For the following set of data, find the population standard deviation, to the nearest hundredth.
Data Frequency
14 55
15 66
17 88
18 88
20 22
23 22
32 88
34 66
To find the population standard deviation, we can follow these steps:
1. Calculate the mean:
Mean = (14*55 + 15*66 + 17*88 + 18*88 + 20*22 + 23*22 + 32*88 + 34*66) / (55 + 66 + 88 + 88 + 22 + 22 + 88 + 66)
Mean = (770 + 990 + 1496 + 1584 + 440 + 506 + 2816 + 2244) / 485
Mean = 9326 / 485
Mean = 19.23
2. Calculate the squared difference between each data point and the mean:
(14-19.23)^2 * 55 + (15-19.23)^2 * 66 + (17-19.23)^2 * 88 + (18-19.23)^2 * 88 + (20-19.23)^2 * 22 + (23-19.23)^2 * 22 + (32-19.23)^2 * 88 + (34-19.23)^2 * 66
3. Sum up all these squared differences and divide by the total frequency (485):
Sum = 652.40
4. Calculate the variance:
Variance = Sum / 485
Variance = 1.35
5. Take the square root of the variance to find the standard deviation:
Standard Deviation ≈ √1.35 ≈ 1.16
Therefore, the population standard deviation for this data set is approximately 1.16 (rounded to the nearest hundredth).