Age distribution of employees:
| Age (Years) | Employees |
| ----------- | --------- |
| 20-30 | 5 |
| 30-40 | 8 |
| 40-50 | 12 |
| 50-60 | 10 |
| 60-70 | 5 |
Calculate the standard deviation using the assumed mean method.
Answer (2)
Calculate the midpoints of each age group.
Determine the assumed mean (A) as 45.
Calculate the deviations and find ∑ f i d i = 20 and ∑ f i d i 2 = 5800 .
Apply the standard deviation formula: S D = 40 5800 − ( 40 20 ) 2 ≈ 12.03 .
Explanation
Understand the problem We are given the age distribution of employees and asked to calculate the standard deviation using the assumed mean method.
Calculate Midpoints First, we need to find the midpoint of each class interval. The midpoint is calculated as the average of the lower and upper limits of the class.
List Midpoints The midpoints are:
20-30: 2 20 + 30 = 25
30-40: 2 30 + 40 = 35
40-50: 2 40 + 50 = 45
50-60: 2 50 + 60 = 55
60-70: 2 60 + 70 = 65
Choose Assumed Mean Next, we choose an assumed mean (A). It's convenient to choose the midpoint of the class with the highest frequency, which is 40-50. So, A = 45.
Calculate Deviations Now, we calculate the deviation ( d i ) for each class interval using the formula: d i = x i − A , where x i is the midpoint of the class.
List Deviations The deviations are:
20-30: 25 − 45 = − 20
30-40: 35 − 45 = − 10
40-50: 45 − 45 = 0
50-60: 55 − 45 = 10
60-70: 65 − 45 = 20
Calculate f_i * d_i We calculate the product of the frequency ( f i ) and the deviation ( d i ) for each class interval: f i ∗ d i .
List f_i * d_i The products are:
20-30: 5 ∗ − 20 = − 100
30-40: 8 ∗ − 10 = − 80
40-50: 12 ∗ 0 = 0
50-60: 10 ∗ 10 = 100
60-70: 5 ∗ 20 = 100
Calculate Sum of f_i * d_i We calculate the sum of the products f i ∗ d i : ∑ f i d i = − 100 − 80 + 0 + 100 + 100 = 20 .
Calculate d_i^2 We calculate the square of the deviation ( d i 2 ) for each class interval: d i 2 .
List d_i^2 The squared deviations are:
20-30: ( − 20 ) 2 = 400
30-40: ( − 10 ) 2 = 100
40-50: ( 0 ) 2 = 0
50-60: ( 10 ) 2 = 100
60-70: ( 20 ) 2 = 400
Calculate f_i * d_i^2 We calculate the product of the frequency ( f i ) and the square of the deviation ( d i 2 ) for each class interval: f i ∗ d i 2 .
List f_i * d_i^2 The products are:
20-30: 5 ∗ 400 = 2000
30-40: 8 ∗ 100 = 800
40-50: 12 ∗ 0 = 0
50-60: 10 ∗ 100 = 1000
60-70: 5 ∗ 400 = 2000
Calculate Sum of f_i * d_i^2 We calculate the sum of the products f i ∗ d i 2 : ∑ f i d i 2 = 2000 + 800 + 0 + 1000 + 2000 = 5800 .
Calculate Total Employees (N) We calculate the total number of employees (N): N = ∑ f i = 5 + 8 + 12 + 10 + 5 = 40 .
Calculate Standard Deviation Finally, we apply the formula for standard deviation using the assumed mean method: S D = N ∑ f i d i 2 − ( N ∑ f i d i ) 2 = 40 5800 − ( 40 20 ) 2 = 145 − ( 0.5 ) 2 = 145 − 0.25 = 144.75 ≈ 12.03
State Final Answer Therefore, the standard deviation of the age distribution is approximately 12.03 years.
Examples
Understanding the standard deviation of employee ages can help companies plan for age-related workforce changes. For example, a high standard deviation might indicate a wide range of experience levels, requiring diverse training programs. Conversely, a low standard deviation could suggest a more uniform age group, potentially influencing retirement planning and succession strategies. By analyzing the spread of ages, businesses can make informed decisions about human resources and talent management.
To find the standard deviation of the age distribution, we calculated midpoints, deviations, and then applied the formula for standard deviation using the assumed mean method. The final result shows that the standard deviation is approximately 12.03 years. This value indicates the spread of employee ages around the assumed mean.
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