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Chapter 2: Problem 11

Briefly explain the three decisions that have to be made to group a data set in the form of a frequency distribution table.

Short Answer

Expert verified
The three decisions for creating a frequency distribution table include: Determining the number of classes (usually around \( \sqrt{n} \)), Deciding on the class intervals (typically equalling size across classes, with interval size calculated as \( \frac{{range of data}}{{number of classes}} \)), and Counting the class frequency i.e, the number of data points in each class.

Step by step solution

01

Decision 1 - Number of Classes

One must decide the number of classes or groups. The choice depends on the size and spread of the data. This number usually ranges from 5 to 20. A common practice is to choose a number around \( \sqrt{n} \) where n is the number of data points.
02

Decision 2 - Class Intervals

Determine the class intervals. Here, the goal is to size each class so that it covers a range of data values. Typically, the intervals are chosen so all have the same size. The size of the interval could be calculated as \( \frac{{range of data}}{{number of classes}} \). This, however, 鈭� for skewed or irregularly distributed data, unequal class intervals can be used.
03

Decision 3 - Class Frequencies

Lastly, count the frequency for each class which is the number of data items falling within that class. This is done by going through the data set and categorizing each data point into its respective class.

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Most popular questions from this chapter

The following data give the money (in dollars) spent on textbooks by 35 students during the \(2009-10\) academic year. $$ \begin{array}{lllllllll} 565 & 728 & 470 & 620 & 345 & 368 & 610 & 765 & 550 \\ 845 & 530 & 705 & 490 & 258 & 320 & 505 & 457 & 787 \\ 617 & 721 & 635 & 438 & 575 & 702 & 538 & 720 & 460 \\ 540 & 890 & 560 & 570 & 706 & 430 & 268 & 638 & \end{array} $$ a. Prepare a stem-and-leaf display for these data using the last two digits as leaves. b. Condense the stem-and-leaf display by grouping the stems as \(2-4,5-6\), and \(7-8\).

The accompanying table lists the 2006-07 median household incomes (rounded to the nearest dollar), for all 50 states and the District of Columbia. $$ \begin{array}{lccc} \hline \text { State } & \begin{array}{c} \text { 2006-07 Median } \\ \text { Household Income } \end{array} & \text { State } & \begin{array}{c} 2006-07 \text { Median } \\ \text { Household Income } \end{array} \\ \hline \text { AL } & 40,620 & \text { MT } & 42,963 \\ \text { AK } & 60,506 & \text { NE } & 49,342 \\ \text { AZ } & 47,598 & \text { NV } & 53,912 \\ \text { AR } & 39,452 & \text { NH } & 65,652 \\ \text { CA } & 56,311 & \text { NJ } & 65,249 \\ \text { CO } & 59,209 & \text { NM } & 42,760 \\ \text { CT } & 64,158 & \text { NY } & 49,267 \\ \text { DE } & 54,257 & \text { NC } & 42,219 \\ \text { D.C. } & 50,318 & \text { ND } & 44,708 \\ \text { FL } & 46,383 & \text { OH } & 48,151 \\ \text { GA } & 49,692 & \text { OK } & 41,578 \\ \text { HI } & 63,104 & \text { OR } & 49,331 \\ \text { ID } & 48,354 & \text { PA } & 49,145 \\ \text { IL } & 51,279 & \text { RI } & 54,735 \\ \text { IN } & 47,074 & \text { SC } & 42,477 \\ \text { IA } & 49,200 & \text { SD } & 46,567 \\ \text { KS } & 47,671 & \text { TN } & 41,521 \\ \text { KY } & 40,029 & \text { TX } & 45,294 \\ \text { LA } & 39,418 & \text { UT } & 54,853 \\ \text { ME } & 47,415 & \text { VT } & 50,423 \\ \text { MD } & 65,552 & \text { VA } & 58,950 \\ \text { MA } & 57,681 & \text { WA } & 57,178 \\ \text { MI } & 49,699 & \text { WV } & 40,800 \\ \text { MN } & 57,932 & \text { WI } & 52,218 \\ \text { MS } & 36,499 & \text { WY } & 48,560 \\ \text { MO } & 45,924 & & \\ \hline \end{array} $$ a. Construct a frequency distribution table. Use the following classes: \(36,000-40,999,41,000-\) \(45,999,46,000-50,999,51,000-55,999,56,000-60,999,61,000-65,999\) b. Calculate the relative frequencies and percentages for all classes. c. Based on the frequency distribution, can you say whether the data are symmetric or skewed? d. What percentage of these states had a median household income of less than \(\$ 56,000 ?\)

A sample of 80 adults was taken, and these adults were asked about the number of credit cards they possess. The following table gives the frequency distribution of their responses. $$ \begin{array}{lc} \hline \text { Number of Credit Cards } & \text { Number of Adults } \\ \hline 0 \text { to } 3 & 18 \\ 4 \text { to } 7 & 26 \\ 8 \text { to } 11 & 22 \\ 12 \text { to } 15 & 11 \\ 16 \text { to } 19 & 3 \\ \hline \end{array} $$ a. Find the class boundaries and class midpoints. b. Do all classes have the same width? If so, what is this width? c. Prepare the relative frequency and percentage distribution columns. d. What percentage of these adults possess 8 or more credit cards?

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

The following frequency distribution table gives the age distribution of drivers who were at fault in auto accidents that occurred during a 1 -week period in a city. $$ \begin{array}{lr} \hline \text { Age (years) } & \boldsymbol{f} \\ \hline \text { 18 to less than } 20 & 7 \\ 20 \text { to less than } 25 & 12 \\ 25 \text { to less than } 30 & 18 \\ 30 \text { to less than } 40 & 14 \\ 40 \text { to less than } 50 & 15 \\ 50 \text { to less than } 60 & 16 \\ 60 \text { and over } & 35 \\ \hline \end{array} $$ a. Draw a relative frequency histogram for this table. b. In what way(s) is this histogram misleading? c. How can you change the frequency distribution so that the resulting histogram gives a clearer picture?
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