1. Frequency Distributions To accompany Hawkes Lesson 2.1 Original content by D.R.S. 1

2. Is your data Qualitative or Quantitative? Qualitative: it’s a category – Blood type – Model of car – Favorite fast food restaurant Quantitative: it’s a numerical measurement – Heart rate, beats per minute – Fuel efficiency, miles per gallon – Dollars spent on meal – My pain, on a scale from 1 to 10 2

3. 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.) 3

4. Categorical Frequency Distributions are the fuel for the “Family Feud” 4 (Photograph borrowed from some web site somewhere; I failed to record the exact source.)

5. 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 5

6. 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 6

7. 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

8. Quantitative Frequency Distribution (data is 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 8

9. 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 9

10. About the Quantitative Frequency Distribution Instead of individual test score values, we GROUPED data into CLASSES Other names for “classes”: “bins”, “buckets” Each class is a low-to-high range of data values Each data value falls into exactly one class May be one or two “open-ended”classes – Like our “50 and higher” 10

11. About the classes CLASS LIMITS are 10-19, 20-29, etc. Classes do not overlap! Classes are usually the same width. CLASS MIDPOINTS are like 14.5, 24.5, etc. (High minus low, divided by 2) 11

12. Class LIMITS vs. Class BOUNDARIES CLASS LIMITS are 10-19, 20-29, etc. CLASS BOUNDARIES split the “gap” between class limits: 9.5-19.5, 19.5-29.5, etc. “9.5-19.5” means 9.5 ≤ x < 19.5 (note ≤ vs. < ) – All values between 9.5 and 19.5 – Including the lower endpoint of 9.5 – But excluding the upper endpoint of 19.5 12

13. A Cumulative Frequency column Placement Test Score How many applicants Cumulative frequency 0-919 10-193857 20-2952109 30-3971180 40-4950230 50 and above 28258  19 + 38 = 57  + 52 = 109  + 71 = 180  + 50 = 230  + 28 = 258 13

14. A Relative Frequency column Placement Test Score How many applicants Relative frequency 0-9197.4% 10-193814.7% 20-295220.2% 30-397127.5% 40-495019.4% 50 and above 2810.9% TOTAL258 14

15. Constructing a Frequency Distribution 1.How many classes should we have? 2.What class width should we use? 3.Find the class limits. 4. Sort your data, find the frequency of each class. Adapted from textbook page 46 © HLS 15

16. Example of Construction Using runners’ times from the Bunny Hop 5K in Cordele, March 31, 2012 – original data downloaded from a link at rungeorgia.com Click link to pdflink to pdf 16

17. 1. How many classes? Between 5 classes and 20 classes is good How many data values do you have? One textbook suggests: if you have < 125 data values, use the square root of the number of data values The Bunny Hop race had 103 finishers. By that rule, we would have 10 or 11 classes. Let’s agree on 10 classes for this example. 17

18. 2. Choose a Class Width  The “range” is the highest data value minus the lowest data value.  Divide the range by the number of classes  Then bump up to the next integer.  That’s just a starting point 18

19. 2. Choose a Class Width  High 66.8000 – Low 20.0167 = Range 45.7833  Divide the range by the number of classes  45.7833 ÷ 10 = 4.57833  Then bump up to the next integer.  Class width is 5  That’s just a starting point  We like it; it sounds good.  Nice “round” kind of a number for our readers 19

20. 3. Find the Class Limits  Start at what value for the first class? The lowest value is 20.0167 Let’s start our first class at 20.0000 Same number of decimal places as the data  The first class has a lower class limit of 20.0000  The lower limit of the next class is 25.0000 Take the lower limit of 20.0000 from previous class + class width of 5 = 25.0000 lower limit for next class 20

21. 3. Find the Class Limits - Lower  The first class has lower class limit = 20.0000  The next class has lower class limit = 25.0000  Etc. for the rest of the 10 classes: 30.0000, 35.0000, 40.0000, 45.0000 minutes, and 50.0000, 55.0000, 60.0000, 65.0000 minutes 21

22. 3. Find the class limits - Upper The first class has lower class limit 20.0000 The second class has lower class limit 25.0000 – So the first class has upper class limit 24.9999 The first class’s class limits: 20.0000 – 24.9999 Then next comes 25.0000 – 29.9999 Then 30.0000 – 34.9999, etc. All the way up through 65.0000-69.9999 22

23. 4. Count the frequency of each class Time (minutes)Frequency 20.0000-24.99999 25.0000-29.999926 30.0000-34.999923 35.0000-39.999914 40.0000-44.99997 45.0000-49.999911 50.0000-54.999910 55.0000-59.99990 60.0000-64.99992 65.0000-69.99991 If tallying unsorted data by hand, hash marks are useful. 23

24. Class Limits and Class Boundaries Class LimitsClass Boundaries 20.0000-24.999919.99995 – 24.99995 25.0000-29.999924.99995 – 29.99995 30.0000-34.999929.99995 – 34.99995 Etc. 55.0000-59.999954.99995 – 59.99995 60.0000-64.999959.99995 – 64.99995 65.0000-69.999964.99995 – 69.99995 24

25. Class Limits and Class Boundaries What to do with the gap between the class limits of adjacent classes? Limits 25.0000-29.9999 and 30.0000-34.9999 There’s gap between 29.99990 and 30.00000 Midway between them is 29.99995 Class Boundaries extend to that midpoint 24.99995 – 29.99995 and 29.99995– 34.99995 25

26. Class Boundaries Example: Class Limits 25.0000 – 29.9999 Class Boundaries 24.99995 – 29.99995 This means 24.99995 ≤ x < 29.99995 Note: including the lower boundary (≤) But not including the upper boundary (<) Because classes must never overlap 26

27. Class Midpoints (Upper Limit + Lower Limit) ÷ 2 Class LimitsClass Midpoints 20.0000-24.999922.49995 25.0000-29.999927.49995 30.0000-34.999932.49995 Etc. 55.0000-59.999957. 49995 60.0000-64.999962. 49995 65.0000-69.999967. 49995 27 = one class width apart

28. Class Limits, Boundaries, and Midpoints for the Placement Test It’s easier with whole numbers as class limits 28 Class Limits Frequency Class Boundaries Class Midpoint 0-919-0.5 – 9.54.5 10-19389.5 – 19.514.5 20-295219.5 – 29.524.5 30-397129.5 – 39.534.5 40-495039.5 – 49.544.5 50 +2849.5 and upNone? Or 54.5?

29. Excel Tools Link: The Excel FREQUENCY function.The Excel FREQUENCY function Link: The Excel COUNTIF function.The Excel COUNTIF function – Need to add info about COUNTIFS function. Also Excel “Histogram” function generates frequency distributions (discussed in the Histogram lesson) 29