
median
0.4, 0.5, 0.3, 0.8, 0.7, 1.0?
mean
892, 954, 1026, 908, 871, 930?
mean
3.12, 3.15, 3.23, 9.34, 3.16, 3.20?
median
Ask pupils to identify the rogue value, if there is one, and to use this to justify their choice of mean or median.
97.85, , , , , ?
mean
87634, 9321, , , , ?
median

31. D2 Processing data Contents D2.1 Finding the mode
D2.2 Calculating the mean
D2.3 Finding the median
D2.4 Finding the range
D2.5 Calculating statistics

32. Finding the range
The range of a set of data is a measure of how the data is spread across the distribution.
To find the range we subtract the lowest value in the set from the highest value.
Range = highest value – lowest value
When the range is small it tells us that the values are similar in size.
What does it mean if the range is small?
When the range is large it tells us that the values vary widely in size.
What does it mean if the range is large?

33. Find the range

34. D2.5 Calculating statistics
Contents
D2 Processing data
D2.1 Finding the mode
D2.2 Calculating the mean
D2.3 Finding the median
D2.4 Finding the range
D2.5 Calculating statistics

35. Remember the three averages and rangeL A R G E S T M
M E D I A N
M I D D L E
M O D E C O M N
M E A N A D
D I V I D E

36. The three averages and range
There are three different types of average:
MEAN
sum of values
number of values
MEDIAN
middle value
MODE
most common
The range is not an average, but tells you how the data is spread out:
RANGE
largest value – smallest value

37. The three averages
Each type of average has its purpose and sometimes one is preferable to an other.
The mode is easy to find and it eliminates some of the
effects of extreme values. It is the only type of average that can be used for categorical (non-numerical) data.
The median is also fairly easy to find and has the advantage of being hardly affected by rogue values or skewed data.
Discuss that advantages and disadvantages of each type of average.
The mean is the most difficult to calculate but takes into account all the values in the data set.

38. Find the mean, median and range
Use the activity to practise finding the mean, median and range.
If there is a modal value for a particular data set, ask pupils to tell you what this is.

39. Find the missing value
Pupils must use the information given to find the missing value.

40. Calculating statistics
Look at the values on these five cards: 2 4 5 8 11
Choose three cards so that:
The mean is bigger than the median.
Establish that for the mean to be bigger than the median, the first two values must be much lower than the third value. For example, the mean of 2, 5 and 11 is 6 and the median is 5.
For the median to be bigger than the mean the second two values must be much greater than the first. For example, the mean of 2, 8 and 11 is 7 and the median is 8.
For the median and the mean to be the same, the difference between the first value and the second value must be the same as the difference between second value and the first value. For example, the mean and the median of 2, 5 and 8 is 5.
Make the activity more challenging by choosing decimal values for the cards, changing the number of cards or asking pupils to choose four cards that satisfy the required conditions.
The median is bigger than the mean.
The mean and the median are the same.

41. Stem-and-leaf diagrams
Sometimes data is arranged in a stem-and-and leaf diagram.
For example, this stem-and-leaf diagram shows the marks scored by 21 pupils in a maths test.
Find the median, mode and range for the data. 1 2 3 4 6 7 5 9 8
stem = tens
leaves = units
There are 21 data values so the median will be the 11th value, that is ___ . 5 25
Start by explaining how to read the stem and leaf diagram. The scores shown in the diagram are 6, 7, 9, 14, 15, 15, 18, 20, 21, 23, 25, 26, 26, 30, 32, 32, 32, 35, 38, 40 and 40. Ask pupils what they think the test might have been out of.
The values used for the stem and leaves depend on the data. For example, if we were showing times in minutes and seconds, we could use the stem to show the minutes and the leaves to show the seconds. If we were showing lengths between 3.0 and 8.0 written to one decimal place, we could use the stem to show the whole number of centimetres and the leaves to show the tenths (or millimetres).
Discuss how to find the median, mode and range.
The mode is ___ . 32
The range is 40 – 6, which is ___ . 34

42. Stem-and-leaf diagrams
Calculate the median, mode and range for each stem-and-leaf diagram.