HS 67 - Intro Health Statistics Describing Distributions with Numbers 4/10/2017 Chapter 2 Describing Distributions with Numbers 4/10/2017 Chapter 2 BPS Chapter 2 1

Numerical Summaries of: HS 67 - Intro Health Statistics 4/10/2017 Numerical Summaries of: Central location mean median Spread Range Quartiles Standard Deviation / variance Shape measures not covered 4/10/2017 Chapter 2 BPS Chapter 2 2

HS 67 - Intro Health Statistics 4/10/2017 Arithmetic Mean Most common measure of central location Notation (“xbar”): Where n is the sample size ∑ is the summation symbol 4/10/2017 Chapter 2 BPS Chapter 2 3

HS 67 - Intro Health Statistics 4/10/2017 Example: Sample Mean Data: Metabolic rates, calories / day: 1792 1666 1362 1614 1460 1867 1439 4/10/2017 Chapter 2 BPS Chapter 2 4

HS 67 - Intro Health Statistics 4/10/2017 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 4/10/2017 Chapter 2 BPS Chapter 2 5

HS 67 - Intro Health Statistics 4/10/2017 Examples: Median Example 1: 2 4 6 Median = 4 Example 2: 2 4 6 8 Median = 5 (average of 4 and 6) Example 3: 6 2 4 Median  2 (Values must first be ordered first 2 4 6 , Median = 4) 4/10/2017 Chapter 2 BPS Chapter 2 6

HS 67 - Intro Health Statistics 4/10/2017 Example: Median The location of the median in ordered array: L(M) = (n + 1) / 2 Data = metabolic rates in slide 4 (n = 7) Ordered array: 1362 1439 1460 1614 1666 1792 1867  median Value of median = 1614 4/10/2017 Chapter 2 BPS Chapter 2 7

The Median is robust to outliers HS 67 - Intro Health Statistics 4/10/2017 The Median is robust to outliers This data set: 1362 1439 1460 1614 1666 1792 1867 has median 1614 and mean 1600 This similar data with high outlier: 1362 1439 1460 1614 1666 1792 9867 still has median 1614 but now has mean 2742.9 4/10/2017 Chapter 2 BPS Chapter 2 8

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 4/10/2017 Chapter 2

HS 67 - Intro Health Statistics 4/10/2017 Spread = Variability Amount of spread around the center! Statistical measures of spread Range Inter-Quartile Range Standard deviation 4/10/2017 Chapter 2 BPS Chapter 2 10

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”)

HS 67 - Intro Health Statistics 4/10/2017 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 4/10/2017 Chapter 2 BPS Chapter 2 12

Example: Quartiles Consider these 10 ages: 05 11 21 24 27 28 30 42 50 52  median The median of the bottom half (Q1) = 21 05 11 21 24 27  The median of the top half (Q3) = 42 28 30 42 50 52  4/10/2017 Chapter 2

HS 67 - Intro Health Statistics 4/10/2017 Example 2: Quartiles, n = 53 Median = 165 L(M)=(53+1) / 2 = 27 4/10/2017 Chapter 2 BPS Chapter 2 14

Bottom half has n* = 26  L(Q1)=(26 + 1) / 2= 13.5 from bottom HS 67 - Intro Health Statistics 4/10/2017 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 4/10/2017 Chapter 2 15 BPS Chapter 2 15

Top half has n* = 26  L(Q3) = 13.5 from the top! HS 67 - Intro Health Statistics 4/10/2017 Example 2: Quartiles, n = 53 Top half has n* = 26  L(Q3) = 13.5 from the top! Q3 = avg(185, 185) = 185 4/10/2017 Chapter 2 16 BPS Chapter 2 16

HS 67 - Intro Health Statistics 4/10/2017 10 0166 11 009 12 0034578 13 00359 14 08 15 00257 16 555 17 000255 18 000055567 19 245 20 3 21 025 22 0 23 24 25 26 0 Example 2 Quartiles Q1 = 127.5 Q2 = 165 Q3 = 185 "5 point summary" = {Min, Q1, Median, Q3, Max} = {100, 127.5, 165, 185, 260} 4/10/2017 Chapter 2 BPS Chapter 2 17

Inter-quartile Range (IQR) HS 67 - Intro Health Statistics 4/10/2017 Inter-quartile Range (IQR) Inter-Quartile Range (IQR) = Q3  Q1 = 185 – 127.5 = 57.5 Q1 = 127.5 Q3 = 185 “spread of middle 50%” 4/10/2017 Chapter 2 BPS Chapter 2 18

Simple Box 5-point summary graphically HS 67 - Intro Health Statistics 4/10/2017 Simple Box 5-point summary graphically Q1 M Q3 min max 100 125 150 175 200 225 250 275 Weight 4/10/2017 Chapter 2 BPS Chapter 2 19

Boxplots are useful for comparing groups 4/10/2017 Chapter 2

Standard Deviation & Variance HS 67 - Intro Health Statistics 4/10/2017 Standard Deviation & Variance Most popular measures of spread Each data value has a deviation, defined as: 4/10/2017 Chapter 2 BPS Chapter 2 21

Example: Deviations Metabolic data (n = 7) 4/10/2017 Chapter 2

HS 67 - Intro Health Statistics 4/10/2017 Variance Find the mean Find the deviation of each value Square the deviations Sum the squared deviations Divide by (n − 1) 4/10/2017 Chapter 2 BPS Chapter 2 23

HS 67 - Intro Health Statistics 4/10/2017 Data Data: Metabolic rates, n = 7 1792 1666 1362 1614 1460 1867 1439 4/10/2017 Chapter 2 BPS Chapter 2 24

HS 67 - Intro Health Statistics 4/10/2017 “Sum of Squares” Obs Deviations Squared deviations 1792 17921600 = 192 (192)2 = 36,864 1666 1666 1600 = 66 (66)2 = 4,356 1362 1362 1600 = -238 (-238)2 = 56,644 1614 1614 1600 = 14 (14)2 = 196 1460 1460 1600 = -140 (-140)2 = 19,600 1867 1867 1600 = 267 (267)2 = 71,289 1439 1439 1600 = -161 (-161)2 = 25,921 11,200 214,870 SUMS 4/10/2017 Chapter 2 BPS Chapter 2 25

HS 67 - Intro Health Statistics 4/10/2017 Variance Sum of Squares 4/10/2017 Chapter 2 BPS Chapter 2 26

HS 67 - Intro Health Statistics 4/10/2017 Standard Deviation Square root of variance 4/10/2017 Chapter 2 BPS Chapter 2 27

Standard Deviation Direct Formula HS 67 - Intro Health Statistics 4/10/2017 Standard Deviation Direct Formula 4/10/2017 Chapter 2 28 BPS Chapter 2 28

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

Choosing Summary Statistics HS 67 - Intro Health Statistics 4/10/2017 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 4/10/2017 Chapter 2 BPS Chapter 2 30

Example: Number of Books Read HS 67 - Intro Health Statistics 4/10/2017 Example: Number of Books Read n = 52 L(M)=(52+1)/2=26.5 M 4/10/2017 Chapter 2 BPS Chapter 2 31

HS 67 - Intro Health Statistics 4/10/2017 Example: Books read, n = 52 5-point summary: 0, 1, 3, 5.5, 99 Highly asymmetric distribution 0 10 20 30 40 50 60 70 80 90 100 Number of books The mean (“xbar” = 7.06) and standard deviation (s = 14.43) give false impressions of location and spread for this distribution and are considered inappropriate. Use the median and 5-point summary instead. 4/10/2017 Chapter 2 BPS Chapter 2 32