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4/12/2015Chapter 21 Describing Distributions with Numbers

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4/12/2015Chapter 22 Numerical Summaries of: Central location –mean –median Spread –Range –Quartiles –Standard Deviation / variance Shape measures not covered

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4/12/2015Chapter 23 Arithmetic Mean Most common measure of central location Notation (“xbar”): Where n is the sample size ∑ is the summation symbol

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4/12/2015Chapter 24 Example: Sample Mean Data: Metabolic rates, calories / day :

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4/12/2015Chapter 25 Median (M) Half the values are less than the median, half are greater If n is odd, the median is the middle ordered value If n is even, the median is the average of the two middle ordered values

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4/12/2015Chapter 26 Examples: Median Example 1: Median = 4 Example 2: Median = 5 (average of 4 and 6) Example 3: Median 2 (Values must first be ordered first 2 4 6, Median = 4)

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4/12/2015Chapter 27 Example: Median Ordered array: median Data = metabolic rates in slide 4 (n = 7) The location of the median in ordered array: L(M) = (n + 1) / 2 Value of median = 1614

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4/12/2015Chapter 2 8 The Median is robust to outliers This data set: has median 1614 and mean 1600 This similar data with high outlier: still has median 1614 but now has mean

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4/12/2015Chapter 29 The skew pulls the mean The average salary at a high tech firm is $250K / year The median salary is $60K What does this tell you? Answer: There are some very highly paid executives, but most of the workers make modest salaries, i.e., there is a positive skew to the distribution

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4/12/2015Chapter 210 Spread = Variability Amount of spread around the center! Statistical measures of spread –Range –Inter-Quartile Range –Standard deviation

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Range and IQR Range = maximum – minimum Easy, but NOT as good as the… Quartiles & Inter-Quartile Range (IQR) –Quartile 1 (Q1) cuts off bottom 25% of data (“25th percentile”) –Quartile 2 (Q2) cuts off two-quarters of data –same as the Median! –Quartile 3 (Q3) cuts off three-quarters of the data (“75th percentile”)

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4/12/2015Chapter 212 Obtaining Quartiles Order data Find the median Look at the lower half of data set –Find “median” of this lower half –This is Q1 Look at the upper half of the data set. –Find “median” of this upper half –This is Q3

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4/12/2015Chapter 213 Example: Quartiles Consider these 10 ages: median The median of the bottom half (Q1) = The median of the top half (Q3) =

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4/12/2015 Chapter 2 14 Example 2: Quartiles, n = 53 L(M)=(53+1) / 2 = 27 Median = 165

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4/12/2015Chapter 215 Example 2: Quartiles, n = 53 Bottom half has n* = 26 L(Q1)=(26 + 1) / 2= 13.5 from bottom Q1 = avg(127, 128) = 127.5

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4/12/2015Chapter 216 Example 2: Quartiles, n = 53 Top half has n* = 26 L(Q3) = 13.5 from the top! Q3 = avg(185, 185) = 185

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4/12/2015 Chapter Example 2 Quartiles Q2 = 165 Q3 = 185 Q1 = "5 point summary" = {Min, Q1, Median, Q3, Max} = {100, 127.5, 165, 185, 260}

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4/12/2015Chapter 218 Inter-quartile Range (IQR) Q 1 = Q 3 = 185 Inter-Quartile Range (IQR) = Q 3 Q 1 = 185 – = 57.5 “spread of middle 50%”

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4/12/2015Chapter 219 M Simple Box 5-point summary graphically Q1Q1 Q3Q3 minmax Weight

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4/12/2015Chapter 220 Boxplots are useful for comparing groups

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4/12/2015Chapter 221 Standard Deviation & Variance Most popular measures of spread Each data value has a deviation, defined as:

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4/12/2015Chapter 222 Example: Deviations Metabolic data (n = 7)

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4/12/2015Chapter 223 Variance Find the mean Find the deviation of each value Square the deviations Sum the squared deviations Divide by (n − 1)

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4/12/2015Chapter 224 Data Data: Metabolic rates, n =

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4/12/2015 Chapter 225 “Sum of Squares” ObsDeviations Squared deviations 1600 = 192 (192) 2 = 36, 1600 = 66 (66) 2 = 4, 1600 = -238 (-238) 2 = 56, 1600 = 14 (14) 2 = 1600 = -140 (-140) 2 = 19, 1600 = 267 (267) 2 = 71, 1600 = -161 (-161) 2 = 25,921 11, ,870 SUMS

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4/12/2015Chapter 226 Variance Sum of Squares

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4/12/2015Chapter 227 Standard Deviation Square root of variance

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4/12/2015Chapter 228 Standard Deviation Direct Formula

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Use calculator to check work! TI-30XIIS sequence: On > CLEAR > 2 nd > STAT > Scroll > Clear Data > Enter 2 nd > STAT > 1-VAR or 2-VAR DATA > “enter data STATVAR key I’m supporting the TI-30XIIS only

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4/12/2015Chapter 230 Choosing Summary Statistics Use the mean and standard deviation to describe symmetrical distributions & distributions free of outliers Use the median and quartiles (IQR) to describe distributions that are skewed or have outliers

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4/12/2015Chapter 231 Example: Number of Books Read M n = 52 L(M)=(52+1)/2=26.5

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4/12/2015Chapter 232 Example: Books read, n = 52 5-point summary: 0, 1, 3, 5.5, 99 Highly asymmetric distribution The mean ( “xbar” = 7.06) and standard deviation ( s = ) give false impressions of location and spread for this distribution and are considered inappropriate. Use the median and 5-point summary instead Number of books

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