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QMS 102 Population vs Sample - Variance and Standard Deviation We only do calculations on samples to estimate corresponding population parameters. ie Therefore the formula for variance and standard deviation must be modified when used on a sample data. As with the range, the sample calculation needs to be modified to get an estim ate of the population Range. QMS 102 Population vs Sample - Variance and Standard Deviation Ex #1 Consider the population with one observation, 2. Clearly :=2 and the F = 0. ))))))0)))))))))x))))))))0))))))))))0))))))))))))) 2 Ex #2 Consider the sam ple with one observation, 3. 0'3 :.3 However there is no evidence about the population variance F. ))))))0)))))))))0)))))))))x)))))))))0))))))))))))) 3 If the population m ean was known [a very unlikely situation] to be 2. Then an estimate of the population variance would be (3-2)2/1=1. ))))))0)))))))))0)))))))))x)))))))))0))))))))))))) : 3 QMS 102 Population vs Sample - Variance and Standard Deviation Ex #3 Consider the sam ple with two observations, 3 and 4. ))))))0)))))))))0)))))))))x)))))))))x))))))))))))) 3 4 =(3+4)/2=3.5 However there is only one actual interval to indicate the population variance [even though we divide this interval into two parts]. Effectively, the equivalent of one observation has been used to estimate the mean, leaving the equivalent of n-1 "gaps" to estimate the population variance. The number of left over observations is often referred to as “degrees of freedom” Sample Standard Deviation: QMS 102 Population vs Sample - Variance and Standard Deviation Use the calculator to compute Sample Mean Firs t Four W eek= Firs t Four W eek = Sample Variance First Four W eek = s 2 First Four W eek = Sample Standard Deviation First Four W eek = s First Four W eek = First Four W eeks: 5,6,12,15,18,4,5,4,14,21,0,7,8,9,16,6,6,12,22,25 Last W eek: 1,3,9,14,16 Sample Mean Last Week = Last Week = Sample Variance Last Week = s 2 Last Week = Sample Standard Deviation Last Week =s Last Week = )