The variance estimated as the average squared difference from the sample mean will always be less than the variance estimated as the average squared difference from the population mean unless the sample mean equals the population mean in which case they will be the same.
The sample mean is the value for which the sum of squared differences and therefore the average squared difference is minimized.
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Try a number of samples and compare the variance estimates. To sample the numbers click the Draw Four Numbers button. The four numbers will be sampled and the mean of the numbers displayed. The two variances estimates will be computed below.
You have a sample of 12 numbers from a population. The mean of the 12 numbers is 8 and the mean of the entire population is 7. Which formula would be better to use as an estimate of the population variance?
It would be better to use the population mean of 7 since you are estimating the average squared deviation from 7. You estimate this with the average of your 12 deviations from 7.
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Formula 1
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Formula 2
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You have a sample of 12 numbers from a population. The mean of the 12 numbers is 8 and the mean of the entire population is 7. You compute an estimate based on each of the formulas.
1. True: The estimate of the variance is more accurate if the population mean is used than if the sample mean is used; 2: False (see 1); 3: True Subtracting the sample mean (8) always gives the lowest estimate. On average it will be too low but in any sample it may be too high; 4: False Not necessarily
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1. The estimate based on Formula 1 will probably but not necessarily be closer to the population variance.
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2. The estimate based on Formula 1 will necessarily be closer to the population variance.
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3. The estimate based on Formula 2 will probably but not necessarily be lower than the population variance.
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4. The estimate based on Formula 2 will necessarily be lower than the population variance.
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Assume you repeatedly sampled 4 numbers from this population with a variance of 2 and, for each sample, estimated the variance using the average squared difference from the sample mean. What would the mean of these variance estimates be?
It would approach 1.5 as the number of samples increases to a very large number. This is equal to (N-1)/N = .75 times the population variance of 2.
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Continue clicking the Draw 4 Numbers button and see what the mean of the variances is. The mean can be found just to the left bottom of the scrolling text. Each entry in the scrolling text is the variance estmate for one of the samples.
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1.5
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Continue clicking the Draw 4 Numbers button and see what the mean of the variances is. The mean can be found just to the left bottom of the scrolling text. Each entry in the scrolling text is the variance estmate for one of the samples.