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5. Calculating standard deviation and variance using the computational formula
Consider a data set containing the following values:
93
83
85
95
97
90
86
91
The mean of the given values is 90. The squares of the scores have been calculated as follows:
8,649
6,889
7,225
9,025
9,409
8,100
7,396
8,281
If the preceding scores are sample data, the sum of squares (SS) is
174
. (
Hint
: Remember that the phrase “sum of squares” or SS refers to
the sum of squared deviations.)
Explanation:
There are two ways to calculate the sum of squares (SS) for a sample of n = 8 scores. You can use the definition SS = Σ(X – M)², or the
computational formula SS = ΣX² – (ΣX)² / n. Here you have been given the squares of the scores, so you should use the computational
formula. However, you could use the definition and would end up with the same answer (or a slightly different value, due to rounding error), but
it would require more work.
To use the computational formula, you simply need to sum the scores to obtain ΣX and sum the squares of the scores to obtain ΣX². Note that
because you have also been given the mean, M = 90 = ΣX / 8, you could also just multiply M by 8 to obtain ΣX.
ΣX = 93 + 83 + 85 + 95 + 97 + 90 + 86 + 91 = 720 = 8 x 90
ΣX² = 8,649 + 6,889 + 7,225 + 9,025 + 9,409 + 8,100 + 7,396 + 8,281 = 64,974
Then, SS = ΣX² – (ΣX)² / 8 = 64,974 – (720)² / 8 = 174.
If the preceding scores are population data, the SS is
174
.
Explanation:
There is no difference between the sum of squares for sample data and population data, except that instead of the sample mean M, you use the
population mean μ, and instead of the sample size n, your sample size is denoted N. However, if this set of scores is considered a population
instead of a sample, what was denoted as M is μ, and what was denoted as n is N. Thus, the sum of squares is identical regardless of whether
this set of scores is considered a sample or a population.
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If the preceding scores are sample data, the sample standard deviation is
4.99 and the sample variance is
24.86 .
Explanation:
The sample standard deviation, s, is the square root of the sum of the squares (SS) divided by (n – 1), expressed by the following formula:
The sum of squares was calculated as SS = 174, and there are n = 8 scores. Therefore, the sample standard deviation is s = √(174/7) =
4.9857, or 4.99.
The sample variance of a set of scores is the square of the sample standard deviation. For sample scores, the variance s² = SS / (n – 1) =
174/7 = 24.8571, or 24.86.
If the preceding scores are population data, the population standard deviation is
4.66 and the population variance is
21.75 .
Explanation:
When the data form a population of size N with mean μ, the population standard deviation is given by the square root of the sum of the squared
deviations from the population mean divided by the population size, as shown in the following formula:
The sum of squares was calculated as SS = 174, and there are N = 8 scores. Therefore, the population standard deviation σ = √(174/8) =
4.6637, or 4.66.
The variance of a set of scores is the square of the standard deviation. For population scores, the variance σ² = SS/N = 174/8 = 21.75.
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