For the following set of data, find the population standard deviation, to the nearest thousandth.
Data Frequency
1717 22
2222 33
2525 88
2626 99
2828 66
3333 99
3434 88
4343 11
4848 22
To calculate the population standard deviation, we need to first calculate the mean of the data set. The mean is calculated as the sum of (Data * Frequency) divided by the total frequency.
Mean = (1717*22 + 2222*33 + 2525*88 + 2626*99 + 2828*66 + 3333*99 + 3434*88 + 4343*11 + 4848*22) / (22 + 33 + 88 + 99 + 66 + 99 + 88 + 11 + 22)
Mean = (37774 + 73326 + 222200 + 259974 + 187008 + 330267 + 302192 + 47773 + 106656) / 528
Mean = 1586289 / 528
Mean = 3008.817
Next, we calculate the sum of the squared differences between each data point and the mean, multiplied by the frequency. Then we sum these values and divide by the total frequency to get the variance.
Variance = ((1717-3008.817)^2 * 22 + (2222-3008.817)^2 * 33 + (2525-3008.817)^2 * 88 + (2626-3008.817)^2 * 99 + (2828-3008.817)^2 * 66 + (3333-3008.817)^2 * 99 + (3434-3008.817)^2 * 88 + (4343-3008.817)^2 * 11 + (4848-3008.817)^2 * 22) / 528
Variance = (272997600.370 + 61034276.840 + 176580444.353 + 133162076.327 + 29240445.571 + 9652116.907 + 23288668.098 + 157808668.127 + 19331921.660) / 528
Variance = 1169815218.655 / 528
Variance = 2218131.522
Finally, the population standard deviation is the square root of the variance.
Standard deviation = √2218131.522
Standard deviation ≈ 1489.532
Therefore, the population standard deviation of the given data set is approximately 1489.532 to the nearest thousandth.