# Measure of central tendency

## Measure of central tendency

Don't be put off, "measure of central tendency" is just a mathematical and rather posh way of saying "averages".

#### The Mode

The mode is the most popular value or values.

It is the piece or pieces of data that occur most often.

Example:

You are given the following pieces of data showing the number of songs on 7 CD's:

#### The Median

The median is the middle piece of data when the data is in numerical order.

Example:

Using 7 pieces of data:

In this case, 25 is the median as it is clearly the value in the middle.

However, if there is an even number of values, there will not be a single value in the middle:

Example:

With 101 pieces of data, odd, we must find just over halfway. In this case, the 51st value. This value will be the median.

With 50 pieces of data, even, we must find halfway and the next value. In this case, the 25th and 26th values. The median will be halfway between these values.

These rules are especially important with large groups of data.

Example:

This example uses a frequency table showing the rainfall in mm over a period of 2 summer months:

 Rainfall (mm): 0 1 2 3 4 5 Frequency (f): 4 12 15 16 9 4

The table shows 60 pieces of data ranging from 0 to 5. As there is an even number the median is found by finding the value halfway between the 30th and 31st pieces of data. In this case, both the 30th and 31st pieces of data are 2's.

Hence the median is 2mm.

#### The Mean

This is the most widely used of all the averages.

The mean of a set of data is the sum of all the values divided by the number of values.

The mean is denoted by x with an overbar, and for n pieces of data, it is calculated by:

Don't be worried, ∑x just means the sum of all the x's - for instance, add all the bits of data together.

Example:

Now let's have a look at a couple of trickier examples when we are given larger sets of data in frequency tables:

Example:

Again, the following example uses a frequency table showing the rainfall in mm over a period of 2 summer months:

 Rainfall (mm): 0 1 2 3 4 5 Frequency (f): 4 12 15 16 9 4

This gives a mean rainfall of 2.4mm

Note: 2 × 15 as there are 15 days with 2mm of rainfall, so 30mm altogether etc...

#### The mean from grouped frequency tables

When dealing with even larger sets of data, we may be given the data in terms of a grouped frequency table.

Example:

In this example, the data given is the height (x) of 100 Premiership footballers:

Height (cm): Frequency (f):
165 ≤ x < 170 8
170 ≤ x < 175 18
175 ≤ x < 180 34
180 ≤ x < 185 23
185 ≤ x < 190 17

To find the exact mean of this data is impossible as we don't know the exact data!

We can however, find an estimate of the mean by assuming each footballer is the height halfway within his interval - for instance, assume the 18 footballers in the interval are all 172.5cm tall! (Just like the median the quickest and easiest way to find halfway is to add up the 2 values and halve.)

Lets look what that does to our table:

Height (cm): Frequency (f): x × f
167.5 8 1340
172.5 18 3105
177.5 34 6035
182.5 23 4197.5
187.5 17 3187.5
Total: 100 17865

Hence the mean height of the 100 footballers is 178.65cm

Note the addition of the last column: This is just the mid-heights multiplied by the frequencies and is an efficient way of showing your workings.