## What you need to know

## Mean, Median, Mode, and Range

The mean**, **median**, **and mode are different types of average and the range tells us how spread out our data is. We use them when have a bunch of numbers, often data that we’ve collected, and we want to get a feel for how big/small that group of numbers in. Are they very high? Are they only a little bigger than zero? Are they closer to 10 or 20 in general? These are the types of questions we answer when finding an average.

**Mean**

The mean is the most popular kind of average. To find the mean we must add up all the numbers we’re finding the average of, and then divide by how many numbers there are in that list.

**Advantages and Disadvantages**

**Advantage**– every bit of data is used in calculating the mean, so it represents all the data.**Disadvantage**– it is highly affected by outliers. An outlier is a piece of data that doesn’t quite fit with the rest of them.

You need to be able to calculate the mean from different types of data, from a list of numbers to frequency tables and grouped frequency tables.

**Median**

The median is often referred to as “the middle”, which is precisely what it is. To find the median of a list of numbers, we put the numbers in order from smallest to largest and find the middle value/middle two values. If there is a middle value, then that is the median; if there are two middle values, then the median is the halfway point between the two.

There are two common ways of finding the middle value(s):

- Cross out the smallest number and the largest number, then cross out the next smallest and largest, keeping going crossing out pairs of number like this until you have one or two left.
- If n is the number of values in the list, then work out the value of \frac{n+1}{2}. If the answer is a whole number such as 5, then the median is the 5th point along the ordered list. If the answer is a decimal (ending in .5) such as 12.5, then the median is the halfway point between the 12th and 13th value along.

**Advantages and Disadvantages**

**Advantage**– it is not affected by outliers.**Disadvantage**– it does not consider all the data. Consider the values 1, 1, 2, 3, 12, 14, 15 – what is the median? Does it actually represent the middle of these numbers?

**Mode**

The mode is the most common value. To find it, look for which value appears most often. There might be two values which are tied for the most appearances, in which case we say the data is **bimodal**, or alternatively there might no repeats at all, in which case there is simply no mode.

**Advantages and Disadvantages**

**Advantage**– it is not affected by outliers.**Disadvantage(s)**– firstly, it sometimes is impossible to find. Secondly, it does not consider all of the data. Consider the values 32, 35, 35, 128, 201, 176, 295 – what is the mode? Does it represent the “average” of the data?

**Range**

The range is not another average – it is a measure of spread. This means the range is a way of telling us how spread out the data is.

To calculate it, we subtract the smallest value from the biggest value.

\text{Biggest value} - \text{Smallest Value}

**Example 1: Finding the Mean, Median and Mode**

9 people take a test. Their scores out of 100 are:

56, 79, 77, 48, 90, 68, 79, 92, 71

Work out the mean, median, and mode of their scores.

First up, the mean. The question tells that there are 9 data points, so we must add the numbers together and divide the result by 9.

\text{Mean } = \dfrac{56 + 79 + 77 + 48 + 90 + 68 + 79 + 92 + 71}{9}=73.3\text{ (1dp)}

So, the mean is 73.3. Next up, the median. Firstly, we have to put the numbers in ascending order. This looks like

48, 56, 68, 71, 77, 79, 79, 90, 92

There are 9 numbers, and \frac{9+1}{2}=5, so the median must be the 5th term along. Counting along the list, we get that the median is 77. Finally, the mode. We can see very clearly from the ordered list that there is only one repeat: 79, so we must have that the mode is 79.

## Example 2: Calculating the Range

Find the range of 12, 8, 4, 16, 15, 15, 5, 15, 10, 8.

A good way to make sure you haven’t missed any numbers in determining the biggest and smallest value is to order them. Doing this, we get

4, 5, 8, 8, 10, 12, 15, 15, 15, 16.

Therefore, the smallest value is 4 and the largest is 16 and 16-4=12, so the range is 12**. **

Sadly, the range also has its disadvantages – it is highly affected by outliers.

*A better way to calculate both mean *and *range is to remove outliers before calculating them. A question may ask you to redo calculations of the mean/range with outliers removed, or it may ask you to identify how these values are affected by outliers. Get to know your outliers.

All that said, if you’re asked to find the mean/range of a bunch of numbers, then don’t go removing any numbers you think might be outliers unless the question asks you to.

## Example 3: Finding the Mean – Applied Questions

There were 5 members of a basketball team who had a mean points score of 12 points per game. One of the team members left, causing the average point score to reduce to 10 points per game. What was the mean score of the player that left?

There are often exam questions that require you to stretch your knowledge and apply hat you know. With examples such as this there are a set of steps you can take which apply to these types of mean questions.

Firstly, find the total for the original number of players 5\times12=60

Secondly, find the total after once the mean has changed, so 4\times10=40

Finally, calculate the difference between these two totals as that difference has been caused by the person who left. 60-40=20

Therefore the mean score of the person who left was 20 points per game. The same method applies if a new person/amount is added, you find the old and new totals and the difference is always due to the thing which caused the change.

### Example Questions

1) Find the mode and range of the list of numbers below:

280, 350, 320, 400, 350, 490, 590, 470, 280, 410, 350.

It is not necessary to order the numbers, but it may help, especially in working out the range. In ascending order, these values are:

280, 280, 320, 350, 350, 350, 400, 410, 470, 490, 590

Since the number 350 occurs 3 times, it is the most common value, so:

\text{mode } = 350.

The range is the difference between the lowest and the highest value. The lowest value is 280 and the highest is 590, so:

\text{range } = 590-280=310.

2) Dani recorded the heights of members of her extended family, to the nearest cm. Find the median of their heights (listed below):

163, 185, 164, 170, 188, 154, 168, 179

First of all, since we have been asked to work out the median, we need to order the set of values:

154, 163, 164, 168, 170, 179, 185, 188

There are 8 values in total, so we need to know which value, or values, we need in order to find the median.

Since there is an even number of values, there is not one single middle value, so you will need to find the two middle values. To find the median value, we can use the following formula:

\dfrac{n + 1}{2} where n represents the total number of values.

In this question, we have 8 values, so:

\frac{8 + 1}{2}=4.5

The answer 4.5 tells us that the median is half-way between the 4th value and the 5th value. The 4th value is 168 and the 5th value is 170, so the median is 169.

**NOTE: **if you struggle to work out the half-way value, add up the two numbers and divide by 2 (in other words, work out the mean of these two values).

3) Below is a list of recorded reaction times in seconds.

0.25, 0.34, 0.39, 0.38, 0.39, 1.67, 0.28, 0.30, 0.42, 0.46

a) Calculate the mean of these values.

b) Identify any outliers and explain how the value of the mean would change if any outliers were to be removed from the calculation of the mean.

a) In order to calculate the mean, we need to add up all the values and divide by 10 (since there are 10 values in total).

\text{Mean }=\dfrac{0.25+0.34+0.39+0.38+0.39+1.67+0.28+0.3+0.42+0.46}{10}=0.488

b) 1.67 is the outlier as it is vastly higher than all the other values.

If this outlier were removed, then the mean would be lower.

4) The mean length of 7 planks of wood is 1.35m. When an extra plank of wood is added, the mean length of a plank of wood increase to 1.4m.

What is the length of the extra plank of wood that was added?

In most questions involving the mean, we are given the total and need to work out the mean from the total. In this question, we have been given the mean, so we are going to have to calculate the total from the mean.

If the mean length of 7 planks of wood is 1.35m, then the total length of all these planks of wood combined can be calculated as follows:

7 \times 1.35\text{ m} = 9.45\text{ m}

When the extra plank of wood is added, the mean length of a plank of wood increases to 1.4m. This means there are now 8 planks of wood, with a combined length of:

8 \times 1.40\text{ m} = 11.2\text{ m}

Therefore, by adding this additional plank of wood, the combined length has increased from 9.45m to 11.2m, so the length of this extra plank of wood is therefore:

11.2\text{ m} - 9.45\text{ m} = 1.75\text{ m}

5) In a rowing team, the weight of 8 women, in kilograms, is:

63, 60, 57, 66, 62, 65, 69, 58

In order to be a more competitive team, the coach has said that each team member should try to increase their overall muscle mass which will result in a 2% gain in overall body weight.

What will be the mean weight of the team if all 8 are successful in precisely meeting this 2% weight gain?

In this question, we do not need to work out a 2% increase in weight for each individual team member (it would not be wrong to do so, just unnecessarily time-consuming).

The combined weight of all 8 members is:

63 + 60+57+66+62+65+69+58 = 500\text{ kg}

If each team member increases their weight by 2%, then this is the same as the team increasing their combined weight by 2%. Therefore, if the team is successful in achieving this 2% weight gain, then the combined weight of the team can be calculated as follows:

1.02\times500 = 510 \text{ kg}

Since there are 8 team members in total, then mean weight following this weight gain is:

510\text{ kg}\div8 =63.75\text{ kg}

### Worksheets and Exam Questions

#### (NEW) Mean, Median, Mode and Range 1 Exam Style Questions - MME

Level 4-5#### (NEW) Mean, Median, Mode and Range 2 Exam Style Questions - MME

Level 4-5#### Mean Median Mode Range - Drill Questions

Level 1-3#### Mean Median Mode Range and IQR - Drill Questions

Level 1-3#### Estimate of Mean - Drill Questions

Level 4-5#### Averages From Frequency Tables - Drill Questions

Level 4-5### Videos

#### Mean, Median, Mode and Range Q1

GCSE MATHS#### Mean, Median, Mode and Range Q2

GCSE MATHS#### Mean, Median, Mode and Range Q3

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