9.2 Summarising data

Siyavula textbooks: grade 10 Page 3 / 3

22	24	48	51	60	72	73	75	80	88	90
		$↓$			$↓$			$↓$
		Lower quartile			Median			Upper quartile
		( $Q_{1}$ )			( $Q_{2}$ )			( $Q_{3}$ )

Method: Calculating the quartiles

Order the data from smallest to largest or from largest to smallest.
Count how many data values there are in the data set.
Divide the number of data values by 4. The result is the number of data values per group.
Determine the data values corresponding to the first, second and third quartiles using the number of data values per quartile.

What are the quartiles of ${3, 5, 1, 8, 9, 12, 25, 28, 24, 30, 41, 50}$ ?

Order the data set from lowest to highest
${1, 3, 5, 8, 9, 12, 24, 25, 28, 30, 41, 50}$
Count the number of data values in the data set
There are 12 values in the data set.
Divide the number of data values by 4 to find the number of data values per quartile.
$12 \div 4 = 3$
Find the data values corresponding to the quartiles.

1 3 5 $∥$ 8 9 12 $∥$ 24 25 28 $∥$ 30 41 50

$Q_{1}$ $Q_{2}$ $Q_{3}$

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.
Answer
The first quartile = 6,5. ( $Q_{1}$ )

The second quartile = 18. ( $Q_{2}$ )

The third quartile = 29. ( $Q_{3}$ )

Inter-quartile range

Inter-quartile Range

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. $Q_{3} - Q_{1}$ .

The semi-interquartile range is half the interquartile range, i.e. $\frac{Q_{3} - Q_{1}}{2}$

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

$Q_{1}$ $M$ $Q_{3}$
The Range
The range = 91 - 20 = 71. This tells us that the marks are quite widely spread.
The median lies between the 6th and 7th mark
i.e. $M = \frac{46 + 53}{2} = \frac{99}{2} = 49, 5$
The lower quartile lies between the 3rd and 4th mark
i.e. $Q_{1} = \frac{40 + 43}{2} = \frac{83}{2} = 41, 5$
The upper quartile lies between the 9th and 10th mark
i.e. $Q_{3} = \frac{63 + 70}{2} = \frac{133}{2} = 66, 5$
Analysing the quartiles
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
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-interquatile Range
The Semi-interquartile Range = $\frac{25}{2}$ = 12,5

Percentiles

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

Order the data from smallest to largest or from largest to smallest.
Count how many data values there are in the data set.
Divide the number of data values by 100. The result is the number of data values per group.
Determine the data values corresponding to the first, second and third quartiles using the number of data values per quartile.

Exercises - summarising data

Three sets of data are given:
1. Data set 1: 9 12 12 14 16 22 24
2. Data set 2: 7 7 8 11 13 15 16 16
3. Data set 3: 11 15 16 17 19 19 22 24 27 For each one find:
  1. the range
  2. the lower quartile
  3. the interquartile range
  4. the semi-interquartile range
  5. the median
  6. the upper quartile
Click here for the solution
There is 1 sweet in one jar, and 3 in the second jar. The mean number of sweets in the first two jars is 2.
1. If the mean number in the first three jars is 3, how many are there in the third jar?
2. If the mean number in the first four jars is 4, how many are there in the fourth jar?
Click here for the solution
Find a set of five ages for which the mean age is 5, the modal age is 2 and the median age is 3 years.
Click here for the solution
Four friends each have some marbles. They work out that the mean number of marbles they have is 10. One of them leaves. She has 4 marbles. How many marbles do the remaining friends have together?
Click here for the solution

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

Calculating the mean

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 $\times$ Freq
41 - 45	(41+45)/2 = 43	7	43 $\times$ 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

Answer
The mean = $\frac{2650}{50} = 53$ .

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.

Times recorded when learners played a game.

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

Click here for the solution
The following data were collected from a group of learners.

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

Click here for the solution

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Source: OpenStax, Siyavula textbooks: grade 10 maths [ncs]. OpenStax CNX. Aug 05, 2011 Download for free at http://cnx.org/content/col11239/1.2

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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

1	3	5	$∥$	8	9	12	$∥$	24	25	28	$∥$	30	41	50
			$Q_{1}$				$Q_{2}$				$Q_{3}$

20	39	40	$∥$	43	43	46	$∥$	53	58	63	$∥$	70	75	91
			$Q_{1}$				$M$				$Q_{3}$