Measure of spread

Range

Definitions

Range

Tells you how spread out a data set is by looking at the two extreme values. It’s the difference between the highest value and the lowest value.

Formula for ungrouped data as is follows:

$$\text{Range}=\text{Highest value}-\text{Lowest value}+1$$

Where data is grouped:

$$\text{Upper interval limit}-\text{lowest interval limit}$$

Standard deviation

Definitions

Standard deviation

Measures how far the data values typically lie from the mean. A larger standard deviation means the data is more spread out around the mean. A smaller standard deviation means the data is more tightly clustered around the mean.

The formula for standard deviation is:

$$\text{Standard deviation}=\sqrt{\frac{\sum F{X}^2}{\sum F}-{\left(\frac{\sum FX}{\sum F}\right)}^2}$$

Let’s use the previous data set. Notice that when the data is grouped into classes, the $X$ values are the midpoints of the classes. If the data is not grouped into classes, then $X$ is simply the value given.

Number of absent days	Frequency
0	5
1	3
2	1
3	3
4	3
5	3
6	1
7	1
Grand total	20

To apply the standard deviation formula, we first calculate the supporting values shown in the table below.

Number of absent days (X)	${\sqrt{X}}^2$	Frequency	FX	F${\sqrt{X}}^2$
0	0	5	0	0
1	1	3	3	3
2	4	1	2	4
3	9	3	9	27
4	16	3	12	48
5	25	3	15	75
6	36	1	6	36
7	49	1	7	49
Grand total		20	54	242

Final calculations:

$$\begin{array}{rl}\frac{242}{20}& =12.1\\ {\left(\frac{54}{20}\right)}^2& =7.29\\ 12.1-7.29& =4.81\\ \sqrt{4.81}& =2.19\end{array}$$

When you do standard deviation questions, make sure you follow the formula carefully and use the correct totals from your table. Remember: a smaller standard deviation means the data is less dispersed, so the mean is based on values that are closer together.

Coefficient of variation

Definitions

Coefficient of variation

Combines the mean and the standard deviation to help you compare the spread of two different data sets, especially when their means are not the same.

Comparing standard deviations directly can be misleading when the data sets are on different scales. For example, salaries in two companies may have different average levels. A company with higher salaries will often have a higher standard deviation simply because the numbers are larger. The coefficient of variation adjusts for this by measuring spread relative to the mean, making the comparison fairer.

A is a management accountant and he has been offered a position by 2 companies in country 7. The following are the average salaries for management accountants in both companies including the standard deviations. Calculate the coefficient of variation and indicate which company the accountant should choose based on your calculations?

Company	Mean	St deviation
KTA	100000	10000
SBA	120000	20000

Solution