1. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Section 2.1 Frequency Distributions With additions and improvements by D.R.S., University of Cordele

2. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Frequency Distributions A distribution is a way to describe the structure of a particular data set or population. A frequency distribution is a display of the values that occur in a data set and how often each value, or range of values, occurs. Frequencies (f) are the numbers of data values in the categories of a frequency distribution. A class is a category of data in a frequency distribution.

3. Are the data measurements Qualitative or Quantitative? – Blood type circle: qualitative or quantitative – Dollars spent on meal qualitative or quantitative – Heart rate, beats/min. qualitative or quantitative – Model of car qualitative or quantitative – Favorite fast food restaurant qualitative or quantitative – Fuel efficiency, mi/gal qualitative or quantitative – My pain, on a scale from 1 to 10 qualitative or quantitative 3

4. Frequency Distribution for Categorical Data CategoryFrequencyRelative Frequency (list the categories here in this column) (put the counts of how many in this column) (this category is what percent of the total sample size?) (What order? Highest frequency down to lowest? Lowest to highest? Alphabetical? It’s your design decision.) 4

5. Categorical Frequency Distributions are the fuel for the “Family Feud” 5 (Photograph borrowed from some web site somewhere; I failed to record the exact source.) What question did they ask to obtain these answers?

6. Categorical (or, Qualitative) Frequency Distribution example “What state did you visit most recently?” State visited (the category)How many (the frequency) Alabama 71 California 18 Florida138 New York 7 South Carolina 48 Tennessee 27 Texas 53 Other states 70 TOTAL432 6

7. Things we do with Categorical Frequency Distributions Sometimes we just leave them as tables of words and numbers for reference and interpretation. We draw pictures of them (future lessons). – Bar graphs – Pie charts – Cutesy repeated icons variation of the bar graph 7

8. A famous categorical frequency distribution we will revisit later Draw this 5-card poker handFrequency Royal Flush 4 Straight Flush (not including Royal Flush) 36 Four of a Kind 624 Full House 3,744 Flush (not including Royal Flush or Straight Flush) 5,108 Straight (not including Royal Flush or Straight Flush) 10,200 Three of a Kind 54,912 Two Pair 123,552 One Pair 1,098,240 Something that’s not special at all 1,302,540 Total 2,598,600

9. Quantitative Frequency Distribution When the data are number measurements ClassesFrequency Each class is a low-to- high range of values These are called the “Class Limits” The frequency column gives a count of how many data values fit in the class 9

10. Quantitative Frequency Distribution (data are number measurements) Placement Test Score How many applicants 0-919 10-1938 20-2952 30-3971 40-4950 50 and above28 10 the 0 to 9 bucket the 10 to 19 bucket the 20 to 29 bucket etc. Each class is a “bucket” containing all the values in its low-to-high limits.

11. Quantitative Frequency Distribution (data are number measurements) Taxable Income How many taxpayers % of taxpayers 11 GPAHow many students % of students How do you know how many rows to put in a frequency distribution? Where are you going to get the raw data from?

12. Quantitative Frequency Distribution (data are number measurements) mphHow many cars % of cars 12 Weight of newborn How many newborns % of newborns How do you pick the low-to-high limits for each of the classes? Will the distribution need to be shown in a graphic form, too, as well as in this tabular form?

13. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Frequency Distributions Constructing a Frequency Distribution 1.Decide how many classes should be in the distribution. There are typically between 5 and 20 classes in a frequency distribution. Several different methods can be used to determine the number of classes that will show the data most clearly, but in this textbook, the number of classes for a given data set will be suggested.

14. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Frequency Distributions Constructing a Frequency Distribution (cont.) 2.Choose an appropriate class width. In some cases, the data set easily lends itself to natural divisions, such as decades or years. At other times, we must choose divisions for ourselves. When starting a frequency distribution from scratch, one method of finding an appropriate class width is to begin by subtracting the lowest number in the data set from the highest number in the data set and dividing the difference by the number of classes.

15. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Frequency Distributions Constructing a Frequency Distribution (cont.) Rounding this number up gives a good starting point from which to choose the class width. You will want to choose a width so that the classes formed present a clear representation of the data and include all members of the data set, so make a sensible choice.

16. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Frequency Distributions Constructing a Frequency Distribution (cont.) 3.Find the class limits. The lower class limit is the smallest number that can belong to a particular class, and the upper class limit is the largest number that can belong to a class. Using the minimum data value, or a smaller number, as the lower limit of the first class is a good place to begin. However, judgment is required. You should choose the first lower limit so that reasonable classes will be produced, and it should have the same number of decimal places as the largest number of decimal places in the data.

17. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Frequency Distributions Constructing a Frequency Distribution (cont.) After choosing the lower limit of the first class, add the class width to it to find the lower limit of the second class. Continue this pattern until you have the desired number of lower class limits. The upper limit of each class is determined such that the classes do not overlap. If, after creating your classes, there are any data values that fall outside the class limits, you must adjust either the class width or the choice for the first lower class limit.

18. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Frequency Distributions Constructing a Frequency Distribution (cont.) 4.Determine the frequency of each class. Make a tally mark for each data value in the appropriate class. Count the marks to find the total frequency for each class.

19. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Frequency Distributions Class width The class width is the difference between the lower limits or upper limits of two consecutive classes of a frequency distribution. The lower class limit is the smallest number that can belong to a particular class. The upper class limit is the largest number that can belong to a particular class.

20. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 2.1: Constructing a Frequency Distribution Create a frequency distribution using five classes for the list of 3-D TV prices given in Table 2.2. Table 2.2: 3-D TV Prices (in Dollars and in an Ordered Array) 1595159916851699 175717871799 188518881899 19841999

21. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 2.1: Constructing a Frequency Distribution (cont.)

22. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 2.1: Constructing a Frequency Distribution (cont.) We will stop here and consider some options. Choosing a class width of $81 does seem perfectly reasonable from a theoretical point of view. However, one should consider the impression created by having TV prices grouped in intervals of $81. Can you imagine presenting this data to a client? 1500 – 1580, 1581 – ______, ______ – _____, etc. Instead, it would be more reasonable to group TV prices by intervals of $100. Therefore, we will choose our class width to be $100.

23. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 2.1: Constructing a Frequency Distribution (cont.) Next, we need to choose a starting point for the classes, that is, the first lower class limit. One should always first consider using the smallest data value for the beginning point. In this case, if we choose the smallest TV price, we would be starting the first class at $_____ with a width of $100. However, given that we’ve chosen a class width of $100, it is more natural to begin the first class at $_____.

24. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 2.1: Constructing a Frequency Distribution (cont.) Now let’s continue building the class limits. Adding the class width of $100 to $1500, we obtain a second lower class limit of $______. The next lower limit is found by adding $____ to $_____. We continue in this fashion until we have five lower class limits, one for each of our five classes. Finally, we need to determine appropriate upper class limits. Again, be reasonable. Remember, too, that the classes are not allowed to overlap.

25. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 2.1: Constructing a Frequency Distribution (cont.) Because the data are in whole dollar amounts, it makes sense to choose upper class limits that are one dollar less than the next lower limit. The classes we have come up with are as follows. 3-D TV Prices ClassFrequency $1500 - $______ $______ - $______

26. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 2.1: Constructing a Frequency Distribution (cont.) Note that the last upper class limit is also the maximum value in the data set. COINCIDENCE! This will not necessarily occur in every frequency table. However, we have included all the data values in our range of classes, so no adjustments to the classes are necessary. (Important for every data value to find a home in one of the classes, nobody’s left out.) Tabulating the number of data values that occur in each class produces the following frequency table.

27. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 2.1: Constructing a Frequency Distribution (cont.) 3-D TV Prices ClassFrequency $1500 - $1599 $1600 - $1699 $1700 - $1799 $1800 - $1899 $1900 - $1999 ____ Note that the sum of the frequency column is ______ and the number of data values in the original list is _____. They are { equal, unequal }. What does that mean? _________________________________

28. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Characteristics of a Frequency Distribution Class boundary A class boundary is the value that lies halfway between the upper limit of one class and the lower limit of the next class. After finding one class boundary, add (or subtract) the class width to find the next class boundary. The boundaries of a class are typically given in interval form: lower boundary–upper boundary.

29. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 2.2: Calculating Class Boundaries

30. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 2.2: Calculating Class Boundaries (cont.) Recall that the class width is 100. Adding 100 to 1599.5 gives the next class boundary. You can repeat this step to find the remaining class boundaries. 3-D TV Prices with Class Boundaries ClassFrequencyClass Boundaries $1500 - $1599 $1600 - $1699 $1700 - $1799 $1800 - $1899 $1900 - $1999 2545425454 _______ - _______ 1599.5 -_______ _______ - _______

31. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Characteristics of a Frequency Distribution Class Midpoint

32. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 2.3: Calculating Class Midpoints

33. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 2.3: Calculating Class Midpoints (cont.) We can use this same calculation to find the midpoints of the remaining classes. Another method is to add 100 (the class width) to the first midpoint, as we did with class boundaries. 3-D TV Prices with Class Boundaries Midpoints ClassFrequencyClass Boundaries Midpoints $1500 - $1599 $1600 - $1699 $1700 - $1799 $1800 - $1899 $1900 - $1999 2545425454 1549.5 __________

34. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Characteristics of a Frequency Distribution Relative Frequency The relative frequency is the fraction or percentage of the data set that falls into a particular class, given by where f is the class frequency, n is the sample size, given by and is the frequency of the class.

35. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 2.4: Calculating Relative Frequencies (cont.) 3-D TV Prices with Relative Frequencies ClassFrequencyRelative Frequency $1500 - $1599 $1600 - $1699 $1700 - $1799 $1800 - $1899 $1900 - $1999 2545425454 The Relative Frequency for a class is: class frequency total of frequency col. Optionally convert it to a percent. Compute these relative frequencies as percents.

36. Calculate relative frequency as decimal Draw this 5-card poker handFrequencyRelative Frequ’cy Royal Flush 4 Straight Flush (not including Royal Flush) 36 Four of a Kind 624 Full House 3,744 Flush (not including Royal Flush or Straight Flush) 5,108 Straight (not including Royal Flush or Straight Flush) 10,200 Three of a Kind 54,912 Two Pair 123,552 One Pair 1,098,240 Something that’s not special at all 1,302,540 Total 2,598,600

37. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Characteristics of a Frequency Distribution Cumulative Frequency The cumulative frequency is the sum of the frequencies of a given class and all previous classes. The cumulative frequency of the last class equals the sample size.

38. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 2.5: Calculating Cumulative Frequencies Calculate the cumulative frequency for each class in the frequency distribution from Example 2.1. Solution 3-D TV Prices with Cumulative Frequencies ClassFrequencyCumulative Frequency $1500 - $1599 $1600 - $1699 $1700 - $1799 $1800 - $1899 $1900 - $1999 2545425454 2 _______

39. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 2.6: Characteristics of a Frequency Distribution Data collected on the numbers of miles that Beta Corp. employees drive to work daily are listed below. Use these data to create a frequency distribution that includes the class boundaries, midpoint, relative frequency, and cumulative frequency of each class. Use six classes. Be sure that your class limits have the same number of decimal places as the largest number of decimal places in the data.

40. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 2.6: Characteristics of a Frequency Distribution (cont.) Class width: max value – min value 6 classes Numbers of Miles Beta Corp. Employees Drive to Work Each Day 3.8 10.2 11.9 2.7 1 5.5 9.3 3.7 4.8 6.5 9.1 7.3 5.8 6.2 9.1 7 11 1.4 Instead of using the rough decimal value, let’s make the class width be a nicer value; let’s make it ________.

41. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 2.6: Characteristics of a Frequency Distribution (cont.) Choose a nice starting point for the lower class limit of the first class. Fill in all the class limits. Check: Will every data value fit in one of the classes? If so, go ahead; count to find the frequency of each class. Numbers of Miles Beta Corp. Employees Drive to Work Each Day ClassFrequency ______ - ______

42. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 2.6: Characteristics of a Frequency Distribution (cont.) Numbers of Miles Beta Corp. Employees Drive to Work Each Day ClassFrequencyClass Boundaries Mid point Relative FrequencyCumulative Frequency 1.0–2.93 3.0–4.93 5.0–6.94 7.0–8.92 9.0–10.94