CIMA BA2 Syllabus D. DECISION MAKING - Measures of Average - Notes 5 / 7
Measures of average and dispersion
Variables can be:
Discrete variables which can only consist of certain values (ungrouped data), for example a company recording the number of complaints received per week over the last year.
This data is unrefined data.
Continuous variables where we group the variables, for example a company recording the total amount paid to employees each week over the last year.
This data has been refined and grouped.
Measures of average for ungrouped data
There are 3 measures of average that we need to be aware of for ungrouped data:
Arithmetic mean
This is calculated by adding up all of the observations and dividing by the number of observations
Median
This is the centrally occurring observation when all of the observations are arranged in order of magnitude
Mode
This is the most frequently occurring observation
Illustration 1
A company has recorded the number of complaints received per week over the past thirteen weeks, and has produced the following table:
Calculate:
1) The arithmetic mean
2) The median
3) The mode
| Number of complaints (Observations) | Frequency (No. of weeks) |
|---|---|
| 0 | 1 |
| 1 | 6 |
| 2 | 4 |
| 3 | 2 |
| 13 weeks |
Solution
The mean is: 20/13 = 1.54 (table below)
Note that when a table is given like this with variables and frequency, it is known as a frequency distribution, the formula that can be used for calculating the mean is: x̄ (sign for the mean)= ∑fx/∑f
| Mean = total number of complaints/total weeks = 20/13 = 1.54 | ||
|---|---|---|
| Number of complaints (observations) | Frequency (weeks) | Total number of complaints (Complaints x Weeks) |
| 0 | 1 | 0 |
| 1 | 6 | 6 |
| 2 | 4 | 8 |
| 3 | 2 | 6 |
| 13 | 20 |
The median is 1, as this the most centrally occurring observation when all of the observations were arranged in order of magnitude (table below)
Here, our data observations total to an odd number, therefore it is easy to find the middle value when the items are arranged in ascending order.
However, if the data observations were an even number, then we would take the 2 middle values when the items are arranged in ascending order and find the average of those 2 numbers.
| Week | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 |
| Complaints (In order of magnitude) | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 3 |
The mode is 1, as this is the most frequently occurring observation.
If there were 2 modes (for example, if 2 was also repeated 6 times, as 1 is - this would be known as bi-modal.
If there were more than 2 modes, this would be known as a multi-modal.
Measures of average for grouped data
We need to be aware of how to calculate the arithmetic mean for grouped data, to do this we must:
1) Find the mid point of our observations
2) Total mid points (mid point x number of times it was observed)
3) Divide by the number of observations.
Illustration 2
A company has recorded the total amount paid to employees each week over the last year in the following table:
Calculate the arithmetic mean
| Total paid ($) | Frequency (weeks) |
|---|---|
| 0 - under 500 | 1 |
| 500 - under 1,000 | 4 |
| 1,000 - under 1,500 | 8 |
| 1,500 - under 2,000 | 19 |
| 2,000 - under 2,500 | 14 |
| 2,500 - under 3,000 | 6 |
| 52 |
Solution
The arithmetic mean = 94,500 / 52 = $1,817 (table below)
| Total paid ($) | Mid point ($) | Frequency (weeks) | Total paid using mid point (mid point x freq) |
|---|---|---|---|
| 0 - under 500 | 250 (0+500)/2 | 1 | 250 |
| 500 - under 1,000 | 750 (500 + 1000) / 2 | 4 | 3,000 |
| 1,000 - under 1,500 | 1,250 | 8 | 10,000 |
| 1,500 - under 2,000 | 1,750 | 19 | 33,250 |
| 2,000 - under 2,500 | 2,250 | 14 | 31,500 |
| 2,500 - under 3,000 | 2,750 | 6 | 16,500 |
| 52 | 94,500 |
How would we find the mode and the median of grouped data?
To find the mode in grouped data, we would need a histogram, go to the highest class of data (the data which takes up the most area under the histogram) and find the point of intersection in this class.
In the exam, you will probably be given a histogram and you will need to estimate it from the histogram.
To find the median in grouped data, you will need to use an ogive graph, work out the median value from the y axis and then read off the value that corresponds on the x axis from the graph.