# Finding a Central Value

When you have two or more numbers it is nice to find a value for the "center".

## 2 Numbers

With just 2 numbers the answer is easy: go half-way between.

### Example: what is the central value for 3 and 7?

Answer: Half-way between, which is 5.

You can calculate it by adding 3 and 7 and then dividing the result by 2:

## 3 or More Numbers

We can use that idea of "adding then dividing" when we have 3 or more numbers:

### Example: what is the central value of 3, 7 and 8?

Answer: You calculate it by adding 3, 7 and 8 and then dividing the results by 3 (because there are 3 numbers):

Notice that we divide by 3 because we have 3 numbers ... very important!

## The Mean

So far we have been calculating the Mean (or the Average):

Mean: Add up the numbers and divide by how many numbers.

But sometimes the Mean can let you down:

### Example: Birthday Activities

Uncle Bob wants to know the average age at the party, to choose an activity.

There will be 6 kids aged 13, and also 5 babies aged 1.

Add up all the ages, and divide by 11 (because there are 11 numbers):

**7.5...**

The mean age is about The 13 year olds are embarrassed, |

The Mean was **accurate**, but in this case it was **not useful**.

## The Median

But you could also use the Median: simply list all numbers in order and choose the middle one:

### Example: Birthday Activities (continued)

List the ages in order:

1, 1, 1, 1, 1, 13, 13, 13, 13, 13, 13

Choose the middle number:

1, 1, 1, 1, 1, **13**, 13, 13, 13, 13, 13

The Median age is **13** ... so let's have a **Disco**!

Sometimes there are **two** middle numbers. Just average those two:

### Example: What is the Median of 3, 4, 7, 9, 12, 15

There are two numbers in the middle:

3, 4, 7, 9, 12, 15

So we average them:

The Median is **8**

## The Mode

The Mode is the value that occurs most often:

### Example: Birthday Activities (continued)

Group the numbers so we can count them:

1, 1, 1, 1, 1, 13, 13, 13, 13, 13, 13

"13" occurs 6 times, "1" occurs only 5 times, so the mode is **13**.

*How to remember? Think "mode is most"*

But Mode can be tricky, there can sometimes be more than one Mode.

### Example: What is the Mode of 3, 4, 4, 5, 6, 6, 7

Well ... 4 occurs twice but 6 **also** occurs twice.

So **both 4 and 6** are modes.

When there are two modes it is called "bimodal", when there are three or more modes we call it "multimodal".

## Outliers

Outliersare values that "**lie** **out**side" the other values.

They can change the mean a lot, so we can either not use them (and say so) or use the median or mode instead.

### Example: 3, 4, 4, 5 and 104

**Mean**: Add them up, and divide by 5 (as there are 5 numbers):

**24**

24 does not represent those numbers well at all!

Without the 104 the mean is:

**4**

But please tell people you are not including the outlier.

**Median**: They are in order, so just choose the middle number, which is **4**:

3, 4, **4**, 5, 104

**Mode**: 4 occurs most often, so the Mode is **4**

3, **4, 4**, 5, 104

## Harmonic Mean

The Harmonic Mean is good at handling large outliers:

### Example: 2, 4, 6 and 100

The arithmetic mean is \frac{2+4+6+100}{4} = **28**

The harmonic mean is 4/(\frac{1}{2} + \frac{1}{4} + \frac{1}{6} + \frac{1}{100}) = **4.32** (to 2 places)

But small outliers will make things worse!

## Conclusion

There are other ways of measuring central values, but **Mean, Median and Mode** are the most common.

Use the one that best suits your data. Or better still, use all three!