# 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.

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

(3+7) / 2 = 10/2 = 5

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

(3+7+8) / 3 = 18/3 = 6

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):

(13+13+13+13+13+13+1+1+1+1+1) / 11 = 7.5...
 The mean age is about 7½, so he gets a Jumping Castle!The 13 year olds are embarrassed, and the 1 year olds can't jump!

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:

(7+9) / 2 = 16/2 = 8

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

Outliers are values that "lie outside" 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):

(3+4+4+5+104) / 5 = 24

24 does not represent those numbers well at all!

Without the 104 the mean is:

(3+4+4+5) / 4 = 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

## Other Means

The mean (average) we have been looking at is more correctly called the Arithmetic Mean.

There are other types of mean! Here are two examples:

The Geometric Mean multiplies the numbers together, then does a square root or cube root etc depending on how many numbers, like in this example:

### Example: The Geometric Mean of 2 and 18

• First we multiply them: 2 × 18 = 36
• Then (as there are two numbers) take the square root: √36 = 6

The Harmonic Mean adds up "1 divided by number" then flips it like this:

### Example: The Harmonic Mean of 2, 4, 5 and 100

With 4 numbers we get:

 4 = 4 = 4.17 (to 2 places) 12 + 14 + 15 + 1100 0.96