15.5. Variance http://www.ck12.org
Concept Problem Revisited
The mean of the test scores is 75. The variance is calculated by taking the difference of each number from the mean,
squaring and summing these differences.
02 + 22 + 32 + 152 + 152 + 242 + 122 + 42 + 52 + 22 = 1228
Since this data is a sample, you divide the sum by one fewer than the number of terms.
101228 − 1 ≈^136.^4444
If you knew the variances for two samples, each from a different class, you could quickly determine which class had
test scores that were more spread out.
Vocabulary
Varianceis a measure of how spread out the data is.
The square root of the variance is thestandard deviation.
Both the variance and the standard deviation can be calculated from asampleor from the wholepopulation. The
formulas are slightly different in each case so it is important to know whether your data is just a sample or is from
the whole population.
Theabsolute deviationis the sum total of how different each number is from the mean.
Themean absolute deviationis an alternate measure of how spread out the data is. While this method might seem
more intuitive, in statistics it has been found to be too limited and is not commonly used.
Guided Practice
- Calculate the standard deviation for the following 6 numbers by hand. Assume the numbers are a population.
2, 4, 6, 8, 12, 19 - Use a spreadsheet to organize your calculations for computing the variance of the following numbers. Assume
these numbers are a true population.
14, 15, 7, 15, 2, 0, 6, 5, 12, 3
Answers:
μ=^16 ( 2 + 4 + 6 + 8 + 12 + 17 ) = 8
σ^2 =^16 (( 8 − 2 )^2 +( 8 − 4 )^2 +( 8 − 6 )^2 + 0 +( 8 − 12 )^2 +( 8 − 17 )^2 )
=^16 ( 62 + 42 + 22 + 42 + 92 )
=^16 ( 36 + 16 + 4 + 16 + 81 )
=^16 ( 153 )
= 25. 5
σ≈ 5. 0498