Answer :
To calculate the arithmetic mean and standard deviation of the age of a group of workers using grouped frequency data, we need to follow these steps:
Step 1: Determine the Midpoints of Each Age Group
The midpoint (or class mark) of each age group can be calculated by averaging the lower and upper boundaries of each class.
- Midpoint for 20-25 = (20 + 25)/2 = 22.5
- Midpoint for 25-30 = (25 + 30)/2 = 27.5
- Midpoint for 30-35 = (30 + 35)/2 = 32.5
- Midpoint for 35-40 = (35 + 40)/2 = 37.5
- Midpoint for 40-45 = (40 + 45)/2 = 42.5
- Midpoint for 45-50 = (45 + 50)/2 = 47.5
- Midpoint for 50-55 = (50 + 55)/2 = 52.5
Step 2: Calculate the Sum of Midpoints Multiplied by Frequencies
Multiply each midpoint by the number of workers in that age group, then sum these products:
[tex]\text{Total} = (22.5 \times 170) + (27.5 \times 110) + (32.5 \times 80) + (37.5 \times 45) + (42.5 \times 40) + (47.5 \times 30) + (52.5 \times 25)[/tex]
Step 3: Calculate the Arithmetic Mean
The arithmetic mean [tex]\bar{x}[/tex] can be calculated by dividing the total sum by the total number of workers:
[tex]\bar{x} = \frac{\text{Total}}{500}[/tex]
where the total number of workers is [tex]170 + 110 + 80 + 45 + 40 + 30 + 25 = 500[/tex].
Step 4: Calculate the Variance and Standard Deviation
For the variance [tex]\sigma^2[/tex], use the formula:
[tex]\sigma^2 = \frac{\sum f(x_i - \bar{x})^2}{N}[/tex]
Where:
- [tex]f[/tex] is the frequency of each group.
- [tex]x_i[/tex] is the midpoint of each group.
- [tex]N[/tex] is the total number of workers (500).
- Calculate [tex](x_i - \bar{x})^2[/tex] for each group.
- Multiply each [tex](x_i - \bar{x})^2[/tex] by the corresponding frequency [tex]f[/tex].
- Sum all these values.
- Divide by the total number of workers to get the variance.
Finally, the standard deviation [tex]\sigma[/tex] is the square root of the variance:
[tex]\sigma = \sqrt{\sigma^2}[/tex]
Conclusion:
Calculating these values step-by-step allows us to understand the distribution and spread of the workers' ages within this group. The arithmetic mean gives us a central value, while the standard deviation provides insights into the variability around that mean.
