Statistics for the Social Sciences

1. Statistics for the Social Sciences Psychology 340 Spring 2005 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 distributions • 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 • Okay, turn these in

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. 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:

9. 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:

10. 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”

11. 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.

12. Our variables • Descrete or continuous? • What level of measurement? • 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 • Now let’s consider how we would describe one of these variables

13. 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

14. 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

15. 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

16. 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

17. 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

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

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

20. 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

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

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

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

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

25. 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

26. 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?

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

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