Standard Deviations and Standard Errors
Standard deviation (computational formula)
s=
(x − x)
2
n −1
Interpreting s:
1. Sample standard deviation is roughly how far the sample observations are from the sample
mean on average.
2. Sample standard deviation is roughly the average distance between observations and the
sample mean.
Standard deviation of a sample proportion
p(1 − p)
n
pˆ = SD( pˆ ) =
Interpretation:
1. This standard deviation is roughly how far sample proportion values would be from the true
proportion on average for random samples of the same size (i.e. n must be fixed).
2. On average, sample proportion values would be roughly this standard deviation away from the
true proportion for random samples of same size.
Standard deviation of a sample mean
x = SD( x ) =
n
Interpretation:
1. This standard deviation is roughly how far sample mean values would be from the true mean on
average for random samples of the same size (i.e. n must be fixed).
2. On average, sample mean values would be roughly this standard deviation away from the true
mean for random samples of the same size.
Note: Standard error is ESTIMATED standard deviation.
Standard error of a sample proportion
pˆ (1 − pˆ )
n
SE ( pˆ ) =
Interpretation:
This is the estimated approximate average distance between sample proportion values and the true
proportion for random samples of the same size.
SD( pˆ ) 0 =
Null standard deviation (error) for a sample proportion
p0 (1 − p0 )
n
This is the estimated approximate average distance between sample proportion values and the true
proportion for random samples of the same size assuming the true proportion is p 0 .
Standard error of a sample mean
SE ( x ) =
s
n
This is the estimated approximate average distance between sample mean values and the true mean
value for random samples of the same size.
Try to interpret yourself (as we get to them):
Standard error of a sample mean difference
Standard error of the difference in two sample means
Standard error of a difference in two sample proportions
Standard Deviations and Standard Errors
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