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Chapter 2: Problem 12

How are the relative frequencies and percentages of classes obtained from the frequencies of classes? Illustrate with the help of an example.

Short Answer

Expert verified
The relative frequency of a class is obtained by dividing the frequency of that class by the total number of observations, and the percentage is then obtained by multiplying this relative frequency by 100.

Step by step solution

01

Understanding Frequency

Frequency of a class refers to the number of times a particular data point or a range (class) occurs in the dataset. So, first, count the frequency of each class in your dataset.
02

Calculation of Relative Frequency

Once the frequency for each class is calculated, find the relative frequency for each class. Relative frequency of a class is calculated by dividing the frequency of a particular class by the total number of observations. For example, if you have a class with a frequency of 7 and there are 50 observations in total, then the relative frequency would be \(\frac{7}{50} = 0.14\).
03

Calculation of Percentage

The last step is to convert this relative frequency to a percentage to understand the distribution better. Simply multiply the relative frequency by 100. Continuing from our previous example, the percentage would be \(0.14 \times 100 = 14\%\). This tells us that 14% of the total observations belong to this class.

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Most popular questions from this chapter

The following data give the waiting times (in minutes) for 25 students at the Student Health Center of a university. \(\begin{array}{lllllllllllll}39 & 19 & 19 & 35 & 18 & 32 & 20 & 15 & 20 & 29 & 25 & 32 & 28 \\ 19 & 42 & 18 & 32 & 31 & 21 & 46 & 27 & 13 & 14 & 15 & 28 & \end{array}\) Create a dotplot for these data.

Consider the following stem-and-leaf display. $$ \begin{array}{l|lllllll} 4 & 3 & 6 & & & & & & \\ 5 & 0 & 1 & 4 & 5 & & & & \\ 6 & 3 & 4 & 6 & 7 & 7 & 7 & 8 & 9 \\ 7 & 2 & 2 & 3 & 5 & 6 & 6 & 9 & \\ 8 & 0 & 7 & 8 & 9 & & & & \end{array} $$ Write the data set that is represented by this display.

Briefly explain the concept of cumulative frequency distribution. How are the cumulative relative frequencies and cumulative percentages calculated?

The following data show the method of payment by 16 customers in a supermarket checkout line. Here, \(\mathrm{C}\) refers to cash, \(\mathrm{CK}\) to check, \(\mathrm{CC}\) to credit card, and \(\mathrm{D}\) to debit card, and \(\mathrm{O}\) stands for other \(\begin{array}{lllllll}\text { C } & \text { CK } & \text { CK } & \text { C } & \text { CC } & \text { D } & \text { O } & \text { C }\end{array}\) CK \(\quad\) CC \(\quad\) D \(\quad\) CC \(\quad\) C \(\begin{array}{lllll}\text { D } & \text { CC } & \text { C } & \text { CK } & \text { CK } & \text { CC }\end{array}\) a. Construct a frequency distribution table. b. Calculate the relative frequencies and percentages for all categories. c. Draw a pie chart for the percentage distribution.

As shown in Exercise \(2.89\), back-to-back stem-and-leaf displays can be used to compare the distribution of a variable for two different groups. Consider the following data, which give the alcohol Flying Dog Brewery: \(\begin{array}{lllllllll}4.7 & 4.7 & 4.8 & 5.1 & 5.5 & 5.5 & 5.6 & 6.0 & 7.1 \\\ 7.4 & 7.8 & 8.3 & 8.3 & 9.2 & 9.9 & 10.2 & 11.5 & \end{array}\) Sierra Nevada Brewery: \(\begin{array}{lllllllllllll}4.4 & 5.0 & 5.0 & 5.6 & 5.6 & 5.8 & 5.9 & 5.9 & 6.7 & 6.8 & 6.9 & 7.0 & 9.6\end{array}\) a. Create a back-to-back stem-and-leaf display of these data. Place the Flying Dog Brewery data to the left of the stems. b. What would you consider to be a typical alcohol content of the beers made by each of the two breweries? c. Does one brewery tend to have higher alcohol content in its beers than the other brewery? If so, which one? Explain how you reach this conclusion by using the stem-and-leaf display. d. Do the alcohol content distributions for the two breweries appear to have the same levels of variability? Explain how you reach this conclusion by using the stem-and-leaf display.
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