An **average **is simply a way of picking out a **typical value** from a set of objects (usually numbers). There are infinitely many different ways to pick a “typical value”, but here are some of my favourites:

**Midrange average:**

The simplest of the lot: just calculate the number that is **half way **between the smallest and largest value. The midrange of the data set pictured is 155 (half way between 4 and 306).

Using the midrange, you can truthfully tell your parents that you did “**better than average**” in the maths exam, **even if you were the second worst in the class!** All you need is for a single person to have dropped more than twice the marks that you did.

Got to love that!

**Mean average:**

The **Arithmetic Mean** (to use its full name) is the average people usually “mean” when they think of the average.

AT GCSE (Foundation): add the numbers up and divide by how many there are.

AT GCSE (Higher): mean = sum of values ÷ number of values.

AT A-LEVEL AND BEYOND: we introduce the symbol $\bar{x}$ pronounced “**x bar**” for the mean:

$\bar{x} = \dfrac{\Sigma x}{n}$ or for grouped data $\dfrac{\Sigma fx}{\Sigma f}$

Unfortunately, the data set pictured has an **outlier** or “extreme value”: the data point 306 pulls the mean up to 51, which most of us would agree is **too big ** to be a sensible “typical value”.

**Geometric Mean average:**

If we can add them up and divide by $n$, why not **multiply them all and take the $n$th root**? This mean would be $\sqrt[7]{4 \times 5 \times 6 \times 8 \times 10 \times 18 \times 306} = 12.68…$

*Why would you want to do that??*

This average allows you to find the average interest rate of the money in your bank account:

e.g. Suppose your money grows at 3%, 6%, 8% in year 1, 2, 3. What interest rate, fixed over the whole 3 years, would this be the same as?

Simply find the geometric mean of the three **multipliers**. The multipliers for 3%, 6%, 8% interest are 1.03, 1.06 and 1.08 (check your GCSE maths to find out why this is). The geometric mean is

$\sqrt[3]{1.03 \times 1.06 \times 1.08} = 1.0564…$ so the result is the same as if the bank had paid us a fixed rate of just over 5.6% each year.

**Median average:**

Put them in order and choose the **middle **one. In our case: 8.

Notice that using the median does not suffer from the same problem as the mean: the **outlier** 306 does not distort it. For this reason the median is usually preferred if your data set has outliers or is **skewed **(lopsided rather than symmetrical).

In 2020 the **average income for adults in full time employment in the UK** was about **£26000** using median but **£32000** using the mean – quite a difference!

AT A-LEVEL: we introduce the symbol **$Q_2$** for the median, because it is the **second quartile **(or two quarters of the way through the data).

**Mode average:**

AT GCSE: “the one that appears the most”.

AT A-LEVEL: the value with the highest **frequency**.

Our data set does not have a mode, since each value appears just once. Some data sets have several modes e.g. 4, 5, 5, 6, 8, 8 (the two modes are 5 and 8).

This average is most useful for **qualitatative** (non-numerical) data such as eye colour or favourite Harry Potter character.

**Harmonic Mean average:**

Yes, it’s another type of mean!

This time you take the **reciprocal **of the **arithmetic mean** of the reciprocals (!).

A REMINDER: the *reciprocal* of $x$ is $\dfrac{1}{x}$

*Why would you want to do that??*

If you make a journey at one speed going out, and a different speed on the return journey, then the Harmonic Mean gives you your average speed for the whole journey – pretty cool!

e.g. 60 mph outbound, 20 mph return, the time for the whole journey will be the same as travelling there *and* back at a speed of:

$\dfrac{1}{(\dfrac{1}{60} + \dfrac{1}{20}) \div 2}$ = 30mph

See if you can check this gives the same overall time (hint: **time = dist ÷ speed**)

**A FINAL WORD:**

No discussion of averages would be complete without the delightful rhyme (sing along if you know the tune…):

Hey Diddle Diddle

The **Median**‘s the Middle

You add and divide for the **Mean**.

The **Mode **is the one that appears the most

and the **Range **[*not* an average but a **measure of spread**!] is the difference between.