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Standard deviation
•
Improving the value of standard deviation
– Sometimes you may have a small set of results,
but the same type of analysis has been done
many times in the past on other samples
– You can use standard deviations calculated from
You can use standard deviations calculated from
historical data to get lower confidence limits
– This is referred to as "pooling" data
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Pooled standard deviation
•
Example. Refer to data sets "A" and "B"
2
Recall: Confidence limits
•
A: 60.53, 60.53, 60.37, 60.37, 60.29, 60.29%
•
B: 60.53, 60.37, 60.29 %
– Calculate the 95% confidence intervals for the
sets of results A and B above
• This means that on repeated measurement you would
•
This means that on repeated measurement, you would
get a result within these ranges 95 times out of 100
•
Only 5% of the time would you get a result outside
this range
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Recall:Confidence limits
4
Pooled standard deviation
– A: 60.53, 60.53, 60.37, 60.37, 60.29, 60.29 %
– Mean = (
∑
x
i
)/
N
= 362.38/6 = 60.40 %
– Standard deviation (s) =
√
([
∑
(
x
i
–
x
)
2
]/
N
-1)
–
=
√
(0.0598/5) =
√
0.01196 = 0.1093 = 0.11 %
– 95% CI (
μ
) =
x
±
t*s
/
√
N
95% CI (
)
t s
/
–
= 60.40 ± 2.571*0.11 /
√
6
–
= 60.40 ± 0.2810/2.449 = 60.40 ± 0.11 %
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Pooled standard deviation
•
B: 60.53, 60.37, 60.29 %
– Mean = (
∑
x
i
)/
N
= 181.19/3 = 60.40 %
– Standard deviation (s) =
√
([
∑
(
x
i
–
x
)
2
]/
N
-1)
–
=
√
(0.0299/2) =
√
0.01495 = 0.1222 = 0.12 %
95% CI (
μ
) =
x
±
t*s
/
√
N
– 95% CI (
) =
/
–
= 60.40 ± 4.303*0.12 /
√
3
–
= 60.40 ± 0.5258/1.732 = 60.40 ± 0.30 %
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