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Summary Statistics

Summary Statistics. Last week we used stemplots and histograms to describe the shape , location , and spread of a distribution. This week we use numerical summaries of location and spread. Main Summary Statistics by Type. Central location Mean Median Mode Spread

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Summary Statistics

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  1. Summary Statistics Last week we used stemplots and histograms to describe the shape, location, and spread of a distribution. This week we use numerical summaries of location and spread. Summary Statistics

  2. Main Summary Statistics by Type • Central location • Mean • Median • Mode • Spread • Variance and standard deviation • Quartiles and Inter Quartile Range (IQR) • Shape • Statistical measures of spread (e.g., skewness and kurtosis) are available but are seldom used in practice (not covered) Summary Statistics

  3. Notation • n sample size • X variable • xi value of individual i •   sum all values (capital sigma) • Illustrative example (sample.sav), data: 21 42 5 11 30 50 28 27 24 52 • n= 10 • X = age • x1= 21, x2= 42, …, x10= 52 • x = 21 + 42 + … + 52 = 290 Summary Statistics

  4. Sample Mean Illustrative example: n = 10 (data & intermediate calculations on prior slide) Summary Statistics

  5. Population Mean • Same operation as sample mean, but based on entire population (N = population size) • Not available in practice, but important conceptually Summary Statistics

  6. Interpretation of xbar • Sample mean used to predict • an observation drawn at random from a sample • an observation drawn at random from the population • the population mean • Gravitational center (balance point) Summary Statistics

  7. Median – a different kind of average • “Middle value” • Covered last week • Order data • Depth of median is (n+1) / 2 • When n is odd  middle value • When n is even  average two middle values • Illustrative example, n = 10  median has depth (10+1) / 2 = 5.5 05 11 21 24 27 28 30 42 50 52median = average of 27 and 28 = 27.5 Summary Statistics

  8. Outlier Median is “robust” Robust  resistant to skews and outliers This data set has a mean (xbar) of 1600: 1362 1439 1460 1614 1666 1792 1867 This data set has an outlier and a mean of 2743: 1362 1439 1460 1614 1666 1792 9867 The median is 1614 in both instances. The median was not influenced by the outlier. Summary Statistics

  9. Mode • Mode  value with greatest frequency • e.g., {4, 7, 7, 7, 8, 8, 9} has mode = 7 • Used only in very large data sets Summary Statistics

  10. Mean, Median, Mode • Symmetrical data: mean = median • positive skew: mean > median [mean gets “pulled” by tail] • negative skew: mean < median Summary Statistics

  11. Spread = Variability • Variability  amount values spread above and below the average • Measures of spread • Range and inter-quartile range • Standard deviation and variance (this week) Summary Statistics

  12. Range = max – min The range is rarely used in practice b/c it tends to underestimate population range and is not robust Summary Statistics

  13. Deviation = Sum of squared deviations = Sample standard deviation = Standard deviation Most common descriptive measure of spread Sample variance = Summary Statistics

  14. Standard deviation (formula) Sample standard deviation s is the unbiased estimator of population standard deviation . Population standard deviation  is rarely known in practice. Summary Statistics

  15. New data set (“Metabolic Rates”)This example is not in your lecture notes Metabolic rates (cal/day), n = 7 1792 1666 1362 1614 1460 1867 1439 Summary Statistics

  16. Metabolic rates showing mean (*) and deviations of first two observations Summary Statistics

  17. Standard Deviation Calculationmetabolic.sav – introduced slide 15 * Sum of deviations will always equal zero Summary Statistics

  18. Standard Deviation Metabolic data (cont.) Variance (s2) Standard deviation (s) Summary Statistics

  19. General rule for rounding means and standard deviations • Report mean to one additional decimals above that of the data • To achieve accuracy, intermediate calculations should carry still an additional decimals • Illustrative example • Suppose data is recorded with one decimal accuracy (i.e., xx.x) • Report mean with two decimal accuracy (i.e., xx.xx) • Carry all intermediate calculations with at least three decimal accuracy (i.e., xx.xxx) Even more important: Always use common sense and judgment. Summary Statistics

  20. TI-30XIIS – about $12 In practice, we often use software or a calculator to check our standard deviation Summary Statistics

  21. Interpretation of Standard Deviation • Larger standard deviation  greater variability • s1 = 15 and s2 = 10  group 1 has more variability • 68-95-99.7 rule – Normal data only • 68% of data with 1 SD of mean, 95% within 2 SD from mean, and 99.7% within 3 SD of mean • e.g., if mean = 30 and SD = 10, then 95% of individuals are in the range 30 ± (2)(10) = 30 ± 20 = (10 to 50) • Chebychev’s rule – All data • at least 75% data within 2 SD of mean • e.g., mean = 30 and SD = 10, then at least 75% of individuals in range 30 ± (2)(10) = (10 to 50) Summary Statistics

  22. Quartiles and IQR • Quartiles divide the ordered data into four equally-sized groups • Q0 = minimum • Q1 = 25th %ile • Q2 = 50th %ile (Median) • Q3 = 75th %ile • Q4 = maximum Summary Statistics

  23. Rule for quartiles • Find the median  Q2 • Middle of lower half of data set  Q1 • Middle of upper half of the data  Q3 Bottom half | Top half 05 11 21 24 27 | 28 30 42 50 52    Q1 Q2 Q3 IQR = Q3 – Q1 = 42 – 21 = 21 gives spread of middle 50% of the data Summary Statistics

  24. 5-Point Summary (sample.sav) • Q0 = 5 (minimum) • Q1 = 21 (lower hinge) • Q2 = 27.5 (median) • Q3 = 42 (upper hinge) • Q4 = 52 (maximum) Best descriptive statistics for skewed data Summary Statistics

  25. Illustrative example (metabolic.sav) 1362 1439 1460 1614 1666 1792 1867 median Bottom half : 1362 1439 1460 1614  Q1 = (1439 + 1460) / 2 = 1449.5 Top half: 1614 1666 1792 1867 Q3 = (1666 + 1792) / 2 = 1729 5-point summary: 1362, 1449.5, 1614, 1729, 1867 Summary Statistics

  26. Box-and-whiskers plot (boxplot) • 5 point summary + “outside values” • Procedure • Determine 5-point summary • Draw box from Q1 to Q3 • Draw line @ Q2 • Calculate IQR = Q3 – Q1 • Calculate fences • FLower = Q1 – 1.5(IQR) • FUpper = Q3 + 1.5(IQR) • Determine if any outside values? If so, plot separately • Determine inside values and draw whiskers from box to inside values Summary Statistics

  27. 60 Upper inside = 52 50 Q3 = 42 40 30 Q2 = 27.5 Q1 = 21 20 10 Lower inside = 5 0 Boxplot example 05 11 21 24 27 28 30 42 50 52 • 5-point: 5, 21, 27.5, 42, 52 • IQR = 42 – 21 = 21 • FU = 42 + (1.5)(21) = 73.5 • No outside above (outside) Upper inside value = 52 • FL = 21 – (1.5)(21) = –10.5 • No values below (outside) • Lower inside value = 5 Summary Statistics

  28. Boxplot example 2 3 21 22 24 25 26 28 29 31 51 • 5-point: 3, 22, 25.5, 29, 51 • IQR = 29 – 22 = 7 • FU = 29 + (1.5)(7) = 39.5 • One outside (51) • Inside value = 31 • FL = 22 – (1.5)(7) = 11.5 • One outside (3) • Inside value = 21 Summary Statistics

  29. Boxplot example 3 (metabolic.sav) 1362 1439 1460 1614 1666 1792 1867 • 5-point: 1362, 1449.5, 1614, 1729, 1867 (slide 30) • IQR = 1729 – 1449.5 = 279.5 • FU = 1729 + (1.5)(279.5) = 2148.25 • None outside • Upper inside = 1867 • FL = 1449.5 – (1.5)(279.5) = 1030.25 • None outside • Lower inside = 1362 Summary Statistics

  30. Interpretation of boxplots • Location • Position of median • Position of box • Spread • Hinge-spread (box length) = IQR • Whisker-to-whisker spread (range or range minus the outside values) • Shape • Symmetry of box • Size of whiskers • Outside values (potential outliers) Summary Statistics

  31. Side-by-side boxplots Boxplots are especially useful for comparing groups: Summary Statistics

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