For the following set of data, find the population standard deviation, to the nearest thousandth.
Data Frequency
33 66
55 33
66 77
77 11
88 11
1111 11
1313 66
1414 99
1515 22
To find the population standard deviation, we need to calculate the weighted mean and then the weighted variance.
Weighted Mean:
(33 * 66 + 55 * 33 + 66 * 77 + 77 * 11 + 88 * 11 + 1111 * 11 + 1313 * 66 + 1414 * 99 + 1515 * 22) / (66 + 33 + 77 + 11 + 11 + 11 + 66 + 99 + 22)
= (2178 + 1815 + 5082 + 847 + 968 + 12221 + 86598 + 140286 + 33330) / 385
= 29665 / 385
= 77.06493506
Weighted Variance:
((33 - 77.06493506)^2 * 66 + (55 - 77.06493506)^2 * 33 + (66 - 77.06493506)^2 * 77 + (77 - 77.06493506)^2 * 11 + (88 - 77.06493506)^2 * 11 + (1111 - 77.06493506)^2 * 11 + (1313 - 77.06493506)^2 * 66 + (1414 - 77.06493506)^2 * 99 + (1515 - 77.06493506)^2 * 22) / 385
= (1732.974558 * 66 + 484.7080821 * 33 + 121.6033998 * 77 + 0.004386832 * 11 + 114.4279426 * 11 + 1371908.446 * 11 + 21587935.09 * 66 + 24069946.85 * 99 + 833497.91 * 22) / 385
= 1143784592.44 / 385
= 2972925.542
Population Standard Deviation:
√2972925.542
≈ 1724.338
Therefore, the population standard deviation is approximately 1724.338 (to the nearest thousandth).