CIMA BA2 Syllabus D. DECISION MAKING - Standard Deviation - Notes 6 / 7
Standard Deviation
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
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)
Measures of dispersion
Dispersion is looking at the spread of the observations.
We need to know the following measures of dispersion:
Range
This is the difference between the highest and lowest of the observations
Variance
The variance is calculated as follows:
1) Find the difference between the observation and the arithmetic mean (For ungrouped data we will use the mid point observation)
2) Square the difference
3) Multiply the squared difference with the number of times that it occurred and find the total
4) Take an average of this totalStandard Deviation
This is the square root of the variance
Coefficient of variation
This is the standard deviation divided by the arithmetic mean
Illustration 3
Using Illustration 1 find the:
1) Range
2) Variance
3) Standard deviation
4) Coefficient of variation
Solution
Range 3 - 0 = 3 (Highest value - lowest value)
The variance (W1) = 9.21/13 = 0.71
Standard deviation √ 0.71 = 0.84
Coefficient of variation = 0.84/1.54 = 0.55
W1:
Number of complaints (C) | No of weeks (W) | C - 1.54 (mean) | (Sq) | WxSq |
---|---|---|---|---|
0 | 1 | -1.54 | 2.37 | 2.37 |
1 | 6 | -0.54 | 0.29 | 1.74 |
2 | 4 | +0.46 | 0.21 | 0.84 |
3 | 2 | +1.46 | 2.13 | 4.26 |
13 | 9.21 |
Formula to calculate SD with a frequency distribution, as seen above
√[∑fx²/ ∑f - X̄²]