Answer :
Sure, let's proceed with finding the mean and standard deviation step by step based on the given frequency table.
The frequency table is provided for the weights of 95 individuals with different mass intervals. Here's the step-by-step solution for calculating the mean and standard deviation:
### 1. Finding the Midpoint of Each Mass Interval
First, we need to determine the midpoint (or the class mark) of each mass interval. The midpoint is calculated as the average of the lower and upper limits of each interval:
- For [tex]$60-62$[/tex] kg: [tex]\((60 + 62) / 2 = 61\)[/tex] kg
- For [tex]$63-65$[/tex] kg: [tex]\((63 + 65) / 2 = 64\)[/tex] kg
- For [tex]$66-68$[/tex] kg: [tex]\((66 + 68) / 2 = 67\)[/tex] kg
- For [tex]$69-71$[/tex] kg: [tex]\((69 + 71) / 2 = 70\)[/tex] kg
- For [tex]$72-74$[/tex] kg: [tex]\((72 + 74) / 2 = 73\)[/tex] kg
### 2. Constructing a Table with Midpoints and Frequencies
Now we list the midpoints and their corresponding frequencies:
| Mass midpoint (kg) | Frequency |
|---------------------|-----------|
| 61 | 3 |
| 64 | 18 |
| 67 | 42 |
| 70 | 24 |
| 73 | 8 |
### 3. Calculating the Mean
The mean ([tex]\(\bar{x}\)[/tex]) is calculated as the sum of the products of each midpoint and its frequency divided by the total frequency:
Let [tex]\(\sum f \)[/tex] be the total frequency and [tex]\(\sum fm \)[/tex] be the sum of the product of frequency and midpoint.
[tex]\[ \sum fm = (61 \times 3) + (64 \times 18) + (67 \times 42) + (70 \times 24) + (73 \times 8) = 183 + 1152 + 2814 + 1680 + 584 = 6413 \][/tex]
[tex]\[ \sum f = 3 + 18 + 42 + 24 + 8 = 95 \][/tex]
[tex]\[ \text{Mean} = \bar{x} = \frac{\sum fm}{\sum f} = \frac{6413}{95} \approx 67.5053 \text{ kg} \][/tex]
### 4. Calculating the Variance
The variance ([tex]\(\sigma^2\)[/tex]) is calculated using the formula:
[tex]\[ \sigma^2 = \frac{\sum f (x - \bar{x})^2}{\sum f} \][/tex]
Where [tex]\(x\)[/tex] is the midpoint and [tex]\(\bar{x}\)[/tex] is the mean. We already have [tex]\(\sum f\)[/tex]:
[tex]\[ \sigma^2 = \frac{1}{95} \left[ 3(61 - 67.5053)^2 + 18(64 - 67.5053)^2 + 42(67 - 67.5053)^2 + 24(70 - 67.5053)^2 + 8(73 - 67.5053)^2 \right] \][/tex]
First, calculate each squared term:
[tex]\[ (61 - 67.5053)^2 \approx 42.2921 \][/tex]
[tex]\[ (64 - 67.5053)^2 \approx 12.2811 \][/tex]
[tex]\[ (67 - 67.5053)^2 \approx 0.2553 \][/tex]
[tex]\[ (70 - 67.5053)^2 \approx 6.2253 \][/tex]
[tex]\[ (73 - 67.5053)^2 \approx 30.1911 \][/tex]
Now, multiply each squared term by the frequency and sum them up:
[tex]\[ \sum f (x - \bar{x})^2 = 3 \cdot 42.2921 + 18 \cdot 12.2811 + 42 \cdot 0.2553 + 24 \cdot 6.2253 + 8 \cdot 30.1911 \][/tex]
[tex]\[ \approx 126.8763 + 221.0598 + 10.7276 + 149.4072 + 241.5288 = 749.5997 \][/tex]
[tex]\[ \sigma^2 \approx \frac{749.5997}{95} \approx 7.8921 \][/tex]
### 5. Calculating the Standard Deviation
The standard deviation ([tex]\(\sigma\)[/tex]) is the square root of the variance:
[tex]\[ \sigma = \sqrt{7.8921} \approx 2.8093 \text{ kg} \][/tex]
### Summary
- Mean: [tex]\(\approx 67.5053\)[/tex] kg
- Standard Deviation: [tex]\(\approx 2.8093\)[/tex] kg
The frequency table is provided for the weights of 95 individuals with different mass intervals. Here's the step-by-step solution for calculating the mean and standard deviation:
### 1. Finding the Midpoint of Each Mass Interval
First, we need to determine the midpoint (or the class mark) of each mass interval. The midpoint is calculated as the average of the lower and upper limits of each interval:
- For [tex]$60-62$[/tex] kg: [tex]\((60 + 62) / 2 = 61\)[/tex] kg
- For [tex]$63-65$[/tex] kg: [tex]\((63 + 65) / 2 = 64\)[/tex] kg
- For [tex]$66-68$[/tex] kg: [tex]\((66 + 68) / 2 = 67\)[/tex] kg
- For [tex]$69-71$[/tex] kg: [tex]\((69 + 71) / 2 = 70\)[/tex] kg
- For [tex]$72-74$[/tex] kg: [tex]\((72 + 74) / 2 = 73\)[/tex] kg
### 2. Constructing a Table with Midpoints and Frequencies
Now we list the midpoints and their corresponding frequencies:
| Mass midpoint (kg) | Frequency |
|---------------------|-----------|
| 61 | 3 |
| 64 | 18 |
| 67 | 42 |
| 70 | 24 |
| 73 | 8 |
### 3. Calculating the Mean
The mean ([tex]\(\bar{x}\)[/tex]) is calculated as the sum of the products of each midpoint and its frequency divided by the total frequency:
Let [tex]\(\sum f \)[/tex] be the total frequency and [tex]\(\sum fm \)[/tex] be the sum of the product of frequency and midpoint.
[tex]\[ \sum fm = (61 \times 3) + (64 \times 18) + (67 \times 42) + (70 \times 24) + (73 \times 8) = 183 + 1152 + 2814 + 1680 + 584 = 6413 \][/tex]
[tex]\[ \sum f = 3 + 18 + 42 + 24 + 8 = 95 \][/tex]
[tex]\[ \text{Mean} = \bar{x} = \frac{\sum fm}{\sum f} = \frac{6413}{95} \approx 67.5053 \text{ kg} \][/tex]
### 4. Calculating the Variance
The variance ([tex]\(\sigma^2\)[/tex]) is calculated using the formula:
[tex]\[ \sigma^2 = \frac{\sum f (x - \bar{x})^2}{\sum f} \][/tex]
Where [tex]\(x\)[/tex] is the midpoint and [tex]\(\bar{x}\)[/tex] is the mean. We already have [tex]\(\sum f\)[/tex]:
[tex]\[ \sigma^2 = \frac{1}{95} \left[ 3(61 - 67.5053)^2 + 18(64 - 67.5053)^2 + 42(67 - 67.5053)^2 + 24(70 - 67.5053)^2 + 8(73 - 67.5053)^2 \right] \][/tex]
First, calculate each squared term:
[tex]\[ (61 - 67.5053)^2 \approx 42.2921 \][/tex]
[tex]\[ (64 - 67.5053)^2 \approx 12.2811 \][/tex]
[tex]\[ (67 - 67.5053)^2 \approx 0.2553 \][/tex]
[tex]\[ (70 - 67.5053)^2 \approx 6.2253 \][/tex]
[tex]\[ (73 - 67.5053)^2 \approx 30.1911 \][/tex]
Now, multiply each squared term by the frequency and sum them up:
[tex]\[ \sum f (x - \bar{x})^2 = 3 \cdot 42.2921 + 18 \cdot 12.2811 + 42 \cdot 0.2553 + 24 \cdot 6.2253 + 8 \cdot 30.1911 \][/tex]
[tex]\[ \approx 126.8763 + 221.0598 + 10.7276 + 149.4072 + 241.5288 = 749.5997 \][/tex]
[tex]\[ \sigma^2 \approx \frac{749.5997}{95} \approx 7.8921 \][/tex]
### 5. Calculating the Standard Deviation
The standard deviation ([tex]\(\sigma\)[/tex]) is the square root of the variance:
[tex]\[ \sigma = \sqrt{7.8921} \approx 2.8093 \text{ kg} \][/tex]
### Summary
- Mean: [tex]\(\approx 67.5053\)[/tex] kg
- Standard Deviation: [tex]\(\approx 2.8093\)[/tex] kg
