Describing Data. Carolyn J. Anderson EdPsych 580 Fall Describing Data p. 1/42

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1 Describing Data Carolyn J. Anderson EdPsych 580 Fall 2005 Describing Data p. 1/42

2 Describing Data Numerical Descriptions Single Variable Relationship Graphical displays Single variable. Relationships in data. Reading: Chapter 3 Describing Data p. 2/42

3 Descriptive Statistics Notation: Lower case letters denote observed values: y 1,y 2,...,y n Measures of central tendency. Measures of variability. When there are two variables Both numerical: correlation Both discrete: odds ratio, phi coefficient Numerical & discrete? Describing Data p. 3/42

4 Measures of Central Tendency Mode: the most frequent number. = Maximizes the number of correct guesses. Median: 50% below and 50% above this value. = Minimizes y guess. Mean: the arithmetic average ȳ = 1 n n i=1 y i. = Minimizes n i=1 (y i guess) 2. Describing Data p. 4/42

5 Properties of the Mean All scores influence the value. Add (or subtract) a constant c to each score 1 n n (y i + c) = 1 n i=1 n i=1 y i + 1 n n i=1 c = ȳ + c Multiply (divide) each score by a constant c 1 n n i=1 cy i = c n n i=1 y i = cȳ Describing Data p. 5/42

6 Properties of the Mean (continued) Sum of deviations about the mean ( n n n n 1 (y i ȳ) = y i ȳ = y i n n i=1 i=1 i=1 i=1 n i=1 y i ) = 0 The mean minimizes the sum of squared deviations ( n ) n min (y i c) 2 = (y i ȳ) 2 i=1 i=1 Describing Data p. 6/42

7 Measures of Dispersion (Variance) Range: = y max y min Variance = the average sum of squared deviations from the mean var(of a sample) var(of a population) = s 2 n = 1 n = σ 2 = 1 n un-biased estimate of σ 2 = ˆσ 2 = s 2 = n (y i ȳ) 2 i=1 n (y i µ) 2 i=1 1 (n 1) n (y i ȳ) 2. Standard Deviation = square root of variance. o Describing Data p. 7/42 i=1

8 Graphical Displays How depends on data type: Discrete/cateogrical (e.g., gender, region of US, ethnic background, grade level). = Bar chart, pie chart Ordinal but still discrete (e.g., responses to Likert item on a survey question). = Bar chart Numerical/continuous (e.g., SAT scores). = Cumulative distribution, histogram, box plot, stem-n-leaf. Describing Data p. 8/42

9 Discrete: Pie Chart Describing Data p. 9/42

10 Discrete: Bar Chart Describing Data p. 10/42

11 Ordinal: Bar Chart Describing Data p. 11/42

12 Ordinal: Bar Chart Describing Data p. 12/42

13 Ordinal: Bar Chart Describing Data p. 13/42

14 Numerical Variable: Stem n Leaf Stem Leaf # Multiply Stem.Leaf by 10**+1 \ consecutive Supermarket 7 shoppers and how much each spent (Data from Moore & 6 McCabe) --- probably made up Describing Data p. 14/42

15 Numerical Variable: Box Plot Average Total SAT Score mean * o Mid-West maximum 75 th percentile 50 th percentile (median) 25 th percentile minimum outlier Describing Data p. 15/42

16 Numerical: Histogram Science scores of high school seniors (HSB data) Describing Data p. 16/42

17 Relationships Between Variables Rarely interested in just one variable,e.g., Weight. Number of deaths per country due to heart disease. Number of men and women participating in a study on career choice. Quantitative SAT scores of applicants to UIUC. State average quantitative SAT scores. Describing Data p. 17/42

18 Association Between Variables Definition (Moore & McCabe, 1999): Two variables measured on the same individual are associated if some values of one variable tend to occur more often with some values of the second variable than with other values of that variable. Describing Data p. 18/42

19 Graphics for Relationships For today, look at the following: 1. Symmetric relation for 2 numerical variables. 2. Numerical response variable with numerical explanatory variable. categorical explanatory variable. numerical and categorical explanatory variables. Describing Data p. 19/42

20 Symmetric Relation for 2 Numerical SAT Scores Deborah Guber, Political Science, U of VT Title: Getting What You Pay For Individuals are 50 states. Our interest (for now) is to look at the nature of the relationship between average verbal SAT and average math SAT. Describing Data p. 20/42

21 Scatter Plots Shows the relationship between 2 numerical variables. The value of one variable shown on a horizontal axis and the value of the other variable shown on a vertical axis. One point for each individual. Describing Data p. 21/42

22 Scatter Plot for Symmetric Describing Data p. 22/42

23 Scatter Plot for Symmetric Describing Data p. 23/42

24 What Have We Learned? Positive association between mean verbal and quantitative SAT scores. Linear relationship. Relatively strong association. Describing Data p. 24/42

25 Asymmetric Relation, Two Numerical Individuals are 50 states. Interested in explaining the state SAT scores (response variable) based on expenditure per pupil (explanatory variable). Since verbal and quantitative SAT scores are strongly related, we will use the state total SAT scores as the response variable. Convention: Response/outcome variable on vertical. Explanatory variable on horizontal. Describing Data p. 25/42

26 Scatter Plot for Asymmetric Describing Data p. 26/42

27 What have we learned? Expenditure per Pupil in doesn t help explain the variability of SAT scores (at least at the state level). Possible outliers in terms of expenditure: NJ, NY, CT, AK. Describing Data p. 27/42

28 Alternative Explanatory Variables Individuals are 50 states. Response variable is the state average total SAT scores. Explanatory variables are (a) Percent of eligible students taking the SAT exam. (b) Region of country. Describing Data p. 28/42

29 Scatter Plot for Asymmetric Describing Data p. 29/42

30 What have we learned? Negative relationship between state average total SAT and the percent of eligible students taking the SAT exam. Approximate linear relationship. Variability in SAT total scores is about the same over percent of eligible students taking the SAT exam. Possibly groups of similar states. Describing Data p. 30/42

31 Asymmetric Numerical response variable and discrete explanatory variable. Plot of the data. Box plots. Mean plots. Describing Data p. 31/42

32 Mean Plot Numerical response and categorical explanatory Average Total SAT Score * * * * * East Mid-West North South West Regions of the United States Describing Data p. 32/42

33 What have we learned? Some of the variability in SAT total scores can be accounted for by region. Mid-western states have the highest average SAT total, followed by Southern and Western states, and Eastern and Northern states have the lowest. Describing Data p. 33/42

34 Box Plot: More Information Average Total SAT Score Mid-West maximum 75 th percentile 50 th percentile (median) 25 th percentile minimum Regions of the United States Describing Data p. 34/42

35 Box Plot Average Total SAT Score mean * o Mid-West maximum 75 th percentile 50 th percentile (median) 25 th percentile minimum outlier Regions of the United States Describing Data p. 35/42

36 Box Plots Average Total SAT Score * * o o * * * East Mid-West North South West Regions of the United States Describing Data p. 36/42

37 What have we learned? Some of the variability in SAT total scores can be accounted for by region. Mid-western states have the highest average SAT totals, followed by Southern and Western states, and Eastern and Northern states have the lowest. Southern states are the most variable, and the Eastern and Northern states have the least variabile. Most of the Western states are in between the mid-western and Eastern & Northern states. Describing Data p. 37/42

38 Data Plot: Numerical & Discrete Response = Total SAT, Explanatory = Region Describing Data p. 38/42

39 Scatter Plot with a Discrete Variable Use different symbols or colors Describing Data p. 39/42

40 Scatter Plot with State Labels Describing Data p. 40/42

41 What have we learned? Negative linear relationship between average total SAT score and percent of eligible students taking the SAT. Relatively few students from mid-west and south-central take SAT. Large percentage of students from North and East and south (i.e., AK, NC, SC, GA & FL)take SAT. Describing Data p. 41/42

42 What to Look for in Graphics Overall pattern. Direction of relationship. Strength of relationship. Variability. Deviations from pattern (e.g., outlier). Anything surprising or noteworthy. o Describing Data p. 42/42

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