√(126539.1 / 171) =
To calculate the appropriate measure of skewness for the given data, we can use the formula:
Skewness = [3(mean - median)] / standard deviation
First, we need to find the mean, median, and standard deviation of the data set.
To find the mean, we use the formula:
mean = (Σ(xi * fi)) / N
where xi is the midpoint of each class, fi is the frequency of each class, and N is the total number of workers.
Calculating the midpoint of each class:
5, 15, 25, 35, 45, 55
Multiplying each midpoint by its corresponding frequency, and summing the results:
(5*10) + (15*12) + (25*25) + (35*35) + (45*40) + (55*50) = 6300
Dividing by the total number of workers:
6300 / 172 = 36.63
So the mean is approximately 36.63.
To find the median, we need to first calculate the cumulative frequency for each class:
Class: 0-10 10-20 20-30 30-40 40-50 50-60
Frequency: 10 12 25 35 40 50
Cumulative: 10 22 47 82 122 172
The median is the middle value of the data set, which is the 86th value in this case. Looking at the cumulative frequencies, we see that the median falls in the 30-40 class. So the median class is 30-40, with a cumulative frequency of 82. The width of the class interval is 10, and the frequency of the median class is 35.
Using the formula:
median = L + [(N/2 - CF) / f] * w
where L is the lower limit of the median class, N is the total number of workers, CF is the cumulative frequency of the class preceding the median class, f is the frequency of the median class, and w is the width of the median class interval.
Substituting in the values, we get:
median = 30 + [(86/2 - 82) / 35] * 10 = 31.71
So the median is approximately 31.71.
To find the standard deviation, we can use the formula:
standard deviation = √[Σ(fi * (xi - mean)²) / (N - 1)]
where xi and fi are as before, mean is the mean of the data set, and N is the total number of workers.
Calculating the squared deviation of each midpoint from the mean, and multiplying by the frequency of each class:
[(5 - 36.63)² * 10] + [(15 - 36.63)² * 12] + [(25 - 36.63)² * 25] + [(35 - 36.63)² * 35] + [(45 - 36.63)² * 40] + [(55 - 36.63)² * 50] = 126539.1
Dividing by (N-1) and taking the square root:
√(126539.1 / 171) =