3. Descriptive statistics

3. Descriptive statistics - 3 Descriptive Statistics...

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3. Descriptive Statistics Describing data with tables and graphs (quantitative or categorical variables) Numerical descriptions of center, variability, position (quantitative variables) Bivariate descriptions (In practice, most studies have several variables)
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Frequency distribution : Lists possible values of variable and number of times each occurs Example : Student survey ( n = 60) www.stat.ufl.edu/~aa/social/data.html “political ideology” measured as ordinal variable with 1 = very liberal, …, 4 = moderate, …, 7 = very conservative 1. Tables and Graphs
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Histogram : Bar graph of frequencies or percentages
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Shapes of histograms (for quantitative variables) Bell-shaped (IQ, SAT, political ideology in all U.S. ) Skewed right (annual income, no. times arrested) Skewed left (score on easy exam) Bimodal (polarized opinions) Ex . GSS data on sex before marriage in Exercise 3.73: always wrong, almost always wrong, wrong only sometimes, not wrong at all category counts 238, 79, 157, 409
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Stem-and-leaf plot (John Tukey, 1977) Example : Exam scores ( n = 40 students) Stem Leaf 3 6 4 5 37 6 235899 7 011346778999 8 00111233568889 9 02238
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2.Numerical descriptions Let y denote a quantitative variable, with observations y 1 , y 2 , y 3 , … , y n a. Describing the center Median : Middle measurement of ordered sample Mean : 1 2 ... n i y y y y y n n + + + Σ = =
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Example : Annual per capita carbon dioxide emissions (metric tons) for n = 8 largest nations in population size Bangladesh 0.3, Brazil 1.8, China 2.3, India 1.2, Indonesia 1.4, Pakistan 0.7, Russia 9.9, U.S. 20.1 Ordered sample: Median = Mean = y
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Example : Annual per capita carbon dioxide emissions (metric tons) for n = 8 largest nations in population size Bangladesh 0.3, Brazil 1.8, China 2.3, India 1.2, Indonesia 1.4, Pakistan 0.7, Russia 9.9, U.S. 20.1 Ordered sample: 0.3, 0.7, 1.2, 1.4, 1.8, 2.3, 9.9, 20.1 Median = Mean = y
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Example : Annual per capita carbon dioxide emissions (metric tons) for n = 8 largest nations in population size Bangladesh 0.3, Brazil 1.8, China 2.3, India 1.2, Indonesia 1.4, Pakistan 0.7, Russia 9.9, U.S. 20.1 Ordered sample: 0.3, 0.7, 1.2, 1.4, 1.8, 2.3, 9.9, 20.1 Median = (1.4 + 1.8)/2 = 1.6 Mean = (0.3 + 0.7 + 1.2 + … + 20.1)/8 = 4.7 y
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Properties of mean and median For symmetric distributions, mean = median For skewed distributions, mean is drawn in direction of longer tail, relative to median Mean valid for interval scales, median for interval or ordinal scales Mean sensitive to “outliers” (median often preferred for highly skewed distributions) When distribution symmetric or mildly skewed or discrete with few values, mean preferred because uses numerical values of observations
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3. Descriptive statistics - 3 Descriptive Statistics...

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