Mean
There are four types of average: mean, mode, median and
range. The mean is what most people mean when they say 'average'. It is found by
adding up all of the numbers you have to find the mean of, and dividing by the
number of numbers. So the mean of 3, 5, 7, 3 and 5 is 23/5 = 4.6 .
Grouped Data
When you are given data which has been grouped, you can't
work out the mean exactly because you don't know what the values are exactly
(you just know that they are between certain values). However, we calculate an
estimate of the mean with the formula: Sfx / Sf , where f is the frequency and x is the
midpoint of the group (S means 'the sum
of').
Example
Work out an estimate for the mean height, when the heights of
23 people are given by the first two columns of this table:
| Height (cm) |
Number of People (f) |
Midpoint (x) |
fx |
| 101-120 |
1 |
110.5 |
110.5 |
| 121-130 |
3 |
125.5 |
376.5 |
| 131-140 |
5 |
135.5 |
677.5 |
| 141-150 |
7 |
145.5 |
1018.5 |
| 151-160 |
4 |
155.5 |
622 |
| 161-170 |
2 |
165.5 |
331 |
| 171-190 |
1 |
180.5 |
180.5 |
In this example, the data is grouped. You couldn't find the
mean the "normal way" (by adding up the numbers and dividing by the number of
numbers) because you don't know what the values are. You know that three people
have heights between 121 and 130cm, for example, but you don't know what the
heights are exactly. So we estimate the mean, using "Sfx / Sf".
A good way of setting out your answer would be to add two
columns to the table, as I have.
"Midpoint" means the midpoint of each of the groups. So the
first entry is the middle of the group 101-120 = 110.5 .
Now,
Sfx (add up
all of the values in the last column) = 3316.5
Sf = 23
So an estimate of the mean is 3316.5/23 = 144cm
(3s.f.)
Mode
The mode is the number in a set of numbers which occurs the
most. So the modal value of 5, 6, 3, 4, 5, 2, 5 and 3 is 5, because there are
more 5s than any other number.
Range
The range is the largest number in a set minus the smallest
number. So the range of 5, 7, 9 and 14 is (14 - 5) = 9.
The Median Value
The median of a group of numbers is the number in the middle,
when the numbers are in order of magnitude. For example, if the set of numbers
is 4, 1, 6, 2, 6, 7, 8, the median is 6:
1, 2, 4, 6, 6, 7, 8 (6
is the middle value when the numbers are in order)
If you have n numbers in a
group, the median is the (n + 1)/2 th value. For example, there are 7 numbers in
the example above, so replace n by 7 and the median is the (7 + 1)/2 th value =
4th value. The 4th value is 6.
Copyright © Matthew Pinkney 2003
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GCSE Maths > Statistics and Probability - Averages