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 $=$ $\square$
failed

Solution

failed
failed

Solution Steps

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

  1. Calculate the midpoints for each class interval.
  2. Multiply each midpoint by its corresponding frequency to find the sum of the products.
  3. Calculate the mean of the distribution.
  4. Use the mean to find the squared differences from the mean for each midpoint.
  5. Multiply each squared difference by its corresponding frequency and sum these values.
  6. Divide by the total number of data points to find the variance.
  7. 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

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

Final Answer

\(\boxed{\text{Variance} = 10.7}\)

\(\boxed{\text{Standard deviation} = 3.3}\)

Was this solution helpful?
failed
Unhelpful
failed
Helpful

Questions asked by other users

failedAnswered
Right Triangle Trigonometry Suppose π/2 ≤ θ ≤ π and we obtain the following right triangle: Compute sin(θ). with a=21, b=20, and c=29. 20/21 -20/21 21/20 -21/20 20/29 -20/29 29/20 -29/20 21/29 -21/29 29/21 -29/21
failedAnswered
Four 0.6 mol samples of argon gas are described in the table below. Rank these samples in order of decreasing rate of collisions between atoms. That is, select " 1 " next to the sample with the most collisions per second between Ar atoms and the walls of the container. Select " 2 " next to the sample with the next most collisions per second between Ar atoms and the walls of the container, and so on. sample volume temperature collisions per second A 42 L -44 °C B 34 L -27 °C C 38 L -27 °C D 42 L -27 °C
failedAnswered
Information relevance refers to . A sufficiency B cost compared to value C timeliness and specifics D both context and subject
failedAnswered
Read this text from paragraph 3. (3) ... Roosevelt met up with John Muir and they spent three days camping alone in the Yosemite high country. Because of the impression Muir made upon Roosevelt, Yosemite National Park would be expanded and control of the park would be tumed back over to the federal government. Roosevelt would later go on to sign into law the Antiquities Act, which gives the President the power to proclaim an area of natural or historical value as public lands. Which excerpt supports the inference that Roosevelt and Captain Charles Young shared similar views about conservation? A. "Campaigns in the American West had imbued Army soldiers with the skills necessary for patrolling the vast wilderness that the new National Parks offered the public." B. 'These were areas set aside by the Federal Government to provide access to an unspoiled wilderness area, a frontier that many Americans would never be able to experience." C. "A journey through this park and the Sierra Forest Reserve to the Mount Whitney country will convince even the least thoughtulf man of the needfulness of preserving these mountains just as they are ..." D. "After making a visit to Mammoth Hot Springs in Yellowstone National Park, it was considered acceptable to chip off a piece of the travertine and to carve your name and address."
failedAnswered
Fill in the blank below. The of sets A and B, written A ∪ B, is the set of elements that are members of either A or B (or both). Using set-builder notation, A ∪ B=x: x is a member of A or x is a member of B. exclusion inclusion complement intersection union difference