Answer :
Sure, let's solve the problem by calculating the mean, variance, and standard deviation step-by-step.
### Given Data:
[tex]\[ \begin{array}{|c|c|} \hline \text{Number of Children (x)} & \text{Frequency (f)} \\ \hline 0 & 10 \\ \hline 1 & 19 \\ \hline 2 & 7 \\ \hline 3 & 7 \\ \hline 4 & 5 \\ \hline 5 & 1 \\ \hline 6 & 1 \\ \hline \end{array} \][/tex]
### Step-by-Step Solution:
Step 1: Calculate the Mean (μ)
The mean is the average number of children per household. It is calculated using the formula:
[tex]\[ \mu = \frac{\sum (x \cdot f)}{\sum f} \][/tex]
First, calculate the total number of households ([tex]\(\sum f\)[/tex]):
[tex]\[ 10 + 19 + 7 + 7 + 5 + 1 + 1 = 50 \][/tex]
Next, calculate the sum of the product of each value of [tex]\( x \)[/tex] and its corresponding frequency [tex]\( f \)[/tex]:
[tex]\[ (0 \cdot 10) + (1 \cdot 19) + (2 \cdot 7) + (3 \cdot 7) + (4 \cdot 5) + (5 \cdot 1) + (6 \cdot 1) = 0 + 19 + 14 + 21 + 20 + 5 + 6 = 85 \][/tex]
Now, substitute these values into the mean formula:
[tex]\[ \mu = \frac{85}{50} = 1.7 \][/tex]
Step 2: Create a Table for Calculating Variance
To find the variance, we need to first determine the squared deviation of each value of [tex]\( x \)[/tex] from the mean ([tex]\( x - \mu \)[/tex]), square these deviations, and then multiply by their corresponding frequencies ([tex]\( f \)[/tex]).
[tex]\[ \begin{array}{|c|c|c|c|} \hline \text{Number of Children (x)} & \text{Frequency (f)} & (x - \mu) & (x - \mu)^2 \cdot f \\ \hline 0 & 10 & 0 - 1.7 = -1.7 & (-1.7)^2 \cdot 10 = 2.89 \cdot 10 = 28.9 \\ \hline 1 & 19 & 1 - 1.7 = -0.7 & (-0.7)^2 \cdot 19 = 0.49 \cdot 19 = 9.31 \\ \hline 2 & 7 & 2 - 1.7 = 0.3 & (0.3)^2 \cdot 7 = 0.09 \cdot 7 = 0.63 \\ \hline 3 & 7 & 3 - 1.7 = 1.3 & (1.3)^2 \cdot 7 = 1.69 \cdot 7 = 11.83 \\ \hline 4 & 5 & 4 - 1.7 = 2.3 & (2.3)^2 \cdot 5 = 5.29 \cdot 5 = 26.45 \\ \hline 5 & 1 & 5 - 1.7 = 3.3 & (3.3)^2 \cdot 1 = 10.89 \cdot 1 = 10.89 \\ \hline 6 & 1 & 6 - 1.7 = 4.3 & (4.3)^2 \cdot 1 = 18.49 \cdot 1 = 18.49 \\ \hline \end{array} \][/tex]
Step 3: Calculate the Variance (σ²)
The variance is calculated using the formula:
[tex]\[ \sigma^2 = \frac{\sum (x - \mu)^2 \cdot f}{\sum f} \][/tex]
Sum the values from the last column of the table:
[tex]\[ 28.9 + 9.31 + 0.63 + 11.83 + 26.45 + 10.89 + 18.49 = 106.5 \][/tex]
Now, substitute into the variance formula:
[tex]\[ \sigma^2 = \frac{106.5}{50} = 2.13 \][/tex]
Step 4: Calculate the Standard Deviation (σ)
The standard deviation is the square root of the variance:
[tex]\[ \sigma = \sqrt{\sigma^2} = \sqrt{2.13} \approx 1.46 \][/tex]
### Final Results:
- Mean (μ): 1.7
- Variance (σ²): 2.13
- Standard Deviation (σ): 1.46
Thus, the mean number of children per household is 1.7, the variance is 2.13, and the standard deviation is approximately 1.46.
### Given Data:
[tex]\[ \begin{array}{|c|c|} \hline \text{Number of Children (x)} & \text{Frequency (f)} \\ \hline 0 & 10 \\ \hline 1 & 19 \\ \hline 2 & 7 \\ \hline 3 & 7 \\ \hline 4 & 5 \\ \hline 5 & 1 \\ \hline 6 & 1 \\ \hline \end{array} \][/tex]
### Step-by-Step Solution:
Step 1: Calculate the Mean (μ)
The mean is the average number of children per household. It is calculated using the formula:
[tex]\[ \mu = \frac{\sum (x \cdot f)}{\sum f} \][/tex]
First, calculate the total number of households ([tex]\(\sum f\)[/tex]):
[tex]\[ 10 + 19 + 7 + 7 + 5 + 1 + 1 = 50 \][/tex]
Next, calculate the sum of the product of each value of [tex]\( x \)[/tex] and its corresponding frequency [tex]\( f \)[/tex]:
[tex]\[ (0 \cdot 10) + (1 \cdot 19) + (2 \cdot 7) + (3 \cdot 7) + (4 \cdot 5) + (5 \cdot 1) + (6 \cdot 1) = 0 + 19 + 14 + 21 + 20 + 5 + 6 = 85 \][/tex]
Now, substitute these values into the mean formula:
[tex]\[ \mu = \frac{85}{50} = 1.7 \][/tex]
Step 2: Create a Table for Calculating Variance
To find the variance, we need to first determine the squared deviation of each value of [tex]\( x \)[/tex] from the mean ([tex]\( x - \mu \)[/tex]), square these deviations, and then multiply by their corresponding frequencies ([tex]\( f \)[/tex]).
[tex]\[ \begin{array}{|c|c|c|c|} \hline \text{Number of Children (x)} & \text{Frequency (f)} & (x - \mu) & (x - \mu)^2 \cdot f \\ \hline 0 & 10 & 0 - 1.7 = -1.7 & (-1.7)^2 \cdot 10 = 2.89 \cdot 10 = 28.9 \\ \hline 1 & 19 & 1 - 1.7 = -0.7 & (-0.7)^2 \cdot 19 = 0.49 \cdot 19 = 9.31 \\ \hline 2 & 7 & 2 - 1.7 = 0.3 & (0.3)^2 \cdot 7 = 0.09 \cdot 7 = 0.63 \\ \hline 3 & 7 & 3 - 1.7 = 1.3 & (1.3)^2 \cdot 7 = 1.69 \cdot 7 = 11.83 \\ \hline 4 & 5 & 4 - 1.7 = 2.3 & (2.3)^2 \cdot 5 = 5.29 \cdot 5 = 26.45 \\ \hline 5 & 1 & 5 - 1.7 = 3.3 & (3.3)^2 \cdot 1 = 10.89 \cdot 1 = 10.89 \\ \hline 6 & 1 & 6 - 1.7 = 4.3 & (4.3)^2 \cdot 1 = 18.49 \cdot 1 = 18.49 \\ \hline \end{array} \][/tex]
Step 3: Calculate the Variance (σ²)
The variance is calculated using the formula:
[tex]\[ \sigma^2 = \frac{\sum (x - \mu)^2 \cdot f}{\sum f} \][/tex]
Sum the values from the last column of the table:
[tex]\[ 28.9 + 9.31 + 0.63 + 11.83 + 26.45 + 10.89 + 18.49 = 106.5 \][/tex]
Now, substitute into the variance formula:
[tex]\[ \sigma^2 = \frac{106.5}{50} = 2.13 \][/tex]
Step 4: Calculate the Standard Deviation (σ)
The standard deviation is the square root of the variance:
[tex]\[ \sigma = \sqrt{\sigma^2} = \sqrt{2.13} \approx 1.46 \][/tex]
### Final Results:
- Mean (μ): 1.7
- Variance (σ²): 2.13
- Standard Deviation (σ): 1.46
Thus, the mean number of children per household is 1.7, the variance is 2.13, and the standard deviation is approximately 1.46.
