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

2.1Introduction to Frequency Distributions A frequency distribution –Shows 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 (number of what is measured), in each category

2.2 Frequency Distribution Tables Structure of Frequency Distribution Table –Categories in a column (usually highest to lowest) –Frequency count next to category To compute ΣX from a table –Convert table back to original score or –Compute ΣfX

Proportions and Percentages Proportions Fraction of the group associated with each score Called relative frequencies because they describe the frequency (f) in relation to the total number (N) Percentages Expresses relative frequency out of 100 Often included as a separate column in a frequency distribution table

Ex. 2.3 Frequency, Proportion and Percent in a Table Xfp = f/Npercent = p(100) 51 1/10 =.1010% 42 2/10 =.2020% 33 3/10 =.3030% 23 3/10 =.3030% 11 1/10 =.1010% Example 2.3

Learning Check Use the Frequency Distribution Table to determine how many subjects were in the study 10 A 15 B 33 C Impossible to determine from table D

Learning Check - Answer Use the Frequency Distribution Table to determine how many subjects were in the study 10 A 15 B 33 C Impossible to determine from table D

Learning Check Is each of these statements True or False for the Frequency Distribution shown? More than 50% of the individuals scored above 3. T/F The proportion of scores in the lowest category was p = 3. T/F

Learning Check - Answer Six out of ten individuals scored above 3 = 60% = more than half. True A proportion is a fractional part; 3 out of 10 scores = 3/10 =.30 False

Grouped Frequency Distribution Tables When a set of data covers large range of values we group the scores into intervals then list the intervals in the table. However, some information is lost when categories are grouped or combined –Individual scores cannot be retrieved –The wider the interval, the more information is lost

Rules for Grouped Frequency Distribution Tables 1. Try for about 10 class intervals 2. Width of interval should be simple number 3. Bottom score in interval should be a multiple of the width of the interval 4. Intervals should be the same width

Real Limits and Frequency Distributions For continuous variables, the score recorded corresponds to an interval on the number line Real limits For Grouped Frequency Distributions, the apparent limits are also smaller than the real limits Real limits apply here also

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

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

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? 9 points A 9.5 points B 10 points C 10.5 points D

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? 9 points A 9.5 points B 10 points (29.5 – 19.5 = 10) C 10.5 points D

2.3 Frequency Distribution Graphs Pictures of data from the frequency tables –All have two axes –X-axis (abscissa): categories of measurement scale increasing left to right –Y-axis (ordinate): 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

Creating a Histogram List all numeric scores (categories) or grouped intervals on the X-axis Draw bars above each score or class interval –Height of bar corresponds to frequency –Width of bar corresponds to real limits

Figure 2.1 Frequency Distribution Histogram

Figure 2.2 Frequency Distribution Histogram for Grouped Data

Figure 2.3 Modified Frequency Distribution Histogram

Frequency Distribution Polygons List all numeric scores/intervals on the X-axis Draw a dot above the center of each score or class interval –Height of dot corresponds to frequency –Connect the dots with a continuous line –Close the polygon with lines to the Y = 0 point

Figure 2.4 Frequency Distribution Polygon

Figure 2.5 Frequency Distribution Polygon for Grouped Data

Graphs for Nominal or Ordinal Data For non-numerical values (e.g., distinct categories), a bar graph is used. –Similar to a histogram, BUT –Spaces between adjacent bars indicates discrete categories without order (nominal) or of un-measurable width (ordinal)

Figure 2.6 Bar graph

Graphs for Population Distributions For large population, scores for each member are not possible, so: –Bar graphs can be based on relative frequency; –Smooth curves on a polygon indicate exact scores were not used The Normal Curve –Symmetric with greatest frequency in the middle –Common structure in data for many variables

Figure 2.7 Bar Graph of Relative Frequencies

Figure 2.8 Population Distribution of IQ: Normal Curve

2.4The Shape of a Frequency Distribution A distribution is either symmetrical or skewed: 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 = positive skew –Tail on the left = negative skew

Figure 2.10 Shapes of Distributions

Figure 2.9 Use and Misuse of Graphs

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

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

Learning Check Decide if each of the following statements is True or False. A treatment center for children measured the marital status of their parents (single, married, divorced, etc.) A histogram would be appropriate for these data. T/F A treatment center for children measured the time they spent playing with other children (in minutes). A histogram would be appropriate for these data. T/F

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

Figure 2.11 Answers to Learning Check