L02_220 (1).pdf - The Economist September 6 2014 1 Histograms Central Tendency and Variability Lecture 2 Reading Sections 5.1 5.6 Includes ALL margin

L02_220 (1).pdf - The Economist September 6 2014 1...

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ECO220Y1Y, Lecture 2, Page 1 of 12 1 The Economist , September 6, 2014 Histograms, Central Tendency, and Variability Reading: Sections 5.1 – 5.6 2 Lecture 2 Includes ALL margin notes and boxes: “For Example,” “Guided Example,” “Notation Alert,” “Just Checking,” “Optional Math Boxes”, “What Can Go Wrong?” and “Ethics in Action” Histogram Histogram graphically describes how a single variable containing interval data is distributed Range of data divided into non-overlapping and equal width classes (bins) that cover range of values 3 0 .1 .2 .3 .4 Fraction 0 20 40 60 Inflation Rate, 2011 n = 174 countries How many bins? Width of bins?
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ECO220Y1Y, Lecture 2, Page 2 of 12 4 0 .1 .2 .3 .4 Fraction 0 20 40 60 Inflation Rate, 2011 n = 174 countries 0 .02 .04 .06 .08 .1 Density 0 20 40 60 Inflation Rate, 2011 n = 174 countries 0 20 40 60 80 Frequency 0 20 40 60 Inflation Rate, 2011 n = 174 countries Frequency histogram: Bar height number of observations in bin Density histogram: Bar area measures the fraction of observations in bin Relative frequency histogram: Bar height fraction of obs. in bin 5 0 .1 .2 .3 .4 Density 0 2 4 6 Inflation Rate, 2011 n = 34 OECD countries 6 0 .1 .2 .3 Density 0 2 4 6 Inflation Rate, 2011 n = 34 OECD countries 0 .1 .2 .3 .4 .5 Density 0 2 4 6 Inflation Rate, 2011 n = 34 OECD countries 0 .5 1 1.5 Density 0 2 4 6 Inflation Rate, 2011 n = 34 OECD countries Number of bins changes the appearance of the histogram Sturges’ formula: # of bins = 1 + 3.3*log(n) [Note log base 10.] OECD inflation: 1 + 3.3*log(34) = 6.05 ≈ 6 [but STATA picked 5]
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ECO220Y1Y, Lecture 2, Page 3 of 12 7 Shape of Things Histogram gives overview of a variable with a single picture Can make informal inferences about the shape of population Symmetric: If draw an imaginary line at center, have mirror image on each side Bell/Normal/Gaussian Positively skewed: long tail to right (aka right skewed) Negatively skewed: long tail to left (aka left skewed) Modality: # major peaks 8 0 .05 .1 Density 10 20 30 40 0 .1 .2 .3 Density 1 2 3 4 5 0 .1 .2 .3 .4 Density 2 3 4 5 6 0 .05 .1 .15 .2 .25 Density 0 2 4 6 8 Four Perfectly Symmetric Histograms 9 Two Perfectly Bell Shaped Histograms But histograms of real data will never be perfect: we always mean approximately 0 .1 .2 .3 .4 Density 10 15 20 25 30 0 .1 .2 .3 .4 Density -4 -2 0 2 4 For example, we’d describe the histogram to the right as Normal (Bell) shaped
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ECO220Y1Y, Lecture 2, Page 4 of 12 10 0 .05 .1 .15 .2 Density 0 5 10 15 0 .1
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