Standard Deviation 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)
01
16
24
32
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 total

  • Standard 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
01-1.542.372.37
16-0.540.291.74
24+0.460.210.84
32+1.462.134.26
139.21

Formula to calculate SD with a frequency distribution, as seen above

√[∑fx²/ ∑f - X̄²]

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