# What are averages?

This guide explains the different types of average (mean, median and mode). It details their use, how to calculate them, and when they can be used most effectively.

Other useful guides: Working with percentages, Measures of variability.

## Introduction

The term average is used frequently in everyday life to express an amount that is typical for a group of people or things. For example, you may read in a newspaper that on average people watch 3 hours of television per day.

We understand from the use of the term average that not everybody watches 3 hours of television each day, but that some watch more and some less.

However, we realize from the use of the term average that the figure of 3 hours per day is a good indicator of the amount of TV watched in general.

Averages are useful because they:

• summarise a large amount of data into a single value; and
• indicate that there is some variability around this single value within the original data.

In everyday language most people have an inherent understanding of what the term average means. However, within the language of mathematics there are three different definitions of average known as the mean, median and mode.

The mean, median and mode are each calculated using different methods and when applied to the same set of original data they often result in different average values.

It is important to understand what each of these mathematical measures of average tells you about the original data and consider which measure, the mean, median or mode, is the most appropriate to calculate should you wish to use an average value to describe a dataset.

### What is the mean?

The mean is the most commonly used mathematical measure of average and is generally what is being referred to when people use the term average in everyday language. The mean is calculated by totalling all the values in a dataset; this total is then divided by the number of values that make up the dataset.

For example, to find out the mean amount borrowed by 6 students in a tutorial group taking out a student loan in 1998/9, the amounts borrowed by each student have been collected. These 6 amounts form the dataset given in table 1. Table 1: Amounts borrowed by 6 students taking out a student loan in 1998/9

In order to find the mean loan, the total amount borrowed (£9,140) is divided by the number of students (6) which equals £1,523.

### Formula for the mean

• Whilst it is not vital to know the mathematical formula for the calculation of the mean you may want to include it at some point in a report or dissertation.
• The formula for the mean is written in the following way:
 _X is the symbol for the mean and is referred to as bar X (ex) Σ is the Greek symbol sigma and simply means sum or add up X refers to each of the individual values that make up the dataset n is the number of values that make up the dataset

Re-writing the equations in words results in “the mean is equal to the sum of the individual values in the dataset, divided by the number of values in the dataset”.

### When to use the mean

The mean is a good measure of the average when a dataset contains values that are relatively evenly spread with no exceptionally high or low values – this was the case with the data on student loans given in table 1.

If a dataset contains one or two very high or very low values the mean will be less typical as it will be adversely influenced by these exceptional value(s). This can be seen in table 2, where the mean salary of 6 graduates who responded to a survey about salaries in their first jobs is calculated to be £23,995 (£143,970 divided by 6). Examining the dataset shows that 5 of the 6 graduates earn less than the mean salary of £23,995 and it is Steve’s exceptionally high salary that produces the high mean value.

In this example, the mean gives a misleading impression of the amount a typical graduate earns in their first job.

For datasets containing extremely high or low values the median (see next section) is a better measure of the average value.

### When not to use the mean

The mean is generally an inappropriate measure of average for data that are measured on ordinal scales. Ordinal data are rated according to a category where a higher score indicates a higher or better rank than a lower score.

Ordinal data are frequently used in surveys that ask people to indicate preference. The final information is relative and the difference between the ranks is not equal.

For example, in response to a question regarding the flavour of a new blend of coffee a score of 10 implies a better taste than a score of 1 but it does not mean that the flavour is ten times as good!

### What is the median?

The median refers to the middle value in a dataset, when the values are arranged in order of magnitude from smallest to largest or vice-versa.  When there are an odd number of values in the dataset the middle value is straightforward to find. When there are an equal number of values, the mid-point between the two central values is the median.

For example, if the prices of seven sandwiches bought on campus are placed in order the median will be the 4th price in the sequence:

£1.10, £1.26, £1.30, [£1.40], £1.45, £1.85, £2.00

When the six starting salaries from example 2 are placed in order of magnitude the median value lies half-way between the 3rd and 4th salaries:

£14,870, £18,750, £19,100,    £21,650, £22,400, £47,200

The median value lies half-way between £19,100 and £21,650 and is £20,375 ((£19,100+£21,650)÷2).

### When to use the median

1. The median is a good measure of the average value when the data include exceptionally high or low values because these have little influence on the outcome.
2. The median is the most suitable measure of average for data classified on an ordinal scale.

3. The median is also easy to calculate but this does not imply that it is an inferior measure to the mean – what is important is to use an appropriate measure to determine the average.

Another area where the median is useful is with frequency data.  Frequency data give the numbers of people or things in particular categories.

For example, the frequency distribution of shoe sizes for a sample of 21 women was collected and is summarised in table 3. Table 3: Frequency distribution of shoe sizes

A common mistake is to think that the median shoe size is 6 since this is the middle value in the first column. This is incorrect since it is the frequency information rather than the category (shoe size) that must be considered.

There are 21 women in the sample so when the shoe sizes are arranged in order of magnitude the median shoe size will be placed 11th  (mid-way) along that list.  The 11th value in the frequency column corresponds with a shoe size of 5.

An alternative way of finding the median shoe size in this case is to re-write the data from the table to show each shoe size and the number of times it occurred in each category and use this to work out the median:

4 ,4 ,4 ,4, 4,5 ,5 ,5 ,5 ,5 , , 6, 6, 6, 6, 6, 6, 6, 7, 7, 8

Using this method it is easier to see that the median shoe size is 5.

### What is the mode?

The mode is the value that occurs with the greatest frequency in a dataset.  It is representative or typical because it is the most common value.

There may be more than one mode in a dataset if several values are equally common; alternatively there may be no mode.

In example 3 (frequency distribution of shoe sizes), size 6 is the mode (or modal class), since this shoe size occurs the most frequently (7 times) in the sample.

### When to use the mode

The mode is the only measure of average that can be used with nominal data. For example, late-night users of the library were classified by faculty as: 14% science students, 32% social science students, and 54% biological sciences students. No median or mean can be calculated but the mode is biological science students as students from this faculty were the most common.

### Part four: Calculating Averages Using Excel

In addition to the manual methods described above, you can also calculate the mean, median and mode in Excel using special commands. The commands are entered into the formula bar towards the top of the spreadsheet and are preceded by =.

This use of = informs Excel that a calculation needs to be performed on the data. The corresponding cells in the spreadsheet show the result of the calculation. Example 4, below shows how Excel can be used to find the mean, median and mode of the student loan data originally given in example 1.

Column A shows the different categories of student, whilst column B shows the amounts borrowed. To calculate a mean, Excel uses the term average and in the example spreadsheet, the command =AVERAGE(B2:B7) has been typed in cell D2.  This command will automatically calculate the mean of the loans in cells 2 to 7 of column B.

Excel performs the calculation instantly and the mean value of £1,523.

33 is immediately shown in cell D2, however, the command used to perform the calculation remains displayed in the formula bar for as long as cell D2 is active (or highlighted) which is indicated by the box surrounding it.

In cell D4 the command for the mode has been entered =MODE(B2:B7); however, as there is no modal value in this dataset the result is given as #N/A.

In cell D6 the command for the median has been entered =MEDIAN (B2:B7)

These examples show the quick method of calculating averages using a cell range. Each of the commands can also be written out in a longer format with each of the different amounts of student loan entered as a separate value.

For example using the command =MEDIAN(1170, 1890, 1530, 1160, 1870, 1520) will produce an identical result to =MEDIAN(B2:B7). However, if the amount of one of the loans in column B is changed, the cell range method will automatically adjust the median, whereas the longer format will require manual adjustment of the command.

### Where next?

This guide has outlined the three different types of mathematical averages (mean, median and mode) and explained how to calculate them both manually and in the popular spreadsheet package Excel. More resources on Excel can be found here.

## The Three Types of Average – Median, Mode and Mean

We use three different types of average in maths: the mean, the mode and the median, each of which describes a different ‘normal’ value. The mean is what you get if you share everything equally, the mode is the most common value, and the median is the value in the middle of a set of data.

Here are some more in-depth definitions:

• Median: In a sense, the median is what you normally mean when you say ‘the average man in the street’. The median is the middle-of-the road number – half of the people are above the median and half are below the median. (In America, it’s literally the middle of the road: Americans call the central reservation of a highway the ‘median’.)
Try remembering ‘medium’ clothes are neither large nor small, but somewhere in between. Goldilocks was a median kind of girl.
• Mode: The mode is the most common result. ‘Mode’ is another word for fashion, so think of it as the most fashionable answer – ‘Everyone’s learning maths this year!’
• Mean: The mean is what you get by adding up all of the numbers and dividing by how many numbers were in the list. Most people think of the mean when they use the word ‘average’ in a mathematical sense.
In some ways the mean is the fairest average –you get the mean if the numbers are all piled together and then distributed equally. But the mean is also the hardest average to work out.

You use the different averages in different situations, depending on what you want to communicate with your sums.

### Find the median

To find the median of a set of numbers, you arrange the numbers into order and then find the number exactly in the middle:

1. If the numbers aren’t in order, sort them out.

You can arrange them either going up or down.

2. Circle the number at each end of the list.

3. Keep circling numbers two at a time (one from each end) until you have only one or two uncircled numbers.

4. If only one number is left, that’s the median.

You’re done!

5. If two numbers are left, find the mean.

Add up the two numbers and divide by two. The answer is the median.

### Find the mode

If you have a list of numbers in order, figuring out which number shows up most often is pretty easy. You simply count the numbers – whichever number you have most of is the mode.

If you have the list 1, 1, 3, 5, 5, 5, 6, 6, 7, 8, 9, 9, 10, you count each number in turn and find you have two 1s, one 3, three 5s, two 6s, one 7, one 8, two 9s and a 10. The number 5 comes up more frequently than any of the others, so the mode of these data is 5.

If the data aren’t in a list, I suggest you set up a tally chart to help you count the numbers. Finding the mode in a table of numbers is very easy: whoever made the table has already done the tally chart for you and counted up the 1s. All you do is find the biggest number in the ‘count’ or ‘frequency’ column. The number labelling that row is the mode. ### Working out the mean of a list of numbers

Here’s how to work out the mean of a set of numbers:

1. Write out a list of all the numbers.

2. Add up all the numbers.

3. Count how many numbers are in the list.

4. Divide the total from Step 2 by the total in Step 3.

The mean should be somewhere between the highest and lowest numbers in your list.

Adding up a long list of numbers is a chore. In real life you may use a calculator or a spreadsheet. But in an exam you may not have access to either of those helpful devices.

You can add up longs lists of numbers by working through the list, adding a pair of numbers at a time, and writing the result in the next row. If you have a number left over at the end of the row, just copy that number into the next row and keep going. ## How To Analyze Data Using the Average

The average is a simple term with several meanings. The type of average to use depends on whether you’re adding, multiplying, grouping or dividing work among the items in your set.

Quick quiz: You drove to work at 30 mph, and drove back at 60 mph. What was your average speed?

Hint: It’s not 45 mph, and it doesn’t matter how far your commute is. Read on to understand the many uses of this statistical tool. ### But what does it mean?

Let’s step back a bit: what is the “average” all about?

To most of us, it’s “the number in the middle” or a number that is “balanced”. I’m a fan of taking multipleviewpoints, so here’s another interpretation of the average:

The average is the value that can replace every existing item, and have the same result. If I could throw away my data and replace it with one “average” value, what would it be?

One goal of the average is to understand a data set by getting a “representative” sample. But the calculation depends on how the items in the group interact. Let’s take a look.

### The Arithmetic Mean

The arithmetic mean is the most common type of average:  Let’s say you weigh 150 lbs, and are in an elevator with a 100lb kid and 350lb walrus. What’s the average weight?

• The real question is “If you replaced this merry group with 3 identical people and want the same load in the elevator, what should each clone weigh?”
• In this case, we’d swap in three people weighing 200 lbs each [(150 + 100 + 350)/3], and nobody would be the wiser.
• Pros:
• It works well for lists that are simply combined (added) together.
• Easy to calculate: just add and divide.
• It’s intuitive — it’s the number “in the middle”, pulled up by large values and brought down by smaller ones.

Cons:

• The average can be skewed by outliers — it doesn’t deal well with wildly varying samples. The average of 100, 200 and -300 is 0, which is misleading.

The arithmetic mean works great 80% of the time; many quantities are added together. Unfortunately, there’s always those 20% of situations where the average doesn’t quite fit.

### Median

The median is “the item in the middle”. But doesn’t the average (arithmetic mean) imply the same thing? What gives?

Humor me for a second: what’s the “middle” of these numbers?

## What is the difference between Mean and Average?

Subject: What is the difference between Mean and Average?

Hi my name is Julie,

What is the difference between Mean and Average? My thinking is the average, is the equal to the sum of all numbers divided by the number of numbers added together. But the mean, I think should be calculated by adding the largest and smallest numbers in the set and them dividing by 2. (which is the point where 1/2 the numbers are higher and 1/2 the numbers are lower)

Thank you very much!

Hi Julie,

The term “average” usually encompasses several ways to measure what value
best represents a sample. There are various measurements that are used and at times the term and measurement used depends on the situation.

A statistician or mathematician would use the terms mean and average to refer to the sum of all values divided by the total number of values, what you have called the average. This especially true if you have a list of numbers. In fact even in mathematics there are different “averages” or “means” and this one is more properly called the arithmetic mean.

The quantity obtained by adding the largest and smallest values and dividing by 2, statisticians call the midrange. There are times however that this is called the mean. For example the weather office records the “mean daily temperature”, which is the sum of the high temperature and the low temperature divided by 2.

The median is the central point of a data set. To find the median, you would list all data points in ascending order and simply pick the entry in the middle of that list.

If this is not enough there is another measure. The mode is the value that occurs most often.

For example, consider the following data points: 1,1,2,3,4

The mean or average is (1+1+2+3+4)/5 = 2.2
The median is “2” (the central value).
The mode is “1” (it occurs most often).

• The midrange is (4+1)/2 = 2.5
• Hope this helps,
Patrick and Penny

Go to Math Central

## What Is an Average in Math?

In mathematics and statistics, average refers to the sum of a group of values divided by n, where n is the number of values in the group. An average is also known as a mean.

Like the median and the mode, the average is a measure of central tendency, meaning it reflects a typical value in a given set. Averages are used quite regularly to determine final grades over a term or semester.

Averages are also used as measures of performance. For example, batting averages express how frequently a baseball player hits when they are up to bat.

Gas mileage expresses how far a vehicle will typically travel on a gallon of fuel.

In its most colloquial sense, average refers to whatever is considered common or typical.

A mathematical average is calculated by taking the sum of a group of values and dividing it by the number of values in the group. It is also known as an arithmetic mean. (Other means, such as geometric and harmonic means, are calculated using the product and reciprocals of the values rather than the sum.)

With a small set of values, calculating the average takes only a few simple steps. For example, let us imagine we want to find the average age among a group of five people. Their respective ages are 12, 22, 24, 27, and 35. First, we add up these values to find their sum:

• 12 + 22 + 24 + 27 + 35 = 120

Then we take this sum and divide it by the number of values (5):

The result, 24, is the average age of the five individuals.

The average, or mean, is not the only measure of central tendency, though it is one of the most common. The other common measures are the median and the mode.

The median is the middle value in a given set, or the value that separates the higher half from the lower half.

In the example above, the median age among the five individuals is 24, the value that falls between the higher half (27, 35) and the lower half (12, 22). In the case of this data set, the median and the mean are the same, but that is not always the case.

For example, if the youngest individual in the group were 7 instead of the 12, the average age would be 23. However, the median would still be 24.

For statisticians, the median can be a very useful measure, especially when a data set contains outliers, or values that greatly differ from the other values in the set. In the example above, all of the individuals are within 25 years of each other.

But what if that were not the case? What if the oldest person were 85 instead of 35? That outlier would bring the average age up to 34, a value greater than 80 percent of the values in the set.

Because of this outlier, the mathematical average is no longer a good representation of the ages in the group. The median of 24 is a much better measure.

The mode is the most frequent value in a data set, or the one that is most likely to appear in a statistical sample. In the example above, there is no mode since each individual value is unique. In a larger sample of people, though, there would likely be multiple individuals of the same age, and the most common age would be the mode.

In an ordinary average, each value in a given data set is treated equally. In other words, each value contributes as much as the others to the final average. In a weighted average, however, some values have a greater effect on the final average than others.

For example, imagine a stock portfolio made up of three different stocks: Stock A, Stock B, and Stock C. Over the last year, Stock A's value grew 10 percent, Stock B's value grew 15 percent, and Stock C's value grew 25 percent.

We can calculate the average percent growth by adding up these values and dividing them by three. But that would only tell us the overall growth of the portfolio if the owner held equal amounts of Stock A, Stock B, and Stock C.

Most portfolios, of course, contain a mix of different stocks, some making up a larger percentages of the portfolio than others.

## Averages: Mean, Median and Mode

The term ‘average’ occurs frequently in all sorts of everyday contexts – you may say ‘I’m having an average day today’ meaning your day is neither particularly good nor bad, it is about normal.  Similarly we may refer to people, objects and other things as ‘average’.

The term 'average' refers to the ‘middle’ or ‘central’ point; when used in mathematics the term average refers to a number that is a typical representation of a group of numbers (or data set). Averages can be calculated in different ways – this page covers the Mean, Median and Mode. We include an averages calculator and explanation and examples of each type of average.

The most widely used method of calculating an average is the ‘mean’.  Most of the time when the term ‘average’ is used in a mathematical sense it refers to the mean average.

Quick Guide:

To calculate the Mean

Add the numbers together and divide by the number of numbers. (The sum of values divided by the number of values).

To determine the Median

Arrange the numbers in order, find the middle number. (The middle value when the values are ranked).

To determine the Mode

Count how many times each value occurs the highest is the mode. (The most frequently occurring value)

Use this calculator to work out the mean, median and mode of a set of numbers.

### Mean

The mathematical symbol or notation for mean is ‘x-bar’. This symbol appears on scientific calculators and in mathematical and statistical notations.

The ‘mean’ or ‘arithmetic mean’ is the most commonly used form of average. In order to calculate the mean average a set of related numbers (or data set) is required.  At least two numbers are needed in order to calculate the mean.

The numbers need to be linked or related to each other in some way to have any meaningful result – for instance, temperature readings, the price of coffee, the number of days in a month, the number of heartbeats per minute, student's test grades etc.

To find the (mean) average price of a loaf of bread in the supermarket, for example, first record the price of each type of loaf:

• White: £1
• Wholemeal: £1.20
• Baguette: £1.10

Next, add (+) the prices together £1 + £1.20 + £1.10 = £3.30

£3.30 ÷ 3 = £1.10.

The average price of a loaf of bread in our example is £1.10.

The same method applies with larger sets of data:

To calculate the average number of days in a month we would first establish how many days there are in each month (assuming that it was not a leap year):

 Month Days January 31 February 28 March 31 April 30 May 31 June 30 July 31 August 31 September 30 October 31 November 30 December 31

Next we add all the numbers together: 31 + 28 + 31 + 30 + 31 + 30 + 31 + 31 + 30 + 31 + 30 + 31 = 365

Finally we divide the answer with the number of values in our data set in this case there are 12 (one for each month counted).

So the mean average is 365 ÷ 12 = 30.42.

The average number of days in a month, therefore, is 30.42.

The same calculation can be used to work out the average of any set of numbers, for example the average salary in an organisation:

Let's assume the organisation has 100 employees on one of 5 grades:

 Grade Annual Salary Number of Employees 1 £20,000 21 2 £25,000 25 3 £30,000 40 4 £50,000 9 5 £80,000 5

In this example we can avoid adding each individual employee’s salary as we know how many are in each category.  So instead of writing out £20,000 twenty-one times we can multiply to get our answers:

 Grade Annual Salary Number of Employees Salary x Employees 1 £20,000 21 £420,000 2 £25,000 25 £625,000 3 £30,000 40 £1,200,000 4 £50,000 9 £450,000 5 £80,000 5 £400,000

Next add the values in the Salary x Employees column to find a total: £3,095,000 and finally divide this number by the number of employees (100) to find the average salary:

£3,095,000 ÷ 100 = £30,950.

• Quick Tip:
• The salaries, in the example above, are all multiples of £1,000 – they all end in ,000.
• You can ignore the ,000's when calculating as long as you remember to add them back on at the end.
• In the first row of the table above we know that twenty-one people get paid a salary of £20,000, instead of working with £20,000 work with 20:
• 21 x 20 = 420 then replace the ,000 to get 420,000.

Sometimes we may know the total of our numbers but not the individual numbers that make up the total.

In this example, assume that £122.50 is made by selling lemonade in a week.

We don’t know how much money was made each day, just the total at the end of the week.

What we can work out is the daily average: £122.50 ÷ 7 (Total money divided by 7 days).

122.5 ÷ 7 = 17.50.

So we can say that on average we made £17.50 a day.

## Averages – Maths GCSE Revision

• This section looks at averages.
• Mean
• There are three main types of average:
• mean –  The mean is what most people mean when they say 'average'. It is found by adding up all of the numbers you have to find the mean of, and dividing by the number of numbers. So the mean of 3, 5, 7, 3 and 5 is 23/5 = 4.6 .
• mode – The mode is the number in a set of numbers which occurs the most. So the modal value of 5, 6, 3, 4, 5, 2, 5 and 3 is 5, because there are more 5s than any other number.
• median – The median of a group of numbers is the number in the middle, when the numbers are in order of magnitude. For example, if the set of numbers is 4, 1, 6, 2, 6, 7, 8, the median is 6
1. This video shows you how to calculate the mean, median and mode
2. Grouped Data

When you are given data which has been grouped, you can't work out the mean exactly because you don't know what the values are exactly (you just know that they are between certain values). However, we calculate an estimate of the mean with the formula: ∑fx / ∑f , where f is the frequency and x is the midpoint of the group (∑ means 'the sum of').

Example

Work out an estimate for the mean height, when the heights of 23 people are given by the first two columns of this table:

 Height (cm) Number of People (f) Midpoint (x) fx 101-120 1 110.5 110.5 121-130 3 125.5 376.5 131-140 5 135.5 677.5 141-150 7 145.5 1018.5 151-160 4 155.5 622 161-170 2 165.5 331 171-190 1 180.5 180.5

In this example, the data is grouped. You couldn't find the mean the “normal way” (by adding up the numbers and dividing by the number of numbers) because you don't know what the values are. You know that three people have heights between 121 and 130cm, for example, but you don't know what the heights are exactly. So we estimate the mean, using “∑fx / ∑f”.

A good way of setting out your answer would be to add two columns to the table, as I have.

“Midpoint” means the midpoint of each of the groups. So the first entry is the middle of the group 101-120 = 110.5 .

• Now, ∑fx (add up all of the values in the last column) = 3316.5
• ∑f = 23

So an estimate of the mean is 3316.5/23 = 144cm (3s.f.)

This short video shows you how to find the mean, mode and median from a frequency table for both discrete and grouped data.

Moving Averages

A moving average is used to compare a set of figures over time. For example, suppose you have measured the weight of a child over an eight year period and have the following figures (in kg): 32, 33 ,35, 38, 43, 53, 63 ,65

Taking the mean doesn't give us much useful information. However, we could take the average of each 3 year period. These are the 3-year moving averages. The first is: (32 + 33 + 35)/3 = 33.3 The second is: (33 + 35 + 38)/3 = 35.3

The third is: (35 + 38 + 43)/3 = 38.7, and so on (there are 3 more!).

To calculate the 4 year moving averages, you'd do 4 years at a time instead, and so on…

Mode

The mode is the number in a set of numbers which occurs the most. So the modal value of 5, 6, 3, 4, 5, 2, 5 and 3 is 5, because there are more 5s than any other number.

Range

The range is the largest number in a set minus the smallest number. So the range of 5, 7, 9 and 14 is (14 – 5) = 9. The range gives you an idea of how spread out the data is.

The Median Value

The median of a group of numbers is the number in the middle, when the numbers are in order of magnitude.

For example, if the set of numbers is 4, 1, 6, 2, 6, 7, 8, the median is 6: 1, 2, 4, 6, 6, 7, 8      (6 is the middle value when the numbers are in order) If you have n numbers in a group, the median is the (n + 1)/2 th value.

For example, there are 7 numbers in the example above, so replace n by 7 and the median is the (7 + 1)/2 th value = 4th value. The 4th value is 6.