How do we find averages if there are lots of values?

• In reality there will be far more data to work with than just a few numbers
• In these cases the data is usually organised in such a way to make it easier to follow and understand – for example in a?table?or?chart
• We can still find the?mean,?median?and?mode?but have to ensure we understand what the table or chart is telling us

What do I need to know?

• Finding the median and mode from tables/charts is fairly straightforward once you understand what the table/chart is telling you so these notes focus mainly on finding the mean

1. Finding the mean from (discrete) data presented in tables

• Tables allow data to be summarised neatly – and quite importantly it puts it into order

Eg. the number of pets owned by 40 pupils in year 11 are summarised in the table below:

Work out the mean and median number of pets per pupil.

The?mean?can be found as you long as understand what the table is telling you:

It tells you:

12 (of the 40) pupils had no pets; 15 of them had 1 pet; 8 had 2 pets

This means you can add up all the 0’s very quickly, all the 1’s very quickly etc. using multiplication:? 12 x 0 = 0, 15 x 1 = 15, etc.

The easiest way to do this is to add another column to the table and adding a?total row?will prove useful in the next stage too:

• You can now see that the total number of pets for the whole of the 40 pupils is 48
• If you hadn’t been told in the question, the total of the frequency column would’ve told you how many pupils there are
• You can now find the mean:

48 ÷ 40 = 1.2

The mean number of pets is 1.2 (pets per pupil)

• You may sometimes see the number of pets called x and the frequency f so the last column would be:f x? (f times x)

2. Median

• The?median?is a little complicated to understand but easy to work out
• Remember you are looking for the middle value when the data is in order
• For the example above the table is in order (0, then 1, 2, etc.) – so you need to work out which of the 40 values is in the median position
• To do this you add one to the number of values and divide by 2

(40 + 1) ÷ 2 = 20.5 – so the median is in the “20.5^th” position

• The table tells you the first twelve numbers on the list are all 0’s, the next 15 are all 1’s – so the “20.5^th” value in the list must be a 1The median number of pets is 1(Note that the “20.5^th” value is referring to halfway between the 20^th?and 21^st?values. Both of these are 1’s so median must be 1 (or use (1 + 1) ÷ 2 = 1 if you’re not convinced!)

3. Mode (modal value)

• The?mode?(or?modal?value) is simple to identify
• Look for the highest frequency – and then you find the corresponding data value

In the example above the highest frequency is 15

Modal number of pets = 1

• Don’t confuse the number of pets with the frequency