Answer :
The coefficient of Karl-Pearson's skewness for the given data is approximately 0.35.
To calculate the coefficient of Karl-Pearson's skewness, calculate the mean, standard deviation, and median of the given data.
Here's how we can proceed:
Age in year (x) | No. of employees (f)
50-55 | 25
45-50 | 30
40-45 | 40
35-40 | 45
30-35 | 80
25-30 | 110
20-25 | 170
Step 1: Calculate the midpoint of each age class
To calculate the midpoint, take the average of the lower and upper limits of each age class.
Midpoint (x) | No. of employees (f)
52.5 | 25
47.5 | 30
42.5 | 40
37.5 | 45
32.5 | 80
27.5 | 110
22.5 | 170
Step 2: Calculate the mean ([tex]\bar{x}[/tex])
The mean is calculated by multiplying each midpoint by its corresponding frequency, summing them up, and dividing by the total number of employees.
[tex]\bar{x}[/tex] = (52.525 + 47.530 + 42.540 + 37.545 + 32.580 + 27.5110 + 22.5*170) / (25 + 30 + 40 + 45 + 80 + 110 + 170)
= 32.85
Step 3: Calculate the standard deviation ()
The standard deviation is calculated using the formula:
= √[∑((− [tex]\bar{x}[/tex][tex])^2[/tex]) / ]
where is the total number of employees.
[tex] = \sqrt(25(52.5-32.85)^2 + 30(47.5-32.85)^2 + 40(42.5-32.85)^2 + 45(37.5-32.85)^2 + 80(32.5-32.85)^2 + 110(27.5-32.85)^2 + 170(22.5-32.85)^2) / (25 + 30 + 40 + 45 + 80 + 110 + 170)][/tex]
≈ 15.43
Step 4: Calculate the median ()
The median is the middle value when the data is arranged in ascending order. In this case, we already have the data sorted by midpoint, so the median is the midpoint corresponding to the cumulative frequency closest to (N + 1)/2.
In this case, (N + 1)/2 = (500 + 1)/2 = 250.5, which falls between the cumulative frequencies of 110 and 170.
Therefore, the median falls within the range of 27.5-32.5. Since the cumulative frequency of 110 corresponds to the midpoint of 27.5, the median () is 27.5.
Step 5: Calculate the coefficient of Karl-Pearson's skewness (SK)
The coefficient of Karl-Pearson's skewness is calculated using the formula:
SK = ([tex]\bar{x}[/tex] − ) /
SK = (32.85 − 27.5) / 15.43
SK≈ 0.35
