High School

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17. (a) The following data relate to the age of a group of workers. Calculate the arithmetic mean and standard deviation.

| Age | 20-25 | 25-30 | 30-35 | 35-40 | 40-45 | 45-50 | 50-55 |
|--------------|-------|-------|-------|-------|-------|-------|-------|
| No. of workers | 170 | 110 | 80 | 45 | 40 | 30 | 25 |

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).

  1. Calculate [tex](x_i - \bar{x})^2[/tex] for each group.
  2. Multiply each [tex](x_i - \bar{x})^2[/tex] by the corresponding frequency [tex]f[/tex].
  3. Sum all these values.
  4. 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.

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