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Measures Of Dispersion

10.4 Measures of dispersion (EMA76)

The central tendency is not the only interesting or useful information about a data set. The two data sets illustrated below have the same mean (\(\text{0}\)), but have different spreads around the mean. Each circle represents one value from the data set (or one datum).

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Dispersion is a general term for different statistics that describe how values are distributed around the centre. In this section we will look at measures of dispersion.

Range (EMA77)

Range

The range of a data set is the difference between the maximum and minimum values in the set.

The most straightforward measure of dispersion is the range. The range simply tells us how far apart the largest and smallest values in a data set are. The range is very sensitive to outliers.

Worked example 10: Range

Find the range of the following data set:

\[\left\{1; 4; 5; 8; 6; 7; 5; 6; 7; 4; 10; 9; 10\right\}\]

What would happen if we removed the first value from the set?

Determine the range

The smallest value in the data set is \(\text{1}\) and the largest value is \(\text{10}\).

The range is \(10 - 1 = 9\)

Remove the first value

If the first value, \(\text{1}\), were to be removed from the set, the minimum value would be \(\text{4}\). This means that the range would change to \(10 - 4 = 6\). \(\text{1}\) is not typical of the other values. It is an outlier and has a big influence on the range.

Percentiles (EMA78)

Percentile

The \(p^{\text{th}}\) percentile is the value, \(v\), that divides a data set into two parts, such that \(p\) percent of the values in the data set are less than \(v\) and \(100 - p\) percent of the values are greater than \(v\). Percentiles can lie in the range \(0\le p\le 100\).

To understand percentiles properly, we need to distinguish between \(\text{3}\) different aspects of a datum: its value, its rank and its percentile:

  • The value of a datum is what we measured and recorded during an experiment or survey.

  • The rank of a datum is its position in the sorted data set (for example, first, second, third, and so on).

  • The percentile at which a particular datum is, tells us what percentage of the values in the full data set are less than this datum.

The table below summarises the value, rank and percentile of the data set:

\[\left\{\text{14,2}; \text{13,9}; \text{19,8}; \text{10,3}; \text{13,0}; \text{11,1}\right\}\]

Value

Rank

Percentile

\(\text{10,3}\)

\(\text{1}\)

\(\text{0}\)

\(\text{11,1}\)

\(\text{2}\)

\(\text{20}\)

\(\text{13,0}\)

\(\text{3}\)

\(\text{40}\)

\(\text{13,9}\)

\(\text{4}\)

\(\text{60}\)

\(\text{14,2}\)

\(\text{5}\)

\(\text{80}\)

\(\text{19,8}\)

\(\text{6}\)

\(\text{100}\)

As an example, \(\text{13,0}\) is at the \(40^{\text{th}}\) percentile since there are \(\text{2}\) values less than \(\text{13,0}\) and \(\text{3}\) values greater than \(\text{13,0}\).

\[\frac{2}{2+3} = \text{0,4} = \text{40}\%\]

In general, the formula for finding the \(p^{\text{th}}\) percentile in an ordered data set with \(n\) values is

\[r = \frac{p}{100}\left(n - 1\right) + 1\]

This gives us the rank, \(r\), of the \(p^{\text{th}}\) percentile. To find the value of the \(p^{\text{th}}\) percentile, we have to count from the first value in the ordered data set up to the \(r^{\text{th}}\) value.

Sometimes the rank will not be an integer. This means that the percentile lies between two values in the data set. The convention is to take the value halfway between the two values indicated by the rank.

The figure below shows the relationship between rank and percentile graphically. We have already encountered three percentiles in this chapter: the median (\(50^{\text{th}}\) percentile), the minimum (\(0^{\text{th}}\) percentile) and the maximum (\(100^{\text{th}}\)). The median is defined as the value halfway in a sorted data set.

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Worked example 11: Using the percentile formula

Determine the minimum, maximum and median values of the following data set using the percentile formula.

\[\left\{14; 17; 45; 20; 19; 36; 7; 30; 8\right\}\]

Sort the values in the data set

Before we can use the rank to find values in the data set, we always have to order the values from the smallest to the largest. The sorted data set is

\[\left\{7; 8; 14; 17; 19; 20; 30; 36; 45\right\}\]

Find the minimum

We already know that the minimum value is the first value in the ordered data set. We will now confirm that the percentile formula gives the same answer. The minimum is equivalent to the \(0^{\text{th}}\) percentile. According to the percentile formula the rank, \(r\), of the \(p = 0^{\text{th}}\) percentile in a data set with \(n = 9\) values is:

\begin{align*} r & = \frac{p}{100}\left(n - 1\right) + 1 \\ & = \frac{0}{100}\left(9 - 1\right) + 1 \\ & = 1 \end{align*}

This confirms that the minimum value is the first value in the list, namely \(\text{7}\).

Find the maximum

We already know that the maximum value is the last value in the ordered data set. The maximum is also equivalent to the \(100^{\text{th}}\) percentile. Using the percentile formula with \(p = 100\) and \(n = 9\), we find the rank of the maximum value is:

\begin{align*} r& = \frac{p}{100}\left(n - 1\right) + 1 \\ & = \frac{100}{100}\left(9 - 1\right) + 1 \\ & = 9 \end{align*}

This confirms that the maximum value is the last (the ninth) value in the list, namely \(\text{45}\).

Find the median

The median is equivalent to the \(50^{\text{th}}\) percentile. Using the percentile formula with \(p = 50\) and \(n = 9\), we find the rank of the median value is:

\begin{align*} r & =\frac{50}{100}\left(n - 1\right) + 1 \\ & = \frac{50}{100}\left(9 - 1\right) + 1 \\ & = \frac{1}{2}\left(8\right) + 1 \\ & = 5 \end{align*}

This shows that the median is in the middle (at the fifth position) of the ordered data set. Therefore the median value is \(\text{19}\).

Quartiles

The quartiles are the three data values that divide an ordered data set into four groups, where each group contains an equal number of data values. The median (\(50^{\text{th}}\) percentile) is the second quartile (\(Q2\)). The \(25^{\text{th}}\) percentile is also called the first or lower quartile (\(Q1\)). The \(75^{\text{th}}\) percentile is also called the third or upper quartile (\(Q3\)).

Worked example 12: Quartiles

Determine the quartiles of the following data set:

\[\left\{7; 45; 11; 3; 9; 35; 31; 7; 16; 40; 12; 6\right\}\]

Sort the data set

\[\left\{3; 6; 7; 7; 9; 11; 12; 16; 31; 35; 40; 45\right\}\]

Find the ranks of the quartiles

Using the percentile formula with \(n = 12\), we can find the rank of the \(25^{\text{th}}\), \(50^{\text{th}}\) and \(75^{\text{th}}\) percentiles:

\begin{align*} {r}_{25} & = \frac{25}{100}\left(12 - 1\right) + 1 \\ & = \text{3,75} \\ {r}_{50} & = \frac{50}{100}\left(12 - 1\right) + 1 \\ & = \text{6,5} \\ {r}_{75} & = \frac{75}{100}\left(12 - 1\right) + 1 \\ & = \text{9,25} \end{align*}

Find the values of the quartiles

Note that each of these ranks is a fraction, meaning that the value for each percentile is somewhere in between two values from the data set.

For the \(25^{\text{th}}\) percentile the rank is \(\text{3,75}\), which is between the third and fourth values. Since both these values are equal to \(\text{7}\), the \(25^{\text{th}}\) percentile is \(\text{7}\).

For the \(50^{\text{th}}\) percentile (the median) the rank is \(\text{6,5}\), meaning halfway between the sixth and seventh values. The sixth value is \(\text{11}\) and the seventh value is \(\text{12}\), which means that the median is \(\frac{11 + 12}{2} = \text{11,5}\). For the \(75^{\text{th}}\) percentile the rank is \(\text{9,25}\), meaning between the ninth and tenth values. Therefore the \(75^{\text{th}}\) percentile is \(\frac{31 + 35}{2} = 33\).

Deciles

The deciles are the nine data values that divide an ordered data set into ten groups, where each group contains an equal number of data values.

For example, consider the ordered data set:

\begin{align*} 28; 33; 35; 45; 57; 59; 61; 68; 69; 72; 75; 78; 80; 83; 86; 91; \\ 92; 95; 101; 105; 111; 117; 118; 125; 127; 131; 137; 139; 141 \end{align*}

The nine deciles are: \(35; 59; 69; 78; 86; 95; 111; 125; 137\).

Percentiles for grouped data (EMA79)

In grouped data, the percentiles will lie somewhere inside a range, rather than at a specific value. To find the range in which a percentile lies, we still use the percentile formula to determine the rank of the percentile and then find the range within which that rank is.

Worked example 13: Percentiles in grouped data

The mathematics marks of \(\text{100}\) grade \(\text{10}\) learners at a school have been collected. The data are presented in the following table:

Percentage mark

Number of learners

\(0 \le x < 20\)

2

\(20 \le x < 30\)

5

\(30 \le x < 40\)

18

\(40 \le x < 50\)

22

\(50 \le x < 60\)

18

\(60 \le x < 70\)

13

\(70 \le x < 80\)

12

\(80 \le x < 100\)

10

  1. Calculate the mean of this grouped data set.

  2. In which intervals are the quartiles of the data set?

  3. In which interval is the \(30^{\text{th}}\) percentile of the data set?

Calculate the mean

Since we are given grouped data rather than the original ungrouped data, the best we can do is approximate the mean as if all the learners in each interval were located at the central value of the interval.

\begin{align*} \text{Mean } & = \frac{2(10) + 5(25) + 18(35) + 22(45) + 18(55) + 13(65) + 12(75) + 10(90)}{100} \\ & = \text{54}\% \end{align*}

Find the quartiles

Since the data have been grouped, they have also already been sorted. Using the percentile formula and the fact that there are \(\text{100}\) learners, we can find the rank of the \(25^{\text{th}}\), \(50^{\text{th}}\) and \(75^{\text{th}}\) percentiles as

\begin{align*} {r}_{25} & =\frac{25}{100}\left(100 - 1\right) + 1 \\ & = \text{24,75} \\ {r}_{50}& = \frac{50}{100}\left(100 - 1\right) + 1 \\ & = \text{50,5} \\ {r}_{75} & = \frac{75}{100}\left(100 - 1\right) + 1 \\ & = \text{75,25} \end{align*}

Now we need to find in which ranges each of these ranks lie.

  • For the lower quartile, we have that there are \(2 + 5 = 7\) learners in the first two ranges combined and \(2 + 5 + 18 = 25\) learners in the first three ranges combined. Since \(7 < {r}_{25} < 25\), this means the lower quartile lies somewhere in the third range: \(30 \le x < 40\).

  • For the second quartile (the median), we have that there are \(2 + 5 + 18 + 22 = 47\) learners in the first four ranges combined. Since \(47 < {r}_{50} < 65\), this means that the median lies somewhere in the fifth range: \(50 \le x < 60\).

  • For the upper quartile, we have that there are \(\text{65}\) learners in the first five ranges combined and \(65 + 13 = 78\) learners in the first six ranges combined. Since \(65 < {r}_{75} < 78\), this means that the upper quartile lies somewhere in the sixth range: \(60 \le x < 70\).

Find the \(30^{\text{th}}\) percentile

Using the same method as for the quartiles, we first find the rank of the \(30^{\text{th}}\) percentile.

\begin{align*} r & = \frac{30}{100}\left(100 - 1\right) + 1 \\ & = \text{30,7} \end{align*}

Now we have to find the range in which this rank lies. Since there are \(\text{25}\) learners in the first \(\text{3}\) ranges combined and \(\text{47}\) learners in the first \(\text{4}\) ranges combined, the \(30^{\text{th}}\) percentile lies in the fourth range: \(40 \le x < 50\)

Ranges (EMA7B)

We define data ranges in terms of percentiles. We have already encountered the full data range, which is simply the difference between the \(100^{\text{th}}\) and the \(0^{\text{th}}\) percentile (that is, between the maximum and minimum values in the data set).

Interquartile range

The interquartile range is a measure of dispersion, which is calculated by subtracting the first quartile (\(Q1\)) from the third quartile (\(Q3\)). This gives the range of the middle half of the data set.

Semi interquartile range

The semi interquartile range is half of the interquartile range.

Exercise 10.5

A group of \(\text{15}\) learners count the number of sweets they each have. This is the data they collect:

\[\begin{array}{c c c c c} 4 & 11 & 14 & 7 & 14 \\ 5 & 8 & 7 & 12 & 12 \\ 5 & 13 & 10 & 6 & 7 \end{array}\]

Calculate the range of values in the data set.

We first need to order the data set:

\[\{4; 5; 5; 6; 7; 7; 7; 8; 10; 11; 12; 12; 13; 14; 14\}\]

Next we find the maximum value in the data set:

\[\text{maximum value } = \text{14}\]

Then we find the minimum value in the data set:

\[\text{minimum value } = \text{4}\]

Finally, we calculate the range of the data set:

\begin{align*} \text{range } & = \text{(maximum value) } - \text{ (minimum value)}\\ & = (\text{14}) - (\text{4}) \\ & = \text{10} \end{align*}

A group of \(\text{10}\) learners count the number of playing cards they each have. This is the data they collect:

\[\begin{array}{c c c c c} 5 & 1 & 3 & 1 & 4 \\ 10 & 1 & 3 & 3 & 4 \end{array}\]

Calculate the range of values in the data set.

We first need to order the data set:

\[\{1; 1; 1; 3; 3; 3; 4; 4; 5; 10\}\]

Next we find the maximum value in the data set:

\[\text{maximum value } = \text{10}\]

Then we find the minimum value in the data set:

\[\text{minimum value } = \text{1}\]

Finally, we calculate the range of the data set:

\begin{align*} \text{range} & = \text{(maximum value) } - \text{ (minimum value)}\\ & = \text{10} - \text{1} \\ & = \text{9} \end{align*}

Find the range of the data set

\[\left\{1; 2; 3; 4; 4; 4; 5; 6; 7; 8; 8; 9; 10; 10\right\}\]

The data set is already ordered.

Firstly, we find the maximum value in the data set:

\[\text{maximum value } = \text{10}\]

Secondly, we find the minimum value in the data set:

\[\text{minimum value } = \text{1}\]

Finally, we calculate the range of the data set:

\begin{align*} \text{range} & = \text{(maximum value) } - \text{ (minimum value)}\\ & = \text{10} - \text{1} \\ & = \text{9} \end{align*}

What are the quartiles of this data set?

\[\left\{3; 5; 1; 8; 9; 12; 25; 28; 24; 30; 41; 50\right\}\]

We first order the data set.

\[\left\{1; 3; 5; 8; 9; 12; 24; 25; 28; 30; 41; 50\right\}\]

Next we find the ranks of the quartiles. Using the percentile formula with \(n = 12\), we can find the rank of the \(25^{\text{th}}\), \(50^{\text{th}}\) and \(75^{\text{th}}\) percentiles:

\begin{align*} {r}_{25} & = \frac{25}{100}\left(12 - 1\right) + 1 \\ & = \text{3,75} \\ {r}_{50} & = \frac{50}{100}\left(12 - 1\right) + 1 \\ & = \text{6,5} \\ {r}_{75} & = \frac{75}{100}\left(12 - 1\right) + 1 \\ & = \text{9,25} \end{align*}

Find the values of the quartiles. Note that each of these ranks is a fraction, meaning that the value for each percentile is somewhere in between two values from the data set.

For the \(25^{\text{th}}\) percentile the rank is \(\text{3,75}\), which is between the third and fourth values. Therefore the \(25^{\text{th}}\) percentile is \(\frac{5 + 8}{2} = \text{6,5}\).

For the \(50^{\text{th}}\) percentile (the median) the rank is \(\text{6,5}\), meaning halfway between the sixth and seventh values. Therefore the median is \(\frac{12 + 24}{2} = \text{18}\). For the \(75^{\text{th}}\) percentile the rank is \(\text{9,25}\), meaning between the ninth and tenth values. Therefore the \(75^{\text{th}}\) percentile is \(\frac{28 + 30}{2} = 29\).

Therefore we get the following values for the quartiles: \(Q_1 = \text{6,5}\); \(Q_2 = 18\); \(Q_3 = 29\).

A class of \(\text{12}\) learners writes a test and the results are as follows:

\[\left\{20; 39; 40; 43; 43; 46; 53; 58; 63; 70; 75; 91\right\}\]

Find the range, quartiles and the interquartile range.

The data set is ordered.

The range is:

\begin{align*} \text{range} & = \text{(maximum value) } - \text{ (minimum value)}\\ & = (\text{91}) - (\text{20}) \\ & = \text{71} \end{align*}

To find the quartiles we start by finding the ranks of the quartiles. Using the percentile formula with \(n = 12\), we can find the rank of the \(25^{\text{th}}\), \(50^{\text{th}}\) and \(75^{\text{th}}\) percentiles:

\begin{align*} {r}_{25} & = \frac{25}{100}\left(12 - 1\right) + 1 \\ & = \text{3,75} \\ {r}_{50} & = \frac{50}{100}\left(12 - 1\right) + 1 \\ & = \text{6,5} \\ {r}_{75} & = \frac{75}{100}\left(12 - 1\right) + 1 \\ & = \text{9,25} \end{align*}

Find the values of the quartiles. Note that each of these ranks is a fraction, meaning that the value for each percentile is somewhere in between two values from the data set.

For the \(25^{\text{th}}\) percentile the rank is \(\text{3,75}\), which is between the third and fourth values. Therefore the \(25^{\text{th}}\) percentile is \(\frac{40 + 43}{2} = \text{41,5}\).

For the \(50^{\text{th}}\) percentile (the median) the rank is \(\text{6,5}\), meaning halfway between the sixth and seventh values. Therefore the median is \(\frac{46 + 53}{2} = \text{49,5}\). For the \(75^{\text{th}}\) percentile the rank is \(\text{9,25}\), meaning between the ninth and tenth values. Therefore the \(75^{\text{th}}\) percentile is \(\frac{63 + 70}{2} = \text{66,5}\).

Therefore we get the following values for the quartiles: \(Q_1 = \text{41,5}\); \(Q_2 = \text{49,5}\); \(Q_3 = \text{66,5}\).

Interquartile range:

\begin{align*} \text{interquartile range } & = \text{quartile 3 } - \text{ quartile 1} \\ & = \text{66,5} - \text{41,5} \\ & = 25 \end{align*}

Three sets of data are given:

\begin{align*} \textbf{Data set 1: } & \left\{9; 12; 12; 14; 16; 22; 24\right\} \\ \textbf{Data set 2: } & \left\{7; 7; 8; 11; 13; 15; 16\right\} \\ \textbf{Data set 3: } & \left\{11; 15; 16; 17; 19; 22; 24\right\} \end{align*}

For each data set find:

the range

All three data sets are ordered. To find the range we subtract the minimum value from the maximum value. Doing so for each data set gives the following values for the range.

Data set 1: \(24 - 9 = 15\)

Data set 2: \(16 - 7 = 9\)

Data set 3: \(24 - 11 = 13\)

the lower quartile

For each data set \(n = 7\). Therefore the rank of the \(25^{\text{th}}\) percentile is the same for each data set: \({r}_{25} = \frac{25}{100}\left(7 - 1\right) + 1 = \text{2,5}\). Therefore for each data set the lower quartile lies between the second and third values.

The lower quartile for each data set is:

Data set 1: \(\text{12}\)

Data set 2: \(\text{7,5}\)

Data set 3: \(\text{15,5}\)

the median

For each data set \(n = 7\). Therefore the rank of the \(50^{\text{th}}\) percentile is the same for each data set: \({r}_{50} = \frac{50}{100}\left(7 - 1\right) + 1 = \text{4}\). Therefore for each data set the median is the fourth value.

The median for each data set is:

Data set 1: \(\text{14}\)

Data set 2: \(\text{11}\)

Data set 3: \(\text{17}\)

the upper quartile

For each data set \(n = 7\). Therefore the rank of the \(75^{\text{th}}\) percentile is the same for each data set: \({r}_{75} = \frac{75}{100}\left(7 - 1\right) + 1 = \text{5,5}\). Therefore for each data set the lower quartile lies between the fifth and sixth values.

The upper quartile for each data set is:

Data set 1: \(\text{19}\)

Data set 2: \(\text{14}\)

Data set 3: \(\text{20,5}\)

the interquartile range

The interquartile range is calculated by subtracting the lower quartile from the upper quartile.

Data set 1: \(19 - 12 = \text{7}\)

Data set 2: \(14 - \text{7,5} = \text{6,5}\)

Data set 3: \(\text{20,5} - \text{15,5} = \text{5}\)

the semi-interquartile range

The semi-interquartile range is half the interquartile range.

Data set 1: \(\frac{7}{2} = \text{3,5}\)

Data set 2: \(\frac{\text{6,5}}{2} = \text{3,25}\)

Data set 3: \(\frac{5}{2} = \text{2,5}\)