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For the following set of data, find the population standard deviation, to the nearest thousandth.
103,107,124,121,123,67,77,107,124

For the following set of data, find the population standard deviation, to the nearest thousandth.103,107,124,121,123,67,77,107,124103,107,124,121,123,67,77,107,124

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Q. For the following set of data, find the population standard deviation, to the nearest thousandth.103,107,124,121,123,67,77,107,124103,107,124,121,123,67,77,107,124
  1. List data set: List the data set and verify the number of data points.\newlineThe data set given is: 103,107,124,121,123,67,77,107,124103, 107, 124, 121, 123, 67, 77, 107, 124.\newlineCount the number of data points to ensure none are missing.\newlineThere are 99 data points in total.
  2. Calculate mean: Calculate the mean (average) of the data set.\newlineThe mean is calculated by adding all the data points together and then dividing by the number of data points.\newlineMean = (103+107+124+121+123+67+77+107+124)/9(103 + 107 + 124 + 121 + 123 + 67 + 77 + 107 + 124) / 9\newlineMean = 953/9953 / 9\newlineMean = 105.888105.888\dots
  3. Subtract and square: Subtract the mean from each data point and square the result.\newlineThis step is part of calculating the variance, which is the average of the squared differences from the Mean.\newline(103105.888)2=8.3472=69.672(103 - 105.888)^2 = 8.347^2 = 69.672\newline(107105.888)2=1.1122=1.237(107 - 105.888)^2 = 1.112^2 = 1.237\newline(124105.888)2=18.1122=328.045(124 - 105.888)^2 = 18.112^2 = 328.045\newline(121105.888)2=15.1122=228.472(121 - 105.888)^2 = 15.112^2 = 228.472\newline(123105.888)2=17.1122=292.922(123 - 105.888)^2 = 17.112^2 = 292.922\newline(67105.888)2=38.8882=1511.111(67 - 105.888)^2 = -38.888^2 = 1511.111\newline(77105.888)2=28.8882=834.672(77 - 105.888)^2 = -28.888^2 = 834.672\newline(107105.888)2=1.1122=1.237(107 - 105.888)^2 = 1.112^2 = 1.237\newline(124105.888)2=18.1122=328.045(124 - 105.888)^2 = 18.112^2 = 328.045
  4. Sum squared differences: Sum the squared differences.\newlineAdd all the squared differences together to get the total sum.\newlineTotal sum = 69.672+1.237+328.045+228.472+292.922+1511.111+834.672+1.237+328.04569.672 + 1.237 + 328.045 + 228.472 + 292.922 + 1511.111 + 834.672 + 1.237 + 328.045\newlineTotal sum = 3595.4133595.413
  5. Calculate variance: Calculate the variance.\newlineSince we are dealing with a population standard deviation, we divide the total sum of squared differences by the number of data points NN.\newlineVariance=Total sumN\text{Variance} = \frac{\text{Total sum}}{N}\newlineVariance=3595.4139\text{Variance} = \frac{3595.413}{9}\newlineVariance=399.490\text{Variance} = 399.490
  6. Calculate standard deviation: Calculate the population standard deviation.\newlineThe standard deviation is the square root of the variance.\newlineStandard deviation = Variance\sqrt{\text{Variance}}\newlineStandard deviation = 399.490\sqrt{399.490}\newlineStandard deviation 19.987\approx 19.987
  7. Round standard deviation: Round the standard deviation to the nearest thousandth.\newlineThe standard deviation rounded to the nearest thousandth is 19.98719.987.

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