1. Graphic Representation “A picture is worth a thousand words” captures the value of using graphs to represent distributions.

2. Types of Graphs There are four common types of graphs used to represent frequency distributions: – bar graph – pie chart – histogram – frequency polygon

3. Types of Graphs Bar graph - a series of rectangles, each representing the frequency or relative frequency of values in an unordered or ordered variable. Pie chart - segmented circle in which each segment represents the frequency or relative frequency in an unordered variable.

4. Bar Graph The bars represent distinct categories and, therefore, do not touch. White Red Green Striped 250 200 150 100 50 0 f Color Preference for Toothpaste

5. Pie Chart The size of each segment is calculated according to the minutes on a clock. Breed of Large Dog Ownership Sheep Dog 29% Golden Retriever 32% St. Bernard 26% Collie 13%

6. Types of Graphs Histogram - a series of rectangles, each representing the frequency or relative frequency of scores from a discrete or continuous variable. Frequency Polygon - a series of connected points, each representing the frequency or relative frequency of scores from a discrete or continuous variable.

7. Histogram The vertical boundaries coincide with the exact limits of each class interval. 30 40 50 60 70 80 90 100 25 20 15 10 5 0 f Midterm History Scores

8. Frequency Polygon Each point is positioned over the midpoint of each class interval. 30 40 50 60 70 80 90 100 25 20 15 10 5 0 f Midterm History Scores

9. Cumulative Percentage Frequency Polygon Each point is positioned over the upper exact limit of each class interval. 30 40 50 60 70 80 90 100 100 80 60 40 20 0 Midterm History Scores The characteristic “S” shape is called an ogive. % cum f

10. Cumulative Percentage Frequency Polygon A cumulative percentage frequency polygon can be used to estimate centiles and centile ranks. 30 40 50 60 70 80 90 100 100 80 60 40 20 0 % cum f Midterm History Scores Centile Rank = 70 Centile = 65

11. Describing Distributions An important part of making sense of data is to describe frequency distributions. There are four characteristics used for that purpose: –shape –kurtosis –central tendency –variability

12. Describing Distributions: Shape Frequency distributions often exhibit regularity of shape: –normal –skewed (positively and negatively) –bimodal –J-shaped

13. Describing Distributions: Kurtosis Kurtosis indicates how peaked is a distribution. –leptokurtic –mesokurtic –platykurtic

14. Describing Distributions: Central Tendency Central tendency refers to the average: –mode –median –mean

15. Describing Distributions: Variability Variability refers to the degree to which scores are clustered together. Each of the distributions below indicates a different degree of variability:

16. Describing Distributions We will now consider “central tendency” and “variability” in greater detail.