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# Chapter 11: Summary statistics

It is often useful to describe data by giving a typical or average value of the data. As you have learnt in previous years, the typical values that are used most often are the median, mode and mean.

Each of these three measures is a different way of working out a value that is close to the centre of a data set, so they are called measures of central tendency. They are a way of summarising the data set, so are also one kind of summary statistics.

measures of central tendency The measures of central tendency summarise a data set by giving a value that is close to the centre of a data set.

In this chapter, you will revise the median, mode and mean. You will also learn about the range, which is another kind of summary statistic that summarises a data set. Then you will compare data sets, using the mean and the range.

## 11.1 Revise median, mode and mean

### The median

The median is the middle value of a set of values when all the values are placed in order of size.

• If there is an odd number of values, there will be one middle value.
• If there is an even number of values, there will be two middle values and the median is taken to be mid-way between the two values.

To find the median, do the following:

• Put the values in ascending order first.
• Find the middle value as follows:
• If you have an odd number of values, count from either end until you end up with the middle value.
• If you have an even number of values, two values share the middle place. To find the median in this case, add the two middle values and then divide by 2.

median The median is the middle value in a data set that has been arranged in ascending order.

Only numbers and measurements can have a middle value or median. Other things such as names, cars, colours, and so on, cannot be put in order of size, so they cannot have a median or middle value.

### Worked example 11.1: Finding the median of an odd number of values

Find the median of the following data set:

1. Step 1: Write the values in ascending order.

2. Step 2: Find the middle value.

We have 7 values. This is an odd number of values, so the middle value is the 4th term.

Median: 6

### Worked example 11.2: Finding the median of an even number of values

Find the median in this set of data:

1. Step 1: Write the values in ascending order.

2. Step 2: Find the middle value.

We have 10 values. This is an even number of values, so the middle value lies between the 5th and 6th values.

The 5th value is 8, and the 6th value is also 8.

To find the median we need to add the two middle values, 8 and 8, and divide by 2.

Median = $\dfrac{\text{8 + 8}}{\text{2}} = \text{8}$.

Median: 8

### Exercise 11.1: Find the median of a set of values

1. Find the median of the following set of data: $\text{6; 10; 8; 6; 11; 5; 10; 11; 6; 7; 5}$.

Write the values in ascending order: $\text{5; 5; 6; 6; 6; 7; 8; 10; 10; 11; 11}$

Find the middle value.

There are 11 values. This is an odd number of values, so the middle term is the 6th term, which is 7.

Median: 7

2. Determine the median of the following set of data: $\text{6; 6; 14; 14; 13; 14; 15; 13; 18; 6}$.

Write the values in ascending order: $\text{6; 6; 6; 13; 13; 14; 14; 14; 15; 18}$

Find the middle value.

There are 10 values. This is an even number of values, so the middle values lie between the 5th and 6th values.

The 5th value is 13, and the 6th value is 14.

Median = $\dfrac{\text{13 + 14}}{\text{2}} = \dfrac{\text{27}}{\text{2}} = \text{13.5}$.

Median: 13.5

### The mode

The mode is the value that occurs most often in a set of values. It is the most common or most popular value.

• In some sets of data, all the data items are different. In these sets, there is no mode.
• Some sets of data have only one mode.
• Some sets of data have more than one mode.

mode The mode is the value that occurs most often in a data set.

You can find a mode for any type of data. The data can be numbers, measurements, names, days of the week, colours, and so on. As long as there is a data item that occurs more than once in the data set, you can find a mode.

### Worked example 11.3: Finding the mode of a set of data with a single mode

Find the mode of the following set of data:

1. Step 1: Write the values in ascending order.

2. Step 2: Find the value that appears most often.

The 7 appears three times, which is more than any other number appears.

Mode: 7

### Worked example 11.4: Finding the mode of a set of data with more than one mode

Find the mode of the following set of data.

1. Step 1: Write the values in ascending order.

2. Step 2: Find the value or values that occurs most often.

The 5, 7 and 11 each appear twice.

Modes: 5; 7; 11

### Exercise 11.2: Find the mode of sets of data

1. Find the mode of the following data set: $\text{18; 4; 4; 6; 4; 11; 4; 8; 11; 4; 4}$.

Write the values in ascending order: $\text{4; 4; 4; 4; 4; 4; 6; 8; 11; 11; 18}$

The mode is the value that appears most often. Find this value (or values).

The 4 appears six times in the data set, which is more often than any other number appears.

Mode: 4

2. Determine the mode of the data set $\text{5; 4; 4; 4; 4; 4; 3; 5; 4; 15}$.

In ascending order: $\text{3; 4; 4; 4; 4; 4; 4; 5; 5; 15}$

The mode is 4 as it occurs six times in the data set.

Mode: 4

3. Determine the mode of the following data set: $\text{7; 6; 10; 7; 8; 6; 9; 11; 10; 11; 9}$.

In ascending order: $\text{6; 6; 7; 7; 8; 9; 9; 10; 10; 11}$

The 6, 7, 9 and 10 each appear twice.

Modes: 6; 7; 9; 11

### The mean

The mean gives the average of a data set.

To work out the mean of a set of values, add up all the values, and then divide by how many values you have.

mean The mean gives the average of a data set. It is calculated by adding the values in the data set and dividing the total by the number of values.

• When the values are measurements, make sure the units are all the same before you carry out the calculation.
• Always check that you have the correct unit, such as cm, kg or ml, in your answer.
• You cannot find the mean of data items such as colours, car makes, and so on. The data set must consist of numbers or measurements such as lengths and masses.

### Worked example 11.5: Sharing out equally to find the mean

Nnenne bought ten different packets of nuts from the market. She counted the nuts in each packet and found that they held $\text{15; 15; 19; 20; 18; 20; 17; 15; 21}$ and $20$ nuts.

She decided to put all the nuts together and to then share this total amount out equally between the ten packets. How many nuts should she put in each packet?

When the nuts are shared out equally, the number of nuts in each packet is the mean, which we calculate in the usual way.

1. Step 1: Add together the number of nuts in each packet to find out how many nuts there are altogether.

Sum of the number of nuts = $\text{15 + 15 + 19 + 20 + 18 + 20 + 17 + 15 + 21 + 20}$ = $\text{180}$

2. Step 2: Note the number of packets.

The number of packets = 10.

3. Step 3: Use the formula to find the mean.

Mean = $\dfrac{\text{sum of the number of nuts}}{\text{number of packets}} = \dfrac{\text{180}}{\text{10}} = \text{18}$ nuts

This means that Nnenne should put 18 nuts into each packet, because 18 is the mean or average number of nuts in a packet.

Nnenne should put 18 nuts in each packet.

### Worked example 11.6: Finding the mean

Consider the following set of data: $\text{7; 11; 7; 11; 5; 10; 7; 10}$.

What is the mean value of the data set?

1. Step 1: Start by finding the sum by adding all the data values.

Sum of the values = $\text{7}$ + $\text{11}$ + $\text{7}$ + $\text{11}$ + $\text{5}$ + $\text{10}$ + $\text{7}$ + $\text{10}$ = $\text{68}$

2. Step 2: Count the number of data values.

There are 8 data values.

3. Step 3: Calculate the mean by dividing the sum of the values by the number of values.

Mean = $\dfrac{\text{sum of the values}}{\text{number of values}} = \dfrac{\text{68}}{\text{8}} = \text{8.5}$

Mean: 8.5

### Exercise 11.3: Find the mean

1. Find the mean of the following data set:

To find the mean, we add all the values of the data and divide by the number of data values.

Sum of the values = $\text{6}$ + $\text{8}$ + $\text{7}$ + $\text{10}$ + $\text{8}$ + $\text{7}$ + $\text{10}$ + $\text{9}$ + $\text{7}$ = $\text{72}$

Number of values = 9

Mean = $\dfrac{\text{sum of the values}}{\text{number of values}} = \dfrac{\text{72}}{\text{9}} = \text{8}$

Mean: 8

2. Find the mean of the following set of data: $\text{11; 6; 8; 7; 9; 9; 10}$. Round your answer to two decimal places.

Sum of the values = $\text{11}$ + $\text{6}$ + $\text{8}$ + $\text{7}$ + $\text{9}$ + $\text{9}$ + $\text{10}$ = $\text{60}$

Number of values = 7

Mean = $\dfrac{\text{sum of the values}}{\text{number of values}} = \dfrac{\text{60}}{\text{7}} = \text{8.57142...} \approx 8.57$

Mean to two decimal places: 8.57

3. Adanna measured the rainfall each day for one week. She recorded her results, in millimetres, in the table below:

Mon Tues Wed Thurs Fri Sat Sun
0.9 1.2 0.8 1.3 1.1 1.7 1.1

Calculate the mean daily rainfall. Round your answer to two decimal places.

Find the sum of the daily rainfall in millimetres.

Sum of the rainfall (in mm) = $\text{0.9}$ + $\text{1.2}$ + $\text{0.8}$ + $\text{1.3}$ + $\text{1.1}$ + $\text{1.7}$ + $\text{1.1}$ = $\text{8.1}$

Number of days = 7

Mean daily rainfall = $\dfrac{\text{sum of the rainfall (in mm)}}{\text{number of days}} = \dfrac{\text{8.1}}{\text{7}} = \text{1.15714...mm} \approx \text{1.16 mm}$

Mean daily rainfall to two decimal places: 1.16 mm

### Working with the mean

If you know the mean and the number of values used to find it, then you can find the total of the values.

We know that:

From the equation above, we can get to the equation below:

### Worked example 11.7: Using the mean to find the total number of values

Seven friends are playing a card game. Each friend has some playing cards. They work out that the average number of playing cards each friend has is 6 playing cards. So the mean of this data set is 6.

Then two friends leave, and take an unknown number of playing cards with them. The remaining five friends work out that they now have a mean of 7.6 playing cards.

How many playing cards were taken away by the two friends?

1. Step 1: Work out the total number of playing cards there were at the beginning of the game.

At the beginning, there were 7 people in the group, and they had a mean of 6 playing cards.

To find the original number of playing cards, we use the formula for the mean:

$\text{Mean}$ = $\dfrac{\text{sum of the values}}{\text{number of values}}$ = $\dfrac{\text{original number of playing cards}}{\text{group size}}$

$\therefore \text{Original number of playing cards}$ = $\text{mean}\times \text{group size}$ = $\text{6}\times\text{7}$ = $\text{42}$

2. Step 2: Find the total number of playing cards after the two friends leave.

After the two people leave, there are 5 people left in the group, and they have a mean of 7.6 playing cards.

To find the final number of playing cards we use the formula for the mean:

$\text{Mean}$ = $\dfrac{\text{final number of playing cards}}{\text{group size}}$

$\therefore \text{Final number of playing cards}$ = $\text{mean}\times \text{group size}$ = $\text{7.6}\times\text{5}$ = $\text{38}$

3. Step 3: Work out the number of playing cards that were taken away by subtracting the second total from the first.

Four playing cards were taken away by the two friends.

### Exercise 11.4: Use the mean to find the total number of values

1. A group of six friends each have some sweets. They work out that they have a mean of 6 sweets each.

Then two friends eat all their sweets. The four friends who still have sweets work out that they now have a mean of 8 sweets each.

How many sweets did the two friends eat?

Before the two friends ate their sweets, the mean number of sweets was 6 and there were 6 people in the group.

$\text{Mean}$ = $\dfrac{\text{original number of sweets}}{\text{group size}}$

$\therefore \text{Original number of sweets}$ = $\text{mean}\times \text{group size}$ = $\text{6}\times\text{6}$ = $\text{36}$

After the two friends ate their sweets, there are four people left with sweets, and the new mean number of sweets is 8.

$\text{Mean}$ = $\dfrac{\text{number of sweets after friends eat their sweets}}{\text{group size}}$

$\therefore \text{Number of sweets after friends eat their sweets}$ = $\text{mean}\times \text{group size}$ = $\text{8}\times\text{4}$ = $\text{32}$

The two friends ate 4 sweets.

2. A group of seven marble players each have some marbles. They work out that they have a mean of 1 marble each.

Then two players leave with an unknown number of marbles.

The remaining five players work out that they now each have a mean of 0.6 marbles.

How many marbles did the two players take away from the group?

In the beginning, the mean number of marbles was 1 and there were 7 people in the group.

$\text{Original number of marbles}$ = $\text{mean}\times \text{group size}$ = $\text{1}\times\text{7}$ = $\text{7}$

After the two people leave, there are 5 people left in the group and they have a mean of 0.6 marbles each.

$\text{Final number of marbles}$ = $\text{mean}\times \text{group size}$ = $\text{0.6}\times\text{5}$ = $\text{3}$

The two players took 4 marbles away from the group.

### Finding all three measures of central tendency

As you have learnt, the three measures of central tendency give slightly different representations of a data set. It is useful to determine all three measures for one data set, as this gives more information, and therefore a better summary of the data set, than only one measure can give.

### Worked example 11.8: Finding three measures of central tendency

Consider the following set of data: $\text{7; 6; 10; 11; 9; 6; 5}$.

Find the three measures of central tendency for this set of data. Give answers to two decimal places where necessary.

1. Step 1: Calculate the mean by adding up the numbers in the set and dividing by how many numbers are in the set.

Sum of the values = $\text{7 + 6 + 10 + 11 + 9 + 6 + 5 = 54 }$

The number of values = 7

$\text{Mean}$ = $\dfrac{\text{sum of all the values}}{\text{number of values}}$ = $\frac{54}{7}$ = $\text{7.71428...}$

Mean: 7.71

2. Step 2: To find the mode and the median, the data need to be arranged in ascending order.

The sorted set of data is $\text{5; 6; 6; 7; 9; 10; 11}$.

3. Step 3: Find the mode, which is the value (or values) that appear the most often.

In this data set the mode is 6, which appears twice.

Mode: 6

4. Step 4: Find the median, which is the value in the exact middle of the ordered data.

Here we have 7 values. This is an odd number of values, so the middle term is the 4th term.

Median: 7

5. Step 5: List the three measures of central tendency.

The three measures of central tendency are: mean 7.71; mode 6; median 7.

### Exercise 11.5: Find three measures of central tendency

Find the three measures of central tendency (the mean, mode and median) of the following sets of data. Round to two decimal places where necessary.

1. Calculate the mean by adding up the numbers in the set and dividing by the number of values in the set.

Mean: number of values = 12

Sum of the values = $\text{7 + 5 + 9 + 6 + 7 + 5 + 11 + 6 + 8 + 11 + 7 + 9 = 91}$

In this data set, the mode is 7, which appears three times.

There are 12 values in the data set, which is an even number of values, so the middle values lie between the 6th and 7th values. The 6th value is 7, and the 7th value is also 7.

Mean: number of values = 7

Sum of the values = $39 + 28 + 55 + 55 + 11 + 18 + 11 = 217$

\begin{array}{ll} \text{Mean} & {= \frac{217}{7}} \ & {= \text{31}} \ \end{array}

In ascending order: $11;\ 11;\ 18;\ 28;\ 39;\ 55;\ 55$

Modes: 11 and 55

Median: 28

The three measures of central tendency are: mean 31; modes 11 and 55; median 28.

## 11.2 The range

A measure of central tendency like the median, mode and mean give you one, two, or sometimes a few values that summarise the data.

A measure of spread is another way of summarising what a set of data tells us. A measure of spread tells us how spread out the data is. So it tells us if the data items are close to each other, or if they are very far apart from each other.

For example, here are two sets of data, arranged in ascending order:

A: $13;\ 13;\ 14;\ 15;\ 15;\ 15;\ 16;\ 17$

B: $4;\ 6;\ 9;\ 9;\ 13;\ 18;\ 24;\ 27;\ 27$

Just by looking at these data sets, you can see that in data set A, the values are quite close to each other. But in data set B, the values are quite spread out. This could tell you that the results in the data in set A were all quite similar, but set B indicates results with bigger differences.

The range is the simplest measure of spread. It is the difference between the largest and smallest values in the data.

We find the range by subtracting the largest data item from the smallest data item. We can use the formula:

For example, for the two data sets above:

A: $\text{Range} = 17 - 13 = 4$

B: $\text{Range} = 27 - 4 = 23$

range The range is the difference between the largest and smallest values in the data set.

### Worked example 11.9: Finding the range of a set of data

Find the range of the following set of data.

1. Step 1: First arrange the data in ascending order.

2. Step 2: To find the range, subtract the lowest value from the highest value in the set.

Range: 6

### Exercise 11.6: Find the range of a set of data

1. Find the range of the following set of data: $6;\ 7;\ 8;\ 10;\ 5;\ 7;\ 9;\ 8$.

First arrange the data in ascending order:

The range is the difference between the highest and lowest values in the set:

Range: 5

2. For the set of data given below, find the range.

In ascending order: $5;\ 5;\ 6;\ 7;\ 7;\ 9;\ 9;\ 10;\ 11$

Range: 6

3. Find the range of this set of data: $16;\ 30;\ 16;\ 30;\ 8;\ 16;\ 27;\ 5$.

In ascending order: $5;\ 8;\ 16;\ 16;\ 16;\ 27;\ 30;\ 30$

Range: 25

4. Find the range of the given data set.

In ascending order: $105;\ 106;\ 106;\ 107;\ 107;\ 108;\ 108;\ 109;\ 110;\ 110;\ 111;\ 111$

Range: 6

### Working with the range of a data set

We can use the value of the range to calculate either the value of the largest data item or the smallest data item.

### Worked example 11.10: Using range to find largest value in a data set

Suppose a data set has a range of 4. If the smallest value in the data set is 49, what is the largest value in the data set?

Remember that the range of a data set tells us how spread out the data are by giving the difference between the largest and smallest data values.

1. Step 1: Study the data and list what we know.

We don't know the individual data values, but we do know that the range of the data set is 4, and the smallest value is 49.

2. Step 2: Use the equation for the range and put in the values we know.

3. Step 3: Find the largest value by solving the equation.

Or, if you use mental arithmetic, the smallest value in the data set is 49 and the range is 4. So the largest value is 4 more than 49.

The largest value in the data set is 53.

### Exercise 11.7: Use range to find smallest or largest value in a data set

1. Suppose a data set has a range of 13. If the largest value in the data set is 51, what is the smallest value in the data set?

Use the equation for the range and substitute the values that you know.

Or, using mental arithmetic, the largest value is 51 and the range is 13. So the smallest value is 13 less than 51.

The smallest value in the data set is 38.

2. Determine the largest value in a data set if the smallest value in the data set is 48 and the range of the data is 7.

Or, using mental arithmetic, the smallest value in the data set is 48 and the range is 7. So the largest value is 7 more than 48.

The largest value in the data set is 55.

## 11.3 Practical applications: comparing data using mean and range

The purpose of summary statistics is to replace a very large data set with just a few numbers that, together, tell us important facts about the whole data set.

These summary statistics can be used to compare two different data sets. The two measures that are often used to compare data sets are the mean and the range. This is demonstrated in the next worked example.

### Worked example 11.11: Comparing data using mean and range

Anouluwapo and Eniola each wrote ten tests. Each test was worth 20 marks.

Anouluwapo's marks were: $\text{ 20; 16; 10; 3; 12; 10; 11; 14; 5; 19 }$

Eniola's marks were: $\text{ 13; 12; 11; 13; 13; 11; 12; 12; 11; 12 }$

The two students each got a mean mark of 12 out of 20.

Work out the range for each student's marks, and then say which student's marks are more consistent or similar.

1. Step 1: Arrange each set of marks in ascending order.

Anouluwapo's marks in ascending order: $\text{ 3; 5; 10; 10; 11; 12; 14; 16; 19; 20 }$

Eniola's marks in ascending order: $\text{ 11; 11; 11; 12; 12; 12; 12; 13; 13; 13 }$

2. Step 2: Work out the range for each student's data.

Anouluwapo: $\text{Range of marks} = 20 - 3 = 17$

Eniola: $\text{Range of marks} = 13 - 11 = 2$

The range of Anouluwapo's marks is 17 and the range of Eniola's marks is 2.

3. Step 3: Look at the mean and the range of each student's marks and use them to explain whose marks are more consistent or similar.

Anouluwapu: Mean = 12 out of 20; Range = 17

Eniola: Mean = 12 out of 20; Range = 2

Anouluwapo's marks have a bigger range and are spread further apart. This means that her marks are sometimes good and sometimes bad.

Eniola's marks have a smaller range and are all nearly the same. This means that her marks are nearly all the same.

Therefore, Aniola's marks are more consistent.

### Exercise 11.8: Compare data using mean and range

1. Twelve boys ran the 100 metres race and their times were recorded.
• Their mean running time was 13.7 seconds.
• The range of their running times was 6.1 seconds.

Twelve girls ran the 100 metres race and their times were recorded.

• The mean of their running times was 14.3 seconds.
• The range of their running times was 5.6 seconds.

We can summarise this information in a table:

Mean time Range
Boys' times in seconds 13.7 6.1
Girls' times in seconds 14.3 5.6
1. Which group was faster?

Compare the mean of the boys' running times and the girls' running times and say which group was faster.

The mean of the boys' running times was 13.7 seconds.

The mean of the girls' running times was 14.3 seconds.

The mean of the boys' running times is less than the mean of the girls' running times.

Therefore, the boys are faster on average than the girls.

1. Which group's times were more consistent or similar?

Compare the ranges of the boys' running times and the girls' running times and say which group had the smallest differences in times.

The range of the boys' running times was 6.1 seconds.

The range of the girls' running times was 5.6 seconds.

The range of the girls' times is less than the range of the boys' times.

This means that the times for the girls are more consistent than the boys' times.

2. This table shows the marks (out of 50) obtained in eight maths tests by Ikenna and Ndidi.

Test 1 Test 2 Test 3 Test 4 Test 5 Test 6 Test 7 Test 8
Ikenna 28 36 41 36 29 30 29 32
Ndidi 11 48 18 49 21 16 15 27
1. Calculate the range of each student's marks.

Arrange each set of marks in ascending order and then find the range.

Ikenna's marks in ascending order: $\text{28; 29; 29; 30; 32; 36; 36; 41 }$

Range of Ikenna's marks: $41 - 28 = 13$

Ndidi's marks in ascending order: $\text{11; 15; 16; 18; 21; 27; 48; 49 }$

Range of Ndidi's marks: $49 - 11 = 38$

1. Which student's marks are more consistent?

The range of Ikenna's marks is less than the range Ndidi's marks. This means that Ikenna's marks are less spread out than Ndidi's marks.

Therefore, Ikenna's marks are more consistent.

1. Calculate each student's mean mark.
3. We can compare different data sets using the mean and the range.
• The mean gives us a value that is typical of each set of data, so we can compare two means.
• Comparing the range tells us which set of data is more spread out, and therefore less consistent.