<< Chapter < Page | Chapter >> Page > |
22 | 24 | 48 | 51 | 60 | 72 | 73 | 75 | 80 | 88 | 90 |
Lower quartile | Median | Upper quartile | ||||||||
( ) | ( ) | ( ) |
Method: Calculating the quartiles
What are the quartiles of ?
There are 12 values in the data set.
1 | 3 | 5 | 8 | 9 | 12 | 24 | 25 | 28 | 30 | 41 | 50 | |||
The first quartile occurs between data position 3 and 4 and is the average of data values 5 and 8. The second quartile occurs between positions 6 and 7 and is the average of data values 12 and 24. The third quartile occurs between positions 9 and 10 and is the average of data values 28 and 30.
The first quartile = 6,5. ( )
The second quartile = 18. ( )
The third quartile = 29. ( )
The inter quartile range is a measure which provides information about the spread of a data set, and is calculated by subtracting the first quartile from the third quartile, giving the range of the middle half of the data set, trimming off the lowest and highest quarters, i.e. .
The semi-interquartile range is half the interquartile range, i.e.
A class of 12 students writes a test and the results are as follows: 20, 39, 40, 43, 43, 46, 53, 58, 63, 70, 75, 91. Find the range, quartiles and the Interquartile Range.
20 | 39 | 40 | 43 | 43 | 46 | 53 | 58 | 63 | 70 | 75 | 91 | |||
The range = 91 - 20 = 71. This tells us that the marks are quite widely spread.
i.e.
i.e.
i.e.
The quartiles are 41,5, 49,5 and 66,5. These quartiles tell us that 25 of the marks are less than 41,5; 50 of the marks are less than 49,5 and 75 of the marks are less than 66,5. They also tell us that 50 of the marks lie between 41,5 and 66,5.
The Interquartile Range = 66,5 - 41,5 = 25. This tells us that the width of the middle 50 of the data values is 25.
The Semi-interquartile Range = = 12,5
Percentiles are the 99 data values that divide a data set into 100 groups.
The calculation of percentiles is identical to the calculation of quartiles, except the aim is to divide the data values into 100 groups instead of the 4 groups required by quartiles.
Method: Calculating the percentiles
Consider the following grouped data and calculate the mean, the modal group and the median group.
Mass (kg) | Frequency |
41 - 45 | 7 |
46 - 50 | 10 |
51 - 55 | 15 |
56 - 60 | 12 |
61 - 65 | 6 |
Total = 50 |
To calculate the mean we need to add up all the masses and divide by 50. We do not know actual masses, so we approximate by choosing the midpoint of each group. We then multiply those midpoint numbers by the frequency. Then we add these numbers together to find the approximate total of the masses. This is show in the table below.
Mass (kg) | Midpoint | Frequency | Midpt Freq |
41 - 45 | (41+45)/2 = 43 | 7 | 43 7 = 301 |
46 - 50 | 48 | 10 | 480 |
51 - 55 | 53 | 15 | 795 |
56 - 60 | 58 | 12 | 696 |
61 - 65 | 63 | 6 | 378 |
Total = 50 | Total = 2650 |
The mean = .
The modal group is the group 51 - 53 because it has the highest frequency.
The median group is the group 51 - 53, since the 25th and 26th terms are contained within this group.
In each data set given, find the mean, the modal group and the median group.
Time in seconds | Frequency |
36 - 45 | 5 |
46 - 55 | 11 |
56 - 65 | 15 |
66 - 75 | 26 |
76 - 85 | 19 |
86 - 95 | 13 |
96 - 105 | 6 |
Mass in kilograms | Frequency |
41 - 45 | 3 |
46 - 50 | 5 |
51 - 55 | 8 |
56 - 60 | 12 |
61 - 65 | 14 |
66 - 70 | 9 |
71 - 75 | 7 |
76 - 80 | 2 |
Notification Switch
Would you like to follow the 'Siyavula textbooks: grade 10 maths [ncs]' conversation and receive update notifications?