Answer :
To find the standard deviation for the given age distribution, we follow these steps:
1. Identify the Age Intervals and Frequencies:
- The age intervals are: 20-25, 25-30, 30-35, 35-40, 40-45, and 45-50.
- The corresponding number of persons (frequencies) in each interval are: 170, 110, 80, 45, 40, and 35.
2. Calculate the Midpoint of Each Interval:
- For each age interval, the midpoint is calculated as [tex]\((\text{Starting Age} + \text{Ending Age}) / 2\)[/tex].
- Midpoints for each interval are:
- [tex]\( (20 + 25)/2 = 22.5 \)[/tex]
- [tex]\( (25 + 30)/2 = 27.5 \)[/tex]
- [tex]\( (30 + 35)/2 = 32.5 \)[/tex]
- [tex]\( (35 + 40)/2 = 37.5 \)[/tex]
- [tex]\( (40 + 45)/2 = 42.5 \)[/tex]
- [tex]\( (45 + 50)/2 = 47.5 \)[/tex]
3. Calculate the Mean of the Distribution:
- The mean is calculated by taking the sum of the products of midpoints and their corresponding frequencies, divided by the total number of persons.
- Mean = [tex]\((22.5 \times 170 + 27.5 \times 110 + 32.5 \times 80 + 37.5 \times 45 + 42.5 \times 40 + 47.5 \times 35) / (170 + 110 + 80 + 45 + 40 + 35)\)[/tex].
- The mean is approximately 30.21 (rounded to two decimal places).
4. Calculate the Variance:
- For each interval, calculate the square of the difference between each midpoint and the mean, then multiply by the corresponding frequency.
- Variance = [tex]\(\left((22.5 - 30.21)^2 \times 170 + (27.5 - 30.21)^2 \times 110 + \ldots + (47.5 - 30.21)^2 \times 35\right) / (170 + 110 + 80 + 45 + 40 + 35)\)[/tex].
- The variance is approximately 62.98.
5. Calculate the Standard Deviation:
- The standard deviation is the square root of the variance.
- Standard deviation = [tex]\(\sqrt{62.98}\)[/tex].
- The standard deviation is approximately 7.94 (rounded to two decimal places).
Therefore, the standard deviation of the age distribution is approximately 7.94.
1. Identify the Age Intervals and Frequencies:
- The age intervals are: 20-25, 25-30, 30-35, 35-40, 40-45, and 45-50.
- The corresponding number of persons (frequencies) in each interval are: 170, 110, 80, 45, 40, and 35.
2. Calculate the Midpoint of Each Interval:
- For each age interval, the midpoint is calculated as [tex]\((\text{Starting Age} + \text{Ending Age}) / 2\)[/tex].
- Midpoints for each interval are:
- [tex]\( (20 + 25)/2 = 22.5 \)[/tex]
- [tex]\( (25 + 30)/2 = 27.5 \)[/tex]
- [tex]\( (30 + 35)/2 = 32.5 \)[/tex]
- [tex]\( (35 + 40)/2 = 37.5 \)[/tex]
- [tex]\( (40 + 45)/2 = 42.5 \)[/tex]
- [tex]\( (45 + 50)/2 = 47.5 \)[/tex]
3. Calculate the Mean of the Distribution:
- The mean is calculated by taking the sum of the products of midpoints and their corresponding frequencies, divided by the total number of persons.
- Mean = [tex]\((22.5 \times 170 + 27.5 \times 110 + 32.5 \times 80 + 37.5 \times 45 + 42.5 \times 40 + 47.5 \times 35) / (170 + 110 + 80 + 45 + 40 + 35)\)[/tex].
- The mean is approximately 30.21 (rounded to two decimal places).
4. Calculate the Variance:
- For each interval, calculate the square of the difference between each midpoint and the mean, then multiply by the corresponding frequency.
- Variance = [tex]\(\left((22.5 - 30.21)^2 \times 170 + (27.5 - 30.21)^2 \times 110 + \ldots + (47.5 - 30.21)^2 \times 35\right) / (170 + 110 + 80 + 45 + 40 + 35)\)[/tex].
- The variance is approximately 62.98.
5. Calculate the Standard Deviation:
- The standard deviation is the square root of the variance.
- Standard deviation = [tex]\(\sqrt{62.98}\)[/tex].
- The standard deviation is approximately 7.94 (rounded to two decimal places).
Therefore, the standard deviation of the age distribution is approximately 7.94.
