6.3 Standard Deviation and zScores
Variance and Standard Deviation
Sometimes we want to find out how far the data values in a data set are from the middle
of a data set. In that case, we can find the variance or standard deviation.
x is the data value, μ is the mean and N is the population size, and n is the sample
size.
Population Variance
σ2 =
∑ (x−μ)2
N
Sample Variance
s2 =
∑ (x− xˉ)2
n−1
Population Standard Deviation
σ=
∑ (x−μ)2
N
=
∑ x2 −N ∗μ2
N
Sample Standard Deviation
s=
(∑ (x− xˉ2 )
n−1
=
∑ x2 −n∗xˉ2
n−1
z-Scores
When we have the standard deviation, we can describe how far data values are from
the mean by determining how many standard deviations they are from the mean. This is
knows the z-score.
The more standard deviation a data value is from the mean, the further away from
the mean it is.
6.3 Standard Deviation and z-Scores
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To get the z-score, we get the data value and subtract it by the mean, then we
divide by the standard deviation.
Population z
=
x−μ
σ
Sample z-score z
=
x−xˉ
s
6.3 Standard Deviation and z-Scores
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