8 - Standard deviation Improving the value of standard...

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1 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 1 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 3
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2 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* / N 95% CI ( ) ts / = 60.40 ± 2.571*0.11 / 6 = 60.40 ± 0.2810/2.449 = 60.40 ± 0.11 % 5 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* / N – 95% CI ( ) = t*s / = 60.40 ± 4.303*0.12 / 3 = 60.40 ± 0.5258/1.732 = 60.40 ± 0.30 % 6
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3 Pooled standard deviation • Example. Refer to data sets "A" and "B" Suppose similar chloride analysis had been done on other samples and a pooled standard deviation calculated at 0.05 based on150 results You can use this value for your set as follows: Use s Æ σ = 0.05 Use t Æ z = 1.96 (bottom value in 95% column) – For the sets of results A and B, what will be the 95% confidence limits now? • Note that we are improving s ( σ ), while N is unchanged 7 Recall:Confidence limits Confidence levels for various values of z Confidence level, % z 50 0.67 68 1.00 80 1.28 8 90 1.64 95 1.96 95.4 2.00 99 2.58 99.7 3.00 99.9 3.29 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 %
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This note was uploaded on 04/12/2010 for the course ENGINEERIN 1ac3 taught by Professor Xxx during the Spring '10 term at McMaster University.

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8 - Standard deviation Improving the value of standard...

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