# 12.1 Summarising data

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Method: Calculating the percentiles

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

## Five number summary

We can summarise a data set by using the five number summary. The five number summary gives the lowest data value, the highest data value, the median, the first (lower) quartile and the third (higher) quartile. Consider the following set of data: 5, 3, 4, 6, 2, 8, 5, 4, 6, 7, 3, 6, 9, 4, 5. We first order the data as follows: 2, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 8, 9. The lowest data value is 2 and the highest data value is 9. The median is 5. The first quartile is 4 and the third quartile is 6. So the five number summary is: 2, 4, 5, 6, 9.

## Box and whisker plots

The five number summary can be shown graphically in a box and whisker plot. The main features of the box and whisker diagram are shown in [link] . The box can lie horizontally (as shown) or vertically. For a horizonatal diagram, the left edge of the box is placed at the first quartile and the right edge of the box is placed at the third quartile. The height ofthe box is arbitrary, as there is no y-axis. Inside the box there is some representation of central tendency, with the median shown with a vertical line dividing the box into two. Additionally, astar or asterix is placed at the mean value, centered in the box in the vertical direction. The whiskers which extend to the sides reach the minimum and maximum values. This is shown for the data set: 5, 3, 4, 6, 2, 8, 5, 4, 6, 7, 3, 6, 9, 4, 5.

Draw a box and whisker diagram for the data set: $x=\left\{1,25;1,5;2,5;2,5;3,1;3,2;4,1;4,25;4,75;4,8;4,95;5,1\right\}$ .

1. $\mathrm{Minimum}=1,25$
$\mathrm{Maximum}=5,10$
The position of first quartile is between 3 and 4.
The position of second quartile is between 6 and 7.
The position of third quartile is between 9 and 10.
The data value between 3 and 4 is: $\frac{1}{2}\left(2,5+2,5\right)=2,5$
The data value between 6 and 7 is: $\frac{1}{2}\left(3,2+4,1\right)=3,65$
The data value between 9 and 10 is: $\frac{1}{2}\left(4,75+4,8\right)=4,775$

## Exercises - summarising data

1. 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
2. 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?
3. 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.
4. 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?
5. Jason is working in a computer store. He sells the following number of computers each month: 27; 39; 3; 15; 43; 27; 19; 54; 65; 23; 45; 16Give a five number summary and a box and whisker plot of his sales.
6. Lisa works as a telesales person. She keeps a record of the number of sales she makes each month. The data below show how much she sells each month. 49; 12; 22; 35; 2; 45; 60; 48; 19; 1; 43; 12Give a five number summary and a box and whisker plot of her sales.
7. Rose has worked in a florists shop for nine months. She sold the following number of wedding bouquets: 16; 14; 8; 12; 6; 5; 3; 5; 7
1. What is the five-number summary of the data?
2. Since there is an odd number of data points what do you observe when calculating the five-numbers?

We can apply the concepts of mean, median and mode to data that has been grouped. Grouped data does not have individual data points, but rather has the data organized into groups or bins. To calculate the mean we need to add up all the frequencies and divide by the total. We do not know what the actual data values are, 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. The modal group is the group with the highest frequency. The median group is the group that contains the middle terms.

Measures of dispersion can also be found for grouped data. The range is found by subtracting the smallest number in the lowest bin from the largest number in the highest bin. The quartiles are found in a similar way to the median.

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
1. 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
2. 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.

## More mean, modal and median group exercises.

In each data set given, find the mean, the modal group and the median group.

1. 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
2. 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

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