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Chapter 4: Summary Statistics May 17 In Chapter 4: The prior chapter used stemplots and histograms to look at the shape, location, and spread of a distribution. This chapter uses numerical summaries for similar purposes. Summary Statistics • Central location – Mean – Median – Mode • Spread – Range and interquartile range (IQR) – Variance and standard deviation • Shape summaries – seldom used in practice – not covered Notation • n sample size • X the variable (e.g., ages of subjects) • xi the value of individual i for variable X • sum all values (capital sigma) • Illustrative data (ages of participants): 21 42 5 11 30 50 28 27 24 n = 10 X = AGE variable x1= 21, x2= 42, …, x10= 52 xi = x1 + x2 + … + x10= 21 + 42 + … + 52 = 290 52 §4.1: Central Location: Sample Mean • • • • “Arithmetic average” Traditional measure of central location Sum the values and divide by n “xbar” refers to the sample mean 1 1 x x1 x 2 xn n n n x i i 1 Example: Sample Mean Ten individuals selected at random have the following ages: 21 42 5 11 30 50 28 27 24 52 Note that n = 10, xi = 21 + 42 + … + 52 = 290, and 1 1 x x n i 10 (290) 29.0 The sample mean is the gravitational center of a distribution 0 10 20 30 Mean = 29 40 50 60 Uses of the Sample Mean The sample mean can be used to predict: • The value of an observation drawn at random from the sample • The value of an observation drawn at random from the population • The population mean Population Mean x 1 N N i x i • Same operation as sample mean except based on entire population (N ≡ population size) • Conceptually important • Usually not available in practice • Sometimes referred to as the expected value §4.2 Central Location: Median The median is the value with a depth of (n+1)/2 When n is even, average the two values that straddle a depth of (n+1)/2 For the 10 values listed below, the median has depth (10+1) / 2 = 5.5, placing it between 27 and 28. Average these two values to get median = 27.5 05 11 21 24 27 28 30 42 50 median Average the adjacent values: M = 27.5 52 More Examples of Medians • Example A: 2 4 6 Median = 4 • Example B: 2 4 6 8 Median = 5 (average of 4 and 6) • Example C: 6 2 4 Median 2 (Values must be ordered first) The Median is Robust The median is more resistant to skews and outliers than the mean; it is more robust. This data set has a mean of 1636: 1362 1439 1460 1614 1666 1792 1867 Here’s the same data set with a data entry error “outlier” (highlighted). This data set has a mean of 2743: 1362 1439 1460 1614 1666 1792 9867 The median is 1614 in both instances, demonstrating its robustness in the face of outliers. §4.3: Mode • The mode is the most commonly encountered value in the dataset • This data set has a mode of 7 {4, 7, 7, 7, 8, 8, 9} • This data set has no mode {4, 6, 7, 8} (each point appears only once) • The mode is useful only in large data sets with repeating values §4.4: Comparison of Mean, Median, Mode Note how the mean gets pulled toward the longer tail more than the median mean = median → symmetrical distrib mean > median → positive skew mean < median → negative skew §4.5 Spread: Quartiles • Two distributions can be quite different yet can have the same mean • This data compares particulate matter in air samples (μg/m3) at two sites. Both sites have a mean of 36, but Site 1 exhibits much greater variability. We would miss the high pollution days if we relied solely on the mean. Site 1| |Site 2 ---------------42|2| 8|2| 2|3|234 86|3|6689 2|4|0 |4| |5| |5| |6| 8|6| ×10 Spread: Range • Range = maximum – minimum • Illustrative example: Site 1 range = 86 – 22 = 64 Site 2 range = 40 – 32 = 8 • Beware: the sample range will tend to underestimate the population range. • Always supplement the range with at least one addition measure of spread Site 1| |Site 2 ---------------42|2| 8|2| 2|3|234 86|3|6689 2|4|0 |4| |5| |5| |6| 8|6| ×10 Spread: Quartiles • Quartile 1 (Q1): cuts off bottom quarter of data = median of the lower half of the data set • Quartile 3 (Q3): cuts off top quarter of data = median of the to half of the data set • Interquartile Range (IQR) = Q3 – Q1 covers the middle 50% of the distribution 05 11 21 Q1 24 27 28 median 30 42 50 Q3 Q1 = 21, Q3 = 42, and IQR = 42 – 21 = 21 52 Quartiles (Tukey’s Hinges) – Example 2 Data are metabolic rates (cal/day), n = 7 1362 1439 1460 1614 1666 1792 1867 median When n is odd, include the median in both halves of the data set. Bottom half: 1362 1439 1460 which has a median of 1449.5 (Q1) Top half: 1614 1666 1792 1867 which has a median of 1729 (Q3) 1614 Five-Point Summary • • • • • Q0 (the minimum) Q1 (25th percentile) Q2 (median) Q3 (75th percentile) Q4 (the maximum) §4.6 Boxplots 1. Calculate 5-point summary. Draw box from Q1 to Q3 w/ line at median 2. Calculate IQR and fences as follows: FenceLower = Q1 – 1.5(IQR) FenceUpper = Q3 + 1.5(IQR) Do not draw fences 3. Determine if any values lie outside the fences (outside values). If so, plot these separately. 4. Determine values inside the fences (inside values) Draw whisker from Q3 to upper inside value. Draw whisker from Q1 to lower inside value Illustrative Example: Boxplot Data: 05 11 21 24 27 28 30 42 50 52 1. 5 pt summary: {5, 21, 27.5, 42, 52}; box from 21 to 42 with line @ 27.5 2. IQR = 42 – 21 = 21. FU = Q3 + 1.5(IQR) = 42 + (1.5)(21) = 73.5 FL = Q1 – 1.5(IQR) = 21 – (1.5)(21) = –10.5 3. None values above upper fence None values below lower fence 4. Upper inside value = 52 Lower inside value = 5 Draws whiskers 60 50 40 Upper inside = 52 Q3 = 42 30 Q2 = 27.5 20 Q1 = 21 10 Lower inside = 5 0 Illustrative Example: Boxplot 2 Data: 3 21 22 24 25 26 28 29 31 51 1. 5-point summary: 3, 22, 25.5, 29, 51: draw box 2. IQR = 29 – 22 = 7 FU = Q3 + 1.5(IQR) = 28 + (1.5)(7) = 39.5 FL = Q1 – 1.5(IQR) = 22 – (1.5)(7) = 11.6 3. One above top fence (51) One below bottom fence (3) 60 50 Outside value (51) 40 Inside value (31) 30 20 Upper hinge (29) Median (25.5) Lower hinge (22) Inside value (21) 4. Upper inside value is 31 Lower inside value is 21 Draw whiskers 10 Outside value (3) 0 Illustrative Example: Boxplot 3 Seven metabolic rates: 1362 1439 1460 1614 1666 1792 1867 1. 5-point summary: 1362, 1449.5, 1614, 1729, 1867 2000 2. IQR = 1729 – 1449.5 = 279.5 FU = Q3 + 1.5(IQR) = 1729 + (1.5)(279.5) = 2148.25 1900 1800 1700 1600 FL = Q1 – 1.5(IQR) = 1449.5 – (1.5)(279.5) = 1030.25 1500 1400 3. None outside 4. Whiskers end @ 1867 and 1362 1300 N= 7 Data source: Moore, Boxplots: Interpretation • Location – Position of median – Position of box • Spread – Hinge-spread (IQR) – Whisker-to-whisker spread – Range • Shape – Symmetry or direction of skew – Long whiskers (tails) indicate leptokurtosis Side-by-side boxplots Boxplots are especially useful when comparing groups §4.7 Spread: Standard Deviation • Most common descriptive measures of spread • Based on deviations around the mean. • This figure demonstrates the deviations of two of its values This data set has a mean of 36. The data point 33 has a deviation of 33 – 36 = −3. The data point 40 has a deviation of 40 – 36 = 4. Variance and Standard Deviation Deviation = xi x Sum of squared deviations = SS x x 2 i SS Sample variance = s n 1 2 Sample standard deviation = s s 2 Standard deviation (formula) Sum of Squares 1 2 s ( xi x ) n 1 Sample standard deviation s is the estimator of population standard deviation . See “Facts About the Standard Deviation” p. 80. Illustrative Example: Standard Deviation (p. 79) Observation xi Deviations Squared deviations xi x 2 xi x 36 36 36 = 0 02 = 0 38 38 36 = 2 22 = 4 39 39 36 = 3 32 = 9 40 40 36 = 4 42 = 16 36 36 36 = 0 02 = 0 34 34 36 = 2 22 = 4 33 33 36 = 3 32 = 9 32 32 36 = 4 42 = 16 0* SS = 58 SUMS * Sum of deviations always equals zero Illustrative Example (cont.) Sample variance (s2) SS 58 s 8.286 ( g/m 3 ) 2 n 1 8 1 2 Standard deviation (s) s s 8.286 2.88 g/m 2 3 Interpretation of Standard Deviation • Measure spread (e.g., if group was s1 = 15 and group 2 s2 = 10, group 1 has more spread, i.e., variability) • 68-95-99.7 rule (next slide) • Chebychev’s rule (two slides hence) 68-95-99.7 Rule Normal Distributions Only! • • • • 68% of data in the range μ ± σ 95% of data in the range μ ± 2σ 99.7% of data the range μ ± 3σ Example. Suppose a variable has a Normal distribution with = 30 and σ = 10. Then: 68% of values are between 30 ± 10 = 20 to 40 95% are between 30 ± (2)(10) = 30 ± 20 = 10 to 50 99.7% are between 30 ± (3)(10) = 30 ± 30 = 0 to 60 Chebychev’s Rule All Distributions • Chebychev’s rule says that at least 75% of the values will fall in the range μ ± 2σ (for any shaped distribution) • Example: A distribution with μ = 30 and σ = 10 has at least 75% of the values in the range 30 ± (2)(10) = 10 to 50 Rules for Rounding • Carry at least four significant digits during calculations. (Click here to learn about significant digits.) • Round as last step of operation • Avoid pseudo-precision • When in doubt, use the APA Publication Manual • Always report units Always use common sense and good judgment. Choosing Summary Statistics • Always report a measure of central location, a measure of spread, and the sample size • Symmetrical mound-shaped distributions report mean and standard deviation • Odd shaped distributions report 5-point summaries (or median and IQR) Software and Calculators Use software and calculators to check work.