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Chapter 2: Problem 3

Pupils Per Teacher The average number of pupils per teacher in each state is shown. Construct a grouped frequency distribution with 6 classes. Draw a histogram, frequency polygon, and ogive. Analyze the distribution. $$ \begin{array}{lllll} 16 & 16 & 15 & 12 & 14 \\ 13 & 16 & 14 & 15 & 14 \\ 18 & 18 & 18 & 12 & 15 \\ 15 & 16 & 16 & 15 & 15 \\ 25 & 19 & 15 & 12 & 22 \\ 18 & 14 & 13 & 17 & 9 \\ 13 & 14 & 13 & 16 & 12 \\ 14 & 16 & 10 & 22 & 20 \\ 12 & 14 & 18 & 15 & 14 \\ 16 & 12 & 12 & 13 & 15 \end{array} $$

Short Answer

Expert verified
The distribution is slightly right-skewed with most states having lower pupil-to-teacher ratios.

Step by step solution

01

Determine the Range

First, find the range of data by subtracting the smallest value from the largest value. The smallest value is 9 and the largest value is 25. Therefore, the range is calculated as follows: \[ \text{Range} = 25 - 9 = 16 \]
02

Calculate the Class Width

Calculate the class width by dividing the range by the number of classes and rounding up. \[ \text{Class Width} = \frac{16}{6} \approx 2.67 \] Since we need to use whole numbers for class width, we round up to 3.
03

Create the Class Intervals

Construct the class intervals starting from the smallest value and increasing by the class width. Use the interval width calculated in Step 2. The classes are: - 9-11 - 12-14 - 15-17 - 18-20 - 21-23 - 24-26
04

Tally the Frequencies

Count how many data points fall into each class interval: - 9-11: 2 - 12-14: 19 - 15-17: 21 - 18-20: 11 - 21-23: 2 - 24-26: 1
05

Draw the Histogram

For a histogram, draw bars for each class interval with heights corresponding to their frequencies: - The horizontal axis should represent the class intervals. - The vertical axis should represent frequencies.
06

Draw the Frequency Polygon

For a frequency polygon, plot a point for the frequency of each class at the midpoint of the interval and connect the points with straight lines: - Midpoints would be 10, 13, 16, 19, 22, and 25 respectively.
07

Draw the Ogive

For the ogive, plot cumulative frequency points at the upper boundary of each class and connect them with straight lines: - The cumulative frequencies will be 2, 21, 42, 53, 55, and 56.
08

Analyze the Distribution

Analyze the distribution by examining the shape, spread, and center: - The distribution is skewed slightly to the right, indicating that most states have a pupil-to-teacher ratio clustered at lower values, with fewer states having such high ratios as 24-26.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Histogram
Histograms are great for visualizing frequency distributions. They allow us to quickly see how data is spread over different intervals. In a histogram, each interval鈥攐r "bin"鈥攊s represented by a bar. The height of each bar indicates the number of data points in that interval.

To create a histogram for the pupil-to-teacher ratio:
  • Start by drawing two perpendicular axes. The horizontal axis will represent the intervals or bins, and the vertical axis will indicate the frequency of data within each interval.
  • Use the class intervals found previously (9-11, 12-14, 15-17, 18-20, 21-23, 24-26).
  • Draw a bar above each interval where the height matches the number of data points (frequencies) in that interval.
This visual representation helps you to easily compare the frequencies between different intervals, giving a clear picture of where most data points lie. For this exercise, the histogram would highlight the concentration of lower pupil-to-teacher ratios.
Frequency Polygon
A frequency polygon is similar to a histogram but provides a line graph version of frequency data. This makes it excellent for showing trends over intervals.

To construct a frequency polygon:
  • Start by finding the midpoint of each class interval. For this example: 10, 13, 16, 19, 22, and 25.
  • Plot these midpoints on a graph along the horizontal axis.
  • Then plot a point at each midpoint at a height that corresponds to the frequency count of that interval.
  • Connect each of these points with straight lines.
  • To close the polygon, extend the graph to the zero frequency point before the first midpoint and after the last midpoint.
The frequency polygon helps to visualize the data trend and distribution shape, displaying how frequencies rise and fall across the intervals. This method can especially highlight changes in frequency that a histogram might miss.
Ogive
An ogive, or cumulative frequency graph, shows the cumulative frequency for each interval. It's useful for determining the number of values below a particular level.

To create an ogive:
  • Use the upper bound of each class interval along the horizontal axis. For this set: 11, 14, 17, 20, 23, 26.
  • Plot points at the cumulative frequencies calculated for each interval's upper boundary: 2, 21, 42, 53, 55, and 56.
  • Join these points with straight lines.
An ogive provides insight into cumulative data trends and is especially helpful for identifying medians and quartiles. The gradual slope depicted in an ogive like the one in this exercise might show that most pupils-per-teacher ratios are below higher values, confirming previous analyses of the distribution being skewed to the right.

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Most popular questions from this chapter

Draw a time series graph for the number (in millions) of drivers in the United States 70 or older. $$ \begin{array}{l|cccc} \text { Year } & 1982 & 1992 & 2002 & 2012 \\ \hline \text { Number } & 10 & 15 & 20 & 23 \end{array} $$

The U.S. health dollar is spent as indicated below. Construct two different types of graphs to represent the data. $$ \begin{array}{lr} \text { Government administration } & 9.7 \% \\ \text { Nursing home care } & 5.5 \\ \text { Prescription drugs } & 10.1 \\ \text { Physician and clinical services } & 20.3 \\ \text { Hospital care } & 30.5 \\ \text { Other (OTC drugs, dental, etc.) } & 23.9 \end{array} $$

Bear Kills The number of bears killed in 2014 for 56 counties in Pennsylvania is shown in the frequency distribution. Construct a histogram, frequency polygon, and ogive for the data. Comment on the skewness of the distribution. How many counties had 75 or fewer bears killed? (The data for this exercise will be used for Exercise 14 of this section.) $$ \begin{array}{rr} \text { Class limits } & \text { Frequency } \\ \hline 1-25 & 16 \\ 26-50 & 14 \\ 51-75 & 9 \\ 76-100 & 8 \\ 101-125 & 5 \\ 126-150 & 0 \\ 151-175 & 1 \\ 176-200 & 1 \\ 201-225 & 0 \\ 226-250 & 0 \\ 251-275 & 2 \\ & \text { Total } 56 \end{array} $$

State which type of graph (Pareto chart, time series graph, or pie graph) would most appropriately represent the data. a. Situations that distract automobile drivers b. Number of persons in an automobile used for getting to and from work each day c. Amount of money spent for textbooks and supplies for one semester d. Number of people killed by tornados in the United States each year for the last 10 years e. The number of pets (dogs, cats, birds, fish, etc.) in the United States this year f. The average amount of money that a person spent for his or her significant other for Christmas for the last 6 years

Show frequency distributions that are incorrectly constructed. State the reasons why they are wrong. $$ \begin{array}{cc} \text { Class } & \text { Frequency } \\ \hline 5-9 & 1 \\ 9-13 & 2 \\ 13-17 & 5 \\ 17-20 & 6 \\ 20-24 & 3 \end{array} $$
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