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2. Slide 2-2 Chapter 2 Organizing Data Section 2.2 Grouping Data

3. Slide 2-3 Terms Used in Grouping The two most common distributions are categorical frequency distribution and the grouped frequency distribution. Classes: are categories for grouping data. Frequency or Count is the number of observations that fall in a class. Frequency distribution i s the organization of raw data in table form, using classes and frequencies. L isting all classes and their frequencies. Relative frequency or Percent is the ratio of the frequency of a class to the total number of observations. Relative-frequency distribution: A listing of all classes and their relative frequencies.

4. Slide 2-4 Terms Used in Grouping When the range of the data is large, the data must be grouped into classes that are more than one unit in width. Exp. Age groups 16-21, 22-24, 25-30. The first column will be the Class Limits. Lower cutpoint or lower class limit : represents the smallest data value that can be included in the class. Exp. 16, 22 and 25. The smallest value that could go in a class. Upper cutpoint or upper class limit: represents the largest value that can be included in the class. Exp. 21, 24 and 30 The smallest value that could go in the next higher class (equivalent to the lower cutpoint of the next higher class). Mark of the class is the average of the lower and upper limits

5. Slide 2-5 Terms Used in Grouping The class boundaries are used to separate the classes so that there are no gaps in the frequency distribution. Exp 15.5-21.5, 21.5-24.5, and 24.5-30.5 This will be the second column. Midpoint or class midpoint: is found by adding the lower and upper boundaries (or limits) and dividing by 2. Exp. 16+21=37/2=18.5 or 15.5+21.5=37/2=18.5 The middle of a class, found by averaging its cutpoints. Width or class width: for a class in a frequency distribution is found by subtracting the lower (or upper) class limit of one class from the the lower (or upper) class limit of the next class. Exp. 16-21, 21-25, 25-30 the width 21 - 16 = 5. The difference between the cutpoints of a class.

6. Slide 2-6 Class Boundaries Significant Figures Cutpoints or Class limits should have the same decimal place value as the data, but the class boundaries have one additional place value and end in a 5. Exp: if the data set is whole numbers the limits may be 31-37, and boundaries 30.5-37.5. Lower limits are -.5 and Upper limits are +.5. If limits are 7.8-8.8 the boundaries 7.75-8.85, lower limit -.05 and upper limit +.05. Rule of Thumb

7. Slide 2-7 Class Rules There should be between 5 and 20 classes. It is Preferable that the class width be an odd number. The classes must be mutually exclusive. (non-overlapping) The classes must be continuous. Even if there are no values. The classes must exhaustive, accommodate all the data. The classes must be equal width. Open ended distribution - no specific beginning or ending value. EXP. 16 and below or 65 and above.

8. Slide 2-8 Types of Frequency Distributions A categorical frequency distribution is used when the data can be places in specific categories (nominal-level or ordinal- level). Exp. Political affiliation, religious affiliation, major field of study or blood-type. A grouped frequency distribution is used when the range is large and classes of several units in width are needed. Exp. Number of hours that batteries last. An ungrouped frequency distribution is used for numerical data and when the range of data is small.

9. Slide 2-9 Why Construct Frequency Distributions? To facilitate computational procedures for measures of average and spread. To enable the researcher to draw charts and graphs for the presentation of the data. To enable the reader to determine the nature or shape of the distribution. To organize the data in a meaningful, intelligible way. To enable the reader to make comparisons among different data sets.

10. Slide 2-10 Example of Categorical Frequency Distribution Thirty-five people, selected at random, were asked to identify the brand of automobile that they currently drove. The raw data is given below. Organize the data into a frequency table and a relative frequency table. FordDodgeHondaChevyBuick DodgeBuickFord Chevy HondaDodgeChevyFord HondaFord DodgeFord DodgeChevyBuickHonda ChevyDodgeFord Chevy FordChevy DodgeFord Qualitative Order does not matter

11. Slide 2-11 CategoryTallyFrequencyRelative Frequency Ford 1111 1111 11 1212/35 =.34 or 34 Dodge1111 1177/35 =.20 or 20 Chevy1111 99/35 =.26 or 26 Honda111144/35 =.11 or 11 Buick11133/35 =.09 or 9 TOTAL35100 The majority of the people drove a Ford. Answer of Categorical Frequency Distribution

12. Slide 2-12 Example of Ungrouped Frequency Distribution Thirty students were administered a 5-point quiz and their raw scores are given below. Organize the data into a frequency table, a relative frequency table, and a cumulative relative frequency table. 45423 33304 31241 34534 23353 15024 Quantitative Should be in ascending or descending order.

13. Slide 2-13 ANSWER - Ungrouped Frequency Distribution The majority of the students scored 3 on the quiz. ScoreTallyFreqCumFreqRelFreqCumRelFreq 011222/30 = 77 1111351017 21111491431 31111 10193364 41111 117262387 5111143014100 TOTAL30101

14. Slide 2-14 Constructing a Grouped Frequency Distribution Determine the classes. Find the highest and lowest value. Find the range. Select the number of classes desired. Find the width by dividing the range by the number of classes and rounding up. Select a starting point, add the width to get the lower limit. Find the upper class limit. Find the boundaries. Tally the data. Find the numerical frequencies from the tallies. Find the cumulative frequencies.

15. Slide 2-15 Example of Grouped Frequency Distribution Thirty students were administered a 100-point examination. Their scores on this examination are provided below. Organize the data into a frequency table, a relative frequency table, and a cumulative relative frequency table. 7275549397 7884797174 8166706883 67100727678 5947826581 9183864969 Quantitative Should be in ascending or descending order. No overlapping Width = 0-9 = 9 Right end – left end

16. Slide 2-16 ANSWER - Grouped Frequency Distribution The majority of the students scored between 70-79 on the exam. ScoreTallyFreqCumFreqRelFreqCumRelFreq 40-4911222/30 = 77 50-591124714 60-691111591731 70-791111 10193364 80-891111 117262387 90-100111143014100 TOTAL30101

17. Slide 2-17 The table displays the number of days to maturity for 40 short- term investments. Getting a clear picture of these data is difficult, but is much easier if we group them into categories, or classes. The first step is to decide on the classes. One convenient way to group these data is by 10s. Table 2.1 Table 2.2 Grouped Frequency Distribution

18. Slide 2-18 Table 2.3 The percentage of a class, expressed as a decimal, is called the relative frequency of the class. A table that provides all classes and their relative frequencies is called a relative-frequency distribution. The table displays a relative- frequency distribution for the days-to-maturity data. Note that the relative frequencies sum to 1 (100%). Grouped Relative Frequency Distribution

19. Slide 2-19 Table 2.4 A table that provides the classes, frequencies, relative frequencies, and midpoints of a data set is called a grouped- data table. The table is a grouped-data table for the days-to- maturity data. Grouped Data Frequency Distribution

20. Slide 2-20 Example 2.7 TVs per Household Trends in Television, published by the Television Bureau of Advertising, provides information on television ownership. The table gives the number of TV sets per household for 50 randomly selected households. Use classes based on a single value to construct a grouped-data table for these data. Table 2.8

21. Slide 2-21 Table 2.9 Solution Example 2.7 grouped-data table

22. Slide 2-22 Table 2.10 Example 2.8 Professor Weiss asked his introductory statistics students to state their political party affiliations as Democratic (D), Republican (R), or Other (O). The responses are given in the table. Determine the frequency and relative-frequency distributions for these data.

23. Slide 2-23 Table 2.11 Solution Example 2.8 grouped-data table