1. Chapter 2 Frequency Distributions
PowerPoint Lecture Slides Essentials of Statistics for the Behavioral Sciences Eighth Edition by Frederick J Gravetter and Larry B. Wallnau

2. Learning Outcomes 1 Understand how frequency distributions are used 2
Organize data into a frequency distribution table… 3
…and into a grouped frequency distribution table 4
Know how to interpret frequency distributions 5
Organize data into frequency distribution graphs 6
Know how to interpret and understand graphs

3. Tools You Will Need Proportions (math review, Appendix A)
Fractions
Decimals
Percentages
Scales of measurement (Chapter 1)
Nominal, ordinal, interval, and ratio
Continuous and discrete variables (Chapter 1)
Real limits (Chapter 1)

4. 2.1 Frequency Distributions
A frequency distribution is
An organized tabulation
Showing the number of individuals located in each category on the scale of measurement
Can be either a table or a graph
Always shows
The categories that make up the scale
The frequency, or number of individuals, in each category
Terminology associated with frequency distributions is one of the least “standardized” across disciplines and texts students might encounter. Instructors may wish to emphasize the importance of being precise with the terms provided by the text authors, but also be aware that in other texts or courses the terms might be slightly different.

5. 2.2 Frequency Distribution Tables
Structure of Frequency Distribution Table
Categories in a column (often ordered from highest to lowest but could be reversed)
Frequency count next to category
Σf = N
To compute ΣX from a table
Convert table back to original scores or
Compute ΣfX

6. Proportions and Percentages
Measures the fraction of the total group that is associated with each score
Called relative frequencies because they describe the frequency ( f ) in relation to the total number (N)
Expresses relative frequency out of 100
Can be included as a separate column in a frequency distribution table
The ability to quickly and comfortably covert between fractions (proportions) , decimal fractions (relative frequency) and percentages is fundamental to success in this course. Some students struggle with reconciling the fact that although these are three distinct metrics, they all point to the same “deep” meaning.

7. Example 2.3 Frequency, Proportion and Percent
p = f/N
percent = p(100) 5 1
1/10 = .10
10% 4 2
2/10 = .20
20% 3
3/10 = .30
30%

8. Learning Check
Use the Frequency Distribution Table to determine how many subjects were in the study X f 5 2 4 3 1 A 10 B 15 C 33 D
Impossible to determine

9. Learning Check - Answer
Use the Frequency Distribution Table to determine how many subjects were in the study X f 5 2 4 3 1 A 10 B 15 C 33 D
Impossible to determine

10. Learning Check X f 5 2 4 3 1
For the frequency distribution shown, is each of these statements True or False?
T/F
More than 50% of the individuals scored above 3
The proportion of scores in the lowest category was p = 3

11. Learning Check - AnswerX f 5 2 4 3 1
For the frequency distribution shown, is each of these statements True or False?
True
Six out of ten individuals scored above 3 = 60% = more than half
False
A proportion is a fractional part; 3 out of 10 scores = 3/10 = .3

12. Grouped Frequency Distribution Tables
If the number of categories is very large they are combined (grouped) to make the table easier to understand
However, information is lost when categories are grouped
Individual scores cannot be retrieved
The wider the grouping interval, the more information is lost

13. “Rules” for Constructing Grouped Frequency Distributions
Requirements (Mandatory Guidelines)
All intervals must be the same width
Make the bottom (low) score in each interval a multiple of the interval width
“Rules of Thumb” (Suggested Guidelines)
Ten or fewer class intervals is typical (but use good judgment for the specific situation)
Choose a “simple” number for interval width
This slide shows a recommended treatment for the four guidelines presented in the text. The text presents it slightly differently by indicating these are “guidelines” rather than absolute requirements. However, violating guideline 4 distorts the information conveyed and violating guideline 3 makes it much more difficult to assimilate the information conveyed by the table. Consequently, each instructor should clarify expectations for her class: are these “guidelines” or “rules?”

14. Discrete Variables in Frequency or Grouped Distributions
Constructing either frequency distributions or grouped frequency distributions for discrete variables is uncomplicated
Individuals with the same recorded score had precisely the same measurements
The score is an exact score

15. Continuous Variables in Frequency Distributions
Constructing frequency distributions for continuous variables requires understanding that a score actually represents an interval
A given “score” actually could have been any value within the score’s real limits
The recorded value was rounded off to the middle value between the score’s real limits
Individuals with the same recorded score probably differed slightly in their actual performance

16. Continuous Variables in Frequency Distributions
Constructing grouped frequency distributions for continuous variables also requires understanding that a score actually represents an interval
Consequently, grouping several scores actually requires grouping several intervals
Apparent limits of the (grouped) class interval are always one unit smaller than the real limits of the (grouped) class interval. (Why?)
Why? Real limits extend ½ unit above and below each score. So the upper apparent score actually include ½ unit above it. Likewise, the lower apparent score acturally extends ½ unit below it. ½ + ½ = 1 “extra” unit included between real limits than is included between apparent limits.

17. Learning Check
A Grouped Frequency Distribution table has categories 0-9, 10-19, 20-29, and What is the width of the interval 20-29? A
9 points B
9.5 points C
10 points D
10.5 points

18. Learning Check - Answer
A Grouped Frequency Distribution table has categories 0-9, 10-19, 20-29, and What is the width of the interval 20-29? A
9 points B
9.5 points C
10 points (29.5 – 19.5 = 10) D
10.5 points

19. Learning Check
Decide if each of the following statements is True or False.
T/F
You can determine how many individuals had each score from a Frequency Distribution Table
You can determine how many individuals had each score from a Grouped Frequency Distribution

20. Learning Check - Answer
True
The original scores can be recreated from the Frequency Distribution Table
False
Only the number of individuals in the class interval is available once the scores are grouped

21. 2.3 Frequency Distribution Graphs
Pictures of the data organized in tables
All have two axes
X-axis (abscissa) typically has categories of measurement scale increasing left to right
Y-axis (ordinate) typically has frequencies increasing bottom to top
General principles
Both axes should have value 0 where they meet
Height should be about ⅔ to ¾ of length

22. Data Graphing Questions
Level of measurement? (nominal; ordinal; interval; or ratio)
Discrete or continuous data?
Describing samples or populations?
The answers to these questions determine which is the appropriate graph
The wide availability of graphing software may led an instructor to dispense with this slide. On the other hand, talking students through this discussion with a concrete example may help them better understand the issues and prevent some of the software-generated nonsense submitted by naïve users.

23. Frequency Distribution Histogram
Requires numeric scores (interval or ratio)
Represent all scores on X-axis from minimum thru maximum observed data values
Include all scores with frequency of zero
Draw bars above each score (interval)
Height of bar corresponds to frequency
Width of bar corresponds to score real limits (or one-half score unit above/below discrete scores)
Instructors may wish to identify one of the most common errors made by beginning students: dropping scores from the histogram if the frequency is zero.

24. Figure 2.1 Frequency Distribution Histogram
FIGURE 2.1 An example of a frequency distribution histogram. The same set of quiz scores is presented in a frequency distribution table and in a histogram.

25. Grouped Frequency Distribution Histogram
Same requirements as for frequency distribution histogram except:
Draw bars above each (grouped) class interval
Bar width is the class interval real limits
Consequence? Apparent limits are extended out one-half score unit at each end of the interval

26. Figure 2.2 Grouped Frequency Distribution Histogram
FIGURE 2.2 An example of a frequency distribution histogram for grouped data. The same set of children’s heights is presented in a frequency distribution table and in a histogram.

27. Block Histogram A histogram can be made a “block” histogram
Create a bar of the correct height by drawing a stack of blocks
Each block represents one per case
Therefore, block histograms show the frequency count in each bar

28. Figure 2.3 Frequency Distribution Block Histogram
FIGURE 2.3 A frequency distribution in which each individual is represented by a block placed directly above the individual’s score. For example, three people had scores of X = 2.

29. Frequency Distribution Polygons
List all numeric scores on the X-axis
Include those with a frequency of f = 0
Draw a dot above the center of each interval
Height of dot corresponds to frequency
Connect the dots with a continuous line
Close the polygon with lines to the Y = 0 point
Can also be used with grouped frequency distribution data
The importance of “closing the polygon” (sometimes called “anchoring the polygon”) can be humorously illustrated by describing the polygon floating away like an escaping helium-filled balloon.

30. Figure 2.4 Frequency Distribution Polygon
FIGURE 2.4 An example of a frequency distribution polygon. The same set of data is presented in a frequency distribution table and in a polygon.

31. Figure 2.5 Grouped Data Frequency Distribution Polygon
FIGURE 2.5 An example of a frequency distribution polygon for grouped data. The same set of data is presented in a grouped frequency distribution table and in a polygon.

32. Graphs for Nominal or Ordinal Data
For non-numerical scores (nominal and ordinal data), use a bar graph
Similar to a histogram
Spaces between adjacent bars indicates discrete categories
without a particular order (nominal)
non-measurable width (ordinal)

33. Figure Bar graph
FIGURE 2.6 A bar graph showing the distribution of personality types in a sample of college students. Because personality type is a discrete variable measured on a nominal scale, the graph is drawn with space between the bars.

34. Population Distribution Graphs
When population is small, scores for each member are used to make a histogram
When population is large, scores for each member are not possible
Graphs based on relative frequency are used
Graphs use smooth curves to indicate exact scores were not used
Normal
Symmetric with greatest frequency in the middle
Common structure in data for many variables

35. Figure 2.7 Bar Graph of Relative Frequencies
FIGURE 2.7 A frequency distribution showing the relative frequency for two types of fish. Notice that the exact number of fish is not reported; the graph simply says that there are twice as many bluegill as there are bass.

36. Figure 2.8 – IQ Population Distribution Shown as a Normal Curve
FIGURE 2.8 The population distribution of IQ scores: an example of a normal distribution.

37. Box 2.1 - Figure 2.9 Use and Misuse of Graphs
FIGURE 2.9 Two graphs showing the number of homicides in a city over a 4-year period. Both graphs show exactly the same data. However, the first graph gives the appearance that the homicide rate is high and rising rapidly. The second graph gives the impression that the homicide rate is low and has not changed over the 4-year period.

38. 2.4 Frequency Distribution Shape
Researchers describe a distribution’s shape in words rather than drawing it
Symmetrical distribution: each side is a mirror image of the other
Skewed distribution: scores pile up on one side and taper off in a tail on the other
Tail on the right (high scores) = positive skew
Tail on the left (low scores) = negative skew
Many students are confused by skewness and focus on the main cluster of scores instead of the tail. Instructors may wish to relay the following visual aid to retaining the correct information. First make a fist with thumbs pointed out and away from the fist. Next rotate wrists so the fingers are visible (rather than the back of the hands) and point thumbs away from the body on both sides. The thumb on the right hand points to the right and the direction of positive numbers on a number line, and represents the tail in a positively skewed distribution. The thumb on the left hand points to the left and the direction of negative numbers on the number line, and represents the tail in a negatively skewed distribution.

39. Figure 2.10 - Distribution Shapes
FIGURE Examples of different shapes for distributions.

40. Learning Check Negatively skewed Positively skewed Symmetrical
What is the shape of this distribution? A
Symmetrical B
Negatively skewed C
Positively skewed D
Discrete

41. Learning Check - Answer
What is the shape of this distribution? A
Symmetrical B
Negatively skewed C
Positively skewed D
Discrete

42. Learning Check
Decide if each of the following statements is True or False.
T/F
It would be correct to use a histogram to graph parental marital status data (single, married, divorced...) from a treatment center for children
It would be correct to use a histogram to graph the time children spent playing with other children from data collected in children’s treatment center

43. Learning Check - Answer
False
Marital Status is a nominal variable; a bar graph is required
True
Time is measured continuously and is an interval variable

44. Figure 2.11- Answers to Learning Check Exercise 1 (p. 51)

45. Any Questions?
Concepts?
Equations?