Statistics for the Social Sciences

1. Statistics for the Social Sciences Psychology 340 Fall 2006 Distributions

2. Outline (for week) • Variables: IV, DV, scales of measurement • Discuss each variable and it’s scale of measurement • Characteristics of Distributions • Using graphs • Using numbers (center and variability) • Descriptive statistics decision tree • Locating scores: z-scores and other transformations

3. Let’s get some data • On a sheet of paper (that you’ll turn in) write out these pieces of information: • Male or female • Height (in inches) • How many pairs of shoes in your closet • Typical number of servings of soda per day • Typical number of servings of water per day

4. Basic Concepts • Variable • A condition or characteristic that can have different values • Value • A possible number or category that a score can have • Score • A particular person’s value on a variable

5. Basic Concepts • Kinds of Variables • A condition or characteristic that can have different values • Experiment: • Independent - manipulated by experimenter • Dependent - measured by experimenter • Observational: • Explanatory - observed variable to do the explaining • Response - variable to be predicted

6. Measurement • Properties of our measurement? • Units of measurement - whether the measurement has a minimum sized unit or not • Levels (Scales) of measurement - the correspondence between the numbers representing the properties that we’re measuring

7. 3, or 2.5 cookies 2, 1 kid or 2 kids , but not 2.5 Units of Measurement • Continuous variables • Variables can take any number and can be infinitely broken down into smaller and smaller units • E.g., For lunch I can have • Discrete variables • Broken into a finite number of discrete categories that can’t be broken down • E.g., In my family I can have

8. Units of Measurement • Continuous or discrete?: • Male or female • Height (in inches) • How many pairs of shoes in your closet • Typical number of servings of soda per day • Typical number of servings of water per day

9. brown, hazel blue, green, Levels (scales) of measurement • Nominal Scale: Consists of a set of categories that have different names. • Measurements on a nominal scale label and categorize observations, but do not make any quantitative distinctions between observations. • Example: • Eye color:

10. Small, Med, Lrg, XL, XXL Levels of measurement • Ordinal Scale: Consists of a set of categories that are organized in an ordered sequence. • Measurements on an ordinal scale rank observations in terms of size or magnitude. • Example: • T-shirt size:

11. Levels of measurement • Interval Scale: Consists of ordered categories where all of the categories are intervals of exactly the same size. • With an interval scale, equal differences between numbers on the scale reflect equal differences in magnitude. • Ratios of magnitudes are not meaningful. • Example: • Fahrenheit temperature scale 40º 20º “Not Twice as hot”

12. Levels of measurement • Ratio scale: An interval scale with the additional feature of an absolute zero point. • With a ratio scale, ratios of numbers DO reflect ratios of magnitude.

13. Levels of measurement • What kind of measurement is used for each of these variables?: • Male or female • Height (in inches) • How many pairs of shoes in your closet • Typical number of servings of soda per day • Typical number of servings of water per day

14. Distributions • The distribution of a variable is a description of all of the tokens of the variable within in sample (or population if you’ve got the data) • A picture of the distribution is usually helpful • Gives a good sense of the properties of the distribution • Many different ways to display distribution • Frequency distribution table • Graphs

15. The values of the variable Steps for Making a Frequency Table(do this for class soda drinking variable) • Make a list down the page of each possible value, from highest to lowest

16. The values of the variable The number of tokens of each variable Steps for Making a Frequency Table • Go one by one through the scores, making a mark for each next to its value on the list, count up how frequently each value appears and include this in the table

17. The values of the variable The percentage of tokens at each value The number of tokens of each variable Steps for Making a Frequency Table • Figure the percentage (or proportion) of scores for each value % = (f/N)*100 N=total

18. The values of the variable The percentage of tokens at each value The number of tokens of each variable Cumulative percentage Steps for Making a Frequency Table • Figure the cumulative percentage (or proportion) of scores for each value % = (f/N)*100 N=total

19. Grouped Frequency Table(do this for class height variable) • A frequency table that uses intervals (range of values) instead of single values

20. Frequency Graphs • Histogram • Plot the different values against the frequency of each value

21. Frequency Graphs • Histogram (create one for class height) • Step 1: make a frequency distribution table (may use grouped frequency tables) • Step 2: put the values along the bottom, left to right, lowest to highest • Step 3: make a scale of frequencies along left edge • Step 4: make a bar above each value with a height for the frequency of that value

22. Frequency Graphs • Frequency polygon - essentially the same, put uses lines instead of bars

23. Properties of distributions • Distributions are typically summarized with three features • Shape • Center • Variability (Spread)

24. Shapes of Frequency Distributions • Unimodal, bimodal, and rectangular

25. Shapes of Frequency Distributions • Symmetrical and skewed distributions • Normal and kurtotic distributions

26. Displaying two variables • Bar graphs • Can be used in a number of ways (including displaying one or more variables) • Best used for categorical variables • Scatterplots • Best used for continuous variables

27. Bar graphs • Plot a bar graph of men and women in the class • Plot a bar graph of shoes in closet crossed with men and women • What should we plot? (and why?) • Total number of shoes for each group? • Average number of shoes for each group?

28. Scatterplot • Plot a scatterplot of soda and bottled water drinking • Useful for seeing the relationship between the variables

29. Next time • In addition to using tables and graphs to describe distributions, we also can provide numerical summaries