Questions: Pupils Per Teacher The frequency distribution shows the average number of pupils per teacher in some states of the United States. Find the variance or standard deviation for the data: Round your answers to one decimal place. Class limits Frequency 10-12 2 13-15 18 16-18 11 19-21 8 22-24 2 25-27 1 Variance = Standard deviation =

Pupils Per Teacher The frequency distribution shows the average number of pupils per teacher in some states of the United States. Find the variance or standard deviation for the data: Round your answers to one decimal place.

Class limits Frequency
10-12 2
13-15 18
16-18 11
19-21 8
22-24 2
25-27 1

Variance =
Standard deviation =

Transcript text: Pupils Per Teacher The frequency distribution shows the average number of pupils per teacher in some states of the United States. Find the variance ar standard deviation for the data: Round your answers to one decimal place. \begin{tabular}{c|c} Class limits & Frequency \\ \hline $10-12$ & 2 \\ $13-15$ & 18 \\ $16-18$ & 11 \\ $19-21$ & 8 \\ $22-24$ & 2 \\ $25-27$ & 1 \end{tabular} Variance $=$ $\square$ Standard deviation $=$ $\square$

Solution

Solution Steps

To find the variance and standard deviation for the given frequency distribution, follow these steps:

Calculate the midpoints for each class interval.
Multiply each midpoint by its corresponding frequency to find the sum of the products.
Calculate the mean of the distribution.
Use the mean to find the squared differences from the mean for each midpoint.
Multiply each squared difference by its corresponding frequency and sum these values.
Divide by the total number of data points to find the variance.
Take the square root of the variance to find the standard deviation.

Step 1: Calculate Midpoints

For each class interval, calculate the midpoint: \[ \text{Midpoints} = \left[ \frac{10+12}{2}, \frac{13+15}{2}, \frac{16+18}{2}, \frac{19+21}{2}, \frac{22+24}{2}, \frac{25+27}{2} \right] = [11.0, 14.0, 17.0, 20.0, 23.0, 26.0] \]

Step 2: Calculate the Sum of the Products of Midpoints and Frequencies

\[ \sum (\text{midpoint} \times \text{frequency}) = 11.0 \times 2 + 14.0 \times 18 + 17.0 \times 11 + 20.0 \times 8 + 23.0 \times 2 + 26.0 \times 1 = 693.0 \]

Step 3: Calculate the Total Number of Data Points

\[ \text{Total data points} = \sum \text{frequencies} = 2 + 18 + 11 + 8 + 2 + 1 = 42 \]

Step 4: Calculate the Mean

\[ \text{Mean} = \frac{\sum (\text{midpoint} \times \text{frequency})}{\text{Total data points}} = \frac{693.0}{42} = 16.5 \]

Step 5: Calculate the Sum of the Squared Differences from the Mean

\[ \sum (\text{frequency} \times (\text{midpoint} - \text{mean})^2) = 2 \times (11.0 - 16.5)^2 + 18 \times (14.0 - 16.5)^2 + 11 \times (17.0 - 16.5)^2 + 8 \times (20.0 - 16.5)^2 + 2 \times (23.0 - 16.5)^2 + 1 \times (26.0 - 16.5)^2 = 448.5 \]

Step 6: Calculate the Variance

\[ \text{Variance} = \frac{\sum (\text{frequency} \times (\text{midpoint} - \text{mean})^2)}{\text{Total data points}} = \frac{448.5}{42} = 10.6786 \]

Step 7: Calculate the Standard Deviation

\[ \text{Standard deviation} = \sqrt{\text{Variance}} = \sqrt{10.6786} = 3.2678 \]