Finding the mean from (discrete) data presented in tables
STEP 1
Add a column to the table and work out "data value" × "frequency"
(This is effectively doing the 'adding up' part of finding the mean in stages)
STEP 2
Find the total of the extra column to give the overall total of the data values
STEP 3
Find the mean by dividing this total by the total of the frequency column
i.e. divide the total of the data values by the number of data values
Finding the median from (discrete) data presented in tables
Finding the mode (or modal value)
The bar chart shows data about the shoe sizes of pupils in class 11A.
Find the mean shoe size for the class,
Although the data is given in a bar chart, this is essentially the same as a table.
Rewrite it as a table but add an extra column to help find the total of all the shoe sizes.
Shoe size ( x ) | Frequency ( f ) | x f |
6 | 1 | 6 × 1 = 6 |
6.5 | 1 | 6.5 × 1 = 6.5 |
7 | 3 | 7 × 3 = 21 |
7.5 | 2 | 7.5 × 2 = 15 |
8 | 4 | 8 × 4 = 32 |
9 | 6 | 9 × 6 = 54 |
10 | 11 | 10 × 11 = 110 |
11 | 2 | 11 × 2 = 22 |
12 | 1 | 12 × 1 = 12 |
Total | 31 | 278.5 |
Mean
Mean = 8.98 (3 s.f.)
Note that the mean does not have to be an actual shoe size.
Find the median shoe size,
The bar chart/table has the data in order already so find the position of the median.
The median is the 16 ^{th} value.
There are 1 + 1 + 3 + 2 + 4 = 11 values in the first five rows of the table.
There are 11 + 6 = 17 values in the first six rows of the table.
Therefore the 16 ^{th} value must be in the sixth row.
Median shoe size is 9
Suggest a reason the shop owner may wish to know the modal shoe size of their customers.
A shop owner would want to know the modal shoe size of their customers as this size will be more likely to sell than other sizes so the shop owner should order more shoes in the modal size to stock the shop with
15 students were timed how long it took them to solve a maths problem. Their times, in seconds, are given below.
12 | 10 | 15 | 14 | 17 |
11 | 12 | 13 | 9 | 21 |
14 | 20 | 19 | 16 | 23 |
Find the mean and median times.
There are a fair amount of numbers so it may be wise to do the adding up in bits - we've used rows.
12 + 10 + 15 + 14 + 17 = 68
11 + 12 + 13 + 9 + 21 = 66
14 + 20 + 19 + 16 + 23 = 92
For the median, the data needs to be in order first.
Mean = 15.1 seconds (3 s.f.)
Median time = 14 seconds
Comment on the mode of the data.
The mode (or lack of) is easiest to see from the data listed in order in the median question above.
There are two modes (bi-modal) - 12 and 14 seconds
Alternatively we could say there is no mode.
STEP 1
Draw an extra two columns on the end of a table of the grouped data
In the first new column write down the midpoint of each class interval
STEP 2
Work out "frequency" × "midpoint" (This is often called fx )
STEP 3
Total the fx column, and if not already done nor mentioned in the question, total the frequency column to find the number of data values involved
STEP 4
Estimate the mean by using its formula; " total of fx
" ÷ " no. of data values"
The weights of 20 three-week-old Labrador puppies were recorded at a vet's clinic.
The results are shown in the table below.
Weight, w kg | Frequency |
3 ≤ w < 3.5 | 3 |
3.5 ≤ w < 4 | 4 |
4 ≤ w < 4.5 | 6 |
4.5 ≤ w < 5 | 5 |
5 ≤ w < 6 | 2 |
Estimate the mean weight of these puppies.
First add two columns to the table and complete the first new column with the midpoints of the class intervals.
Complete the second extra column by calculating " fx".
A total row would also be useful.
Weight, w kg | Frequency | Midpoint | " fx" |
3 ≤ w < 3.5 | 3 | 3.25 | 3 × 3.25 = 9.75 |
3.5 ≤ w < 4 | 4 | 3.75 | 4 × 3.75 = 15 |
4 ≤ w < 4.5 | 6 | 4.25 | 6 × 4.25 = 25.5 |
4.5 ≤ w < 5 | 5 | 4.75 | 5 × 4.75 = 23.75 |
5 ≤ w < 6 | 2 | 5.5 | 2 × 5.5 = 11 |
Total | 20 | 85 |
Now we can find the mean.
Mean
An estimate of the mean weight of the puppies is 4.25 kg
Write down the modal class.
Looking for the highest frequency in the table we can see it is 6.
This corresponds to the interval 4 ≤ w < 4.5.
The modal class is 4 ≤ w < 4.5
A common error is to write down 6 as the mode (modal value).