# 12.1 Summarising data

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What is the median of $\left\{11,10,14,86,2,68,99,1\right\}$ ?

1. 1,2,10,11,14,68,85,99

2. There are 8 points in the data set.

3. The central position of the data set is between positions 4 and 5.

4. 11 is in position 4 and 14 is in position 5.

5. $\therefore$ the median of the data set $\left\{1,2,10,11,14,68,85,99\right\}$ is

$\left(11+14\right)÷2=12,5$

## Mode

Mode

The mode is the data value that occurs most often, i.e. it is the most frequent value or most common value in a set.

Method: Calculating the mode Count how many times each data value occurs. The mode is the data value that occurs the most.

The mode is calculated from grouped data, or single data items.

Find the mode of the data set $x=\left\{1,2,3,4,4,4,5,6,7,8,8,9,10,10\right\}$

1.  data value frequency data value frequency 1 1 6 1 2 1 7 1 3 1 8 2 4 3 9 1 5 1 10 2
2. 4 occurs most often.

3. The mode of the data set $x=\left\{1,2,3,4,4,4,5,6,7,8,8,9,10,10\right\}$ is 4. Since the number 4 appears the most frequently.

A data set can have more than one mode. For example, both 2 and 3 are modes in the set 1, 2, 2, 3, 3. If all points in a data set occur with equal frequency, it is equally accurate to describe the data set as having many modes or no mode.

## Measures of dispersion

The mean, median and mode are measures of central tendency, i.e. they provide information on the central data values in a set. When describing data it is sometimes useful (and in some cases necessary) to determine the spread of a distribution. Measures of dispersion provide information on how the data values in a set are distributed around the mean value. Some measures of dispersion are range, percentiles and quartiles.

## Range

Range

The range of a data set is the difference between the lowest value and the highest value in the set.

Method: Calculating the range

1. Find the highest value in the data set.
2. Find the lowest value in the data set.
3. Subtract the lowest value from the highest value. The difference is the range.

Find the range of the data set $x=\left\{1,2,3,4,4,4,5,6,7,8,8,9,10,10\right\}$

1. 10 is the highest value and 1 is the lowest value.

2. $10-1=9$
3. For the data set $x=\left\{1,2,3,4,4,4,5,6,7,8,8,9,10,10\right\}$ , the range is 9.

## Quartiles

Quartiles

Quartiles are the three data values that divide an ordered data set into four groups containing equal numbers of data values. The median is the second quartile.

The quartiles of a data set are formed by the two boundaries on either side of the median, which divide the set into four equal sections. The lowest 25% of the data being found below the first quartile value, also called the lower quartile. The median, or second quartile divides the set into two equal sections. The lowest 75% of the data set should be found below the third quartile, also called the upper quartile. For example:

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