For the following set of data, find the population standard deviation, to the nearest hundredth.
Data Frequency
13 77
14 99
28 66
29 44
30 66
32 66
39 66
40 33
50 22
To find the population standard deviation, we first need to calculate the mean of the data set.
Mean = Σ(x * f) / Σf, where x is the data point and f is the frequency.
Mean = (13*77 + 14*99 + 28*66 + 29*44 + 30*66 + 32*66 + 39*66 + 40*33 + 50*22) / (77 + 99 + 66 + 44 + 66 + 66 + 66 + 33 + 22)
Mean = (1001 + 1386 + 1848 + 1276 + 1980 + 2112 + 2574 + 1320 + 1100) / 483
Mean = 13597 / 483
Mean = 28.16 (approximately)
Next, we need to calculate the sum of squared differences from the mean (Σ((x - mean)^2 * f)).
Sum of squared differences = (13-28.16)^2 * 77 + (14-28.16)^2 * 99 + (28-28.16)^2 * 66 + (29-28.16)^2 * 44 + (30-28.16)^2 * 66 + (32-28.16)^2 * 66 + (39-28.16)^2 * 66 + (40-28.16)^2 * 33 + (50-28.16)^2 * 22
Sum of squared differences = (221.36 * 77) + (198.76 * 99) + (0.0272 * 66) + (1.3729 * 44) + (3.3144 * 66) + (12.5424 * 66) + (112.8496 * 66) + (352.2976 * 33) + (478.2464 * 22)
Sum of squared differences = 17056.72 + 19689.24 + 1.79 + 60.39 + 218.76 + 827.98 + 7456.18 + 11635.77 + 10501.42
Sum of squared differences = 67543.38
Now, we can calculate the population standard deviation by dividing the sum of squared differences by the total frequency and taking the square root of the result.
Population standard deviation = √(Sum of squared differences / Σf)
Population standard deviation = √(67543.38 / 483)
Population standard deviation = √139.98
Population standard deviation = 11.83 (approximately)
Therefore, the population standard deviation for the given data set is approximately 11.83.