Answer :
Sure, let's find the mean and the median of the given age distribution for the 70 employees in a company.
### Step-by-Step Solution:
#### a) Mean Calculation
1. Class Intervals and Frequencies:
- Class intervals: 19-20, 21-25, 26-35, 36-40
- Frequencies: 2, 10, 18, 40
2. Midpoints of Each Class Interval:
To find the midpoint (also called class mark) of each interval, use the formula \((\text{lower boundary} + \text{upper boundary}) / 2\):
- Midpoint of 19-20 = \((19 + 20) / 2 = 19.5\)
- Midpoint of 21-25 = \((21 + 25) / 2 = 23\)
- Midpoint of 26-35 = \((26 + 35) / 2 = 30.5\)
- Midpoint of 36-40 = \((36 + 40) / 2 = 38\)
3. Total Frequency:
The total frequency is the sum of all frequencies:
[tex]\[ 2 + 10 + 18 + 40 = 70 \][/tex]
4. Mean Calculation:
The mean can be found using the formula:
[tex]\[ \text{Mean} = \frac{\sum (\text{midpoint} \times \text{frequency})}{\text{total frequency}} \][/tex]
Plugging in the values:
[tex]\[ \sum (\text{midpoint} \times \text{frequency}) = (19.5 \times 2) + (23 \times 10) + (30.5 \times 18) + (38 \times 40) \][/tex]
Simplifying:
[tex]\[ = 39 + 230 + 549 + 1520 = 2338 \][/tex]
Finally, the mean:
[tex]\[ \text{Mean} = \frac{2338}{70} \approx 33.4 \][/tex]
#### b) Median Calculation
1. Cumulative Frequency:
To find the median, we first need the cumulative frequency for each class:
- Cumulative frequency of 19-20 = 2
- Cumulative frequency of 21-25 = 2 + 10 = 12
- Cumulative frequency of 26-35 = 12 + 18 = 30
- Cumulative frequency of 36-40 = 30 + 40 = 70
2. Determine the Median Class:
The median class is the class where the cumulative frequency exceeds half of the total frequency. Here, half of 70 is 35.
- Cumulative frequencies:
- 2 (first interval)
- 12 (second interval)
- 30 (third interval)
- 70 (fourth interval)
The median class is therefore 36-40 as its cumulative frequency is the first to exceed 35.
3. Median Calculation:
For the median, use the formula:
[tex]\[ \text{Median} = L + \left(\frac{\frac{N}{2} - CF}{f}\right) \times h \][/tex]
Where:
- \(L\) = lower boundary of the median class (36)
- \(N\) = total frequency (70)
- \(CF\) = cumulative frequency before the median class (30)
- \(f\) = frequency of the median class (40)
- \(h\) = class width (\(40 - 36 = 4\))
Plugging in the values:
[tex]\[ \text{Median} = 36 + \left(\frac{35 - 30}{40}\right) \times 4 \][/tex]
Simplifying:
[tex]\[ \text{Median} = 36 + \left(\frac{5}{40}\right) \times 4 \][/tex]
[tex]\[ \text{Median} = 36 + 0.5 = 36.5 \][/tex]
### Final Results:
- Mean: 33.4
- Median: 36.5
### Step-by-Step Solution:
#### a) Mean Calculation
1. Class Intervals and Frequencies:
- Class intervals: 19-20, 21-25, 26-35, 36-40
- Frequencies: 2, 10, 18, 40
2. Midpoints of Each Class Interval:
To find the midpoint (also called class mark) of each interval, use the formula \((\text{lower boundary} + \text{upper boundary}) / 2\):
- Midpoint of 19-20 = \((19 + 20) / 2 = 19.5\)
- Midpoint of 21-25 = \((21 + 25) / 2 = 23\)
- Midpoint of 26-35 = \((26 + 35) / 2 = 30.5\)
- Midpoint of 36-40 = \((36 + 40) / 2 = 38\)
3. Total Frequency:
The total frequency is the sum of all frequencies:
[tex]\[ 2 + 10 + 18 + 40 = 70 \][/tex]
4. Mean Calculation:
The mean can be found using the formula:
[tex]\[ \text{Mean} = \frac{\sum (\text{midpoint} \times \text{frequency})}{\text{total frequency}} \][/tex]
Plugging in the values:
[tex]\[ \sum (\text{midpoint} \times \text{frequency}) = (19.5 \times 2) + (23 \times 10) + (30.5 \times 18) + (38 \times 40) \][/tex]
Simplifying:
[tex]\[ = 39 + 230 + 549 + 1520 = 2338 \][/tex]
Finally, the mean:
[tex]\[ \text{Mean} = \frac{2338}{70} \approx 33.4 \][/tex]
#### b) Median Calculation
1. Cumulative Frequency:
To find the median, we first need the cumulative frequency for each class:
- Cumulative frequency of 19-20 = 2
- Cumulative frequency of 21-25 = 2 + 10 = 12
- Cumulative frequency of 26-35 = 12 + 18 = 30
- Cumulative frequency of 36-40 = 30 + 40 = 70
2. Determine the Median Class:
The median class is the class where the cumulative frequency exceeds half of the total frequency. Here, half of 70 is 35.
- Cumulative frequencies:
- 2 (first interval)
- 12 (second interval)
- 30 (third interval)
- 70 (fourth interval)
The median class is therefore 36-40 as its cumulative frequency is the first to exceed 35.
3. Median Calculation:
For the median, use the formula:
[tex]\[ \text{Median} = L + \left(\frac{\frac{N}{2} - CF}{f}\right) \times h \][/tex]
Where:
- \(L\) = lower boundary of the median class (36)
- \(N\) = total frequency (70)
- \(CF\) = cumulative frequency before the median class (30)
- \(f\) = frequency of the median class (40)
- \(h\) = class width (\(40 - 36 = 4\))
Plugging in the values:
[tex]\[ \text{Median} = 36 + \left(\frac{35 - 30}{40}\right) \times 4 \][/tex]
Simplifying:
[tex]\[ \text{Median} = 36 + \left(\frac{5}{40}\right) \times 4 \][/tex]
[tex]\[ \text{Median} = 36 + 0.5 = 36.5 \][/tex]
### Final Results:
- Mean: 33.4
- Median: 36.5
