Introduction to Statistics for the Social Sciences SBS200 - Lecture Section 001, Fall 2016 Room 150 Harvill Building 10:00 - 10:50 Mondays, Wednesdays & Fridays. Welcome http://www.youtube.com/watch?v=oSQJP40PcGI

By the end of lecture today 9/14/16 Use this as your study guide By the end of lecture today 9/14/16 Characteristics of a distribution Central Tendency Dispersion Measures of variability Range Standard deviation Variance Memorizing the four definitional formulae

Homework Assignments 7 & 8 Please complete the homework worksheets Calculating Descriptive Statistics And Presenting Findings in a Memorandum Due: Friday, September 16th

Schedule of readings Before next exam (September 23rd) Please read chapters 1 - 5 in OpenStax textbook Please read Appendix D, E & F online On syllabus this is referred to as online readings 1, 2 & 3 Please read Chapters 1, 5, 6 and 13 in Plous Chapter 1: Selective Perception Chapter 5: Plasticity Chapter 6: Effects of Question Wording and Framing Chapter 13: Anchoring and Adjustment

Labs continue this week Lab sessions Everyone will want to be enrolled in one of the lab sessions Labs continue this week

Overview Frequency distributions The normal curve Challenge yourself as we work through characteristics of distributions to try to categorize each concept as a measure of 1) central tendency 2) dispersion or 3) shape Mean, Median, Mode, Trimmed Mean Skewed right, skewed left unimodal, bimodal, symmetric

A little more about frequency distributions An example of a normal distribution

A little more about frequency distributions An example of a normal distribution

A little more about frequency distributions An example of a normal distribution

A little more about frequency distributions An example of a normal distribution

A little more about frequency distributions An example of a normal distribution

Measure of central tendency: describes how scores tend to Measure of central tendency: describes how scores tend to cluster toward the center of the distribution Normal distribution In all distributions: mode = tallest point median = middle score mean = balance point In a normal distribution: mode = mean = median

Measure of central tendency: describes how scores tend to Measure of central tendency: describes how scores tend to cluster toward the center of the distribution Positively skewed distribution In all distributions: mode = tallest point median = middle score mean = balance point In a positively skewed distribution: mode < median < mean Note: mean is most affected by outliers or skewed distributions

Measure of central tendency: describes how scores tend to Measure of central tendency: describes how scores tend to cluster toward the center of the distribution Negatively skewed distribution In all distributions: mode = tallest point median = middle score mean = balance point In a negatively skewed distribution: mean < median < mode Note: mean is most affected by outliers or skewed distributions

Mode: The value of the most frequent observation Bimodal distribution: Distribution with two most frequent observations (2 peaks) Example: Ian coaches two boys baseball teams. One team is made up of 10-year-olds and the other is made up of 16-year-olds. When he measured the height of all of his players he found a bimodal distribution

Overview Frequency distributions The normal curve Mean, Median, Mode, Trimmed Mean Standard deviation, Variance, Range Mean Absolute Deviation Skewed right, skewed left unimodal, bimodal, symmetric

Frequency distributions The normal curve

Some distributions are more Variability What might this be? Some distributions are more variable than others Let’s say this is our distribution of heights of men on U of A baseball team 5’ 5’6” 6’ 6’6” 7’ 5’ 5’6” 6’ 6’6” 7’ Mean is 6 feet tall What might this be? 5’ 5’6” 6’ 6’6” 7’

Dispersion: Variability Some distributions are more variable than others 6’ 7’ 5’ 5’6” 6’6” A The larger the variability the wider the curve tends to be The smaller the variability the narrower the curve tends to be B Range: The difference between the largest and smallest observations C Range for distribution A? Range for distribution B? Range for distribution C?

84” – 71” = 13” Wildcats Basketball team: Tallest player = 84” (same as 7’0”) (Lauri Markkanen and Dusan Ristic) Shortest player = 71” (same as 5’11”) (Parker Jackson-Cartwritght) Fun fact: Mean is 78 Range: The difference between the largest and smallest scores 84” – 71” = 13” xmax - xmin = Range Range is 13”

No reference is made to numbers between the min and max Baseball Fun fact: Mean is 72 Wildcats Baseball team: Tallest player = 77” (same as 6’5”) (Kevin Ginkel) Shortest player = 68” (same as 5’8”) (Justin Behnke and Cody Ramer & Zach Gibbons) Range: The difference between the largest and smallest score 77” – 68” = 9” xmax - xmin = Range Range is 9” (77” – 68” ) Please note: No reference is made to numbers between the min and max

Let’s build it up again… U of A Baseball team Deviation scores Let’s build it up again… U of A Baseball team Diallo is 0” Diallo is 6’0” Diallo’s deviation score is 0 6’0” – 6’0” = 0 Diallo 5’8” 5’10” 6’0” 6’2” 6’4”

Let’s build it up again… U of A Baseball team Deviation scores Diallo is 0” Let’s build it up again… U of A Baseball team Preston is 2” Diallo is 6’0” Diallo’s deviation score is 0 Preston is 6’2” Preston Preston’s deviation score is 2” 6’2” – 6’0” = 2 5’8” 5’10” 6’0” 6’2” 6’4”

Let’s build it up again… U of A Baseball team Deviation scores Diallo is 0” Let’s build it up again… U of A Baseball team Preston is 2” Mike is -4” Hunter is -2 Diallo is 6’0” Diallo’s deviation score is 0 Hunter Preston is 6’2” Mike Preston’s deviation score is 2” Mike is 5’8” Mike’s deviation score is -4” 5’8” – 6’0” = -4 5’8” 5’10” 6’0” 6’2” 6’4” Hunter is 5’10” Hunter’s deviation score is -2” 5’10” – 6’0” = -2

Let’s build it up again… U of A Baseball team Deviation scores Diallo is 0” Let’s build it up again… U of A Baseball team Preston is 2” Mike is -4” Hunter is -2 Shea is 4 David is 0” Diallo’s deviation score is 0 David Preston’s deviation score is 2” Mike’s deviation score is -4” Shea Hunter’s deviation score is -2” Shea is 6’4” Shea’s deviation score is 4” 5’8” 5’10” 6’0” 6’2” 6’4” 6’4” – 6’0” = 4 David is 6’ 0” David’s deviation score is 0 6’ 0” – 6’0” = 0

Let’s build it up again… U of A Baseball team Deviation scores Diallo is 0” Let’s build it up again… U of A Baseball team Preston is 2” Mike is -4” Hunter is -2 Shea is 4 David is 0” Diallo’s deviation score is 0 David Preston’s deviation score is 2” Mike’s deviation score is -4” Shea Hunter’s deviation score is -2” Shea’s deviation score is 4” David’s deviation score is 0” 5’8” 5’10” 6’0” 6’2” 6’4”

Let’s build it up again… U of A Baseball team Deviation scores Diallo is 0” Let’s build it up again… U of A Baseball team Preston is 2” Mike is -4” Hunter is -2 Shea is 4 David is 0” 5’8” 5’10” 6’0” 6’2” 6’4”

Standard deviation: The average amount Deviation scores Standard deviation: The average amount by which observations deviate on either side of their mean Diallo is 0” Preston is 2” Mike is -4” Hunter is -2 Shea is 4 David is 0” 5’8” 5’10” 6’0” 6’2” 6’4”

Standard deviation: The average amount Deviation scores Standard deviation: The average amount by which observations deviate on either side of their mean Diallo is 0” Preston is 2” Mike is -4” Hunter is -2 Shea is 4 David is 0” 5’8” 5’10” 6’0” 6’2” 6’4”

Standard deviation: The average amount Deviation scores Standard deviation: The average amount by which observations deviate on either side of their mean Diallo is 0” Preston is 2” Mike is -4” Hunter is -2 Shea is 4 David is 0” 5’8” 5’10” 6’0” 6’2” 6’4”

How far away is each score from the mean? Diallo is 0” Mike is -4” Hunter is -2 Shea is 4 David is 0” Preston is 2” Deviation scores (x - µ) Deviation scores: The amount by which observations deviate on either side of their mean (x - µ) How far away is each score from the mean? Mean Diallo Deviation score Mike Preston Shea (x - µ) = ? Hunter Mike 5’8” - 6’0” = - 4” 5’9” - 6’0” = - 3” 5’10’ - 6’0” = - 2” 5’11” - 6’0” = - 1” 6’0” - 6’0 = 0 6’1” - 6’0” = + 1” 6’2” - 6’0” = + 2” 6’3” - 6’0” = + 3” 6’4” - 6’0” = + 4” Diallo How do we find each deviation score? (x - µ) Preston Hunter Diallo Mike Preston Find distance of each person from the mean (subtract their score from mean)

How far away is each score from the mean? Standard deviation: The average amount by which observations deviate on either side of their mean Deviation scores (x - µ) Diallo is 0” Preston is 2” How far away is each score from the mean? Mike is -4” Hunter is -2 Shea is 4 Mean David is 0” Add up Deviation scores Diallo Preston Σ (x - µ) = ? Shea Mike 5’8” - 6’0” = - 4” 5’9” - 6’0” = - 3” 5’10’ - 6’0” = - 2” 5’11” - 6’0” = - 1” 6’0” - 6’0 = 0 6’1” - 6’0” = + 1” 6’2” - 6’0” = + 2” 6’3” - 6’0” = + 3” 6’4” - 6’0” = + 4” How do we find the average height? N Σx = average height How do we find the average spread? Σ(x - x) = 0 Σ(x - µ) N = average deviation Σ(x - µ) = 0

How far away is each score from the mean? Standard deviation: The average amount by which observations deviate on either side of their mean Deviation scores (x - µ) Diallo is 0” Preston is 2” How far away is each score from the mean? Mike is -4” Hunter is -2 Shea is 4 Mean David is 0” Diallo Preston Σ (x - µ) = ? Shea Mike 5’8” - 6’0” = - 4” 5’9” - 6’0” = - 3” 5’10’ - 6’0” = - 2” 5’11” - 6’0” = - 1” 6’0” - 6’0 = 0 6’1” - 6’0” = + 1” 6’2” - 6’0” = + 2” 6’3” - 6’0” = + 3” 6’4” - 6’0” = + 4” Square the deviations Big problem Σ(x - x) 2 2 Σ(x - x) = 0 Σ(x - µ) N Σ(x - µ) 2 Σ(x - µ) = 0

These would be helpful to know by heart – please memorize Standard deviation: The average amount by which observations deviate on either side of their mean These would be helpful to know by heart – please memorize these formula

What do these two formula have in common? Standard deviation: The average amount by which observations deviate on either side of their mean What do these two formula have in common?

What do these two formula have in common? Standard deviation: The average amount by which observations deviate on either side of their mean What do these two formula have in common?

How do these formula differ? Standard deviation: The average amount by which observations deviate on either side of their mean “n-1” is Degrees of Freedom” How do these formula differ?

“Sum of Squares” “Sum of Squares” “Sum of Squares” “Sum of Squares” Standard deviation: The average amount by which observations deviate on either side of their mean “Sum of Squares” “Sum of Squares” “Sum of Squares” “Sum of Squares” Diallo is 0” Mike is -4” Hunter is -2 Shea is 4 David 0” Preston is 2” Deviation scores Remember, it’s relative to the mean “n-1” is “Degrees of Freedom” “n-1” is “Degrees of Freedom” Generally, (on average) how far away is each score from the mean? Based on difference from the mean Mean Remember, We are thinking in terms of “deviations” Diallo Please memorize these Preston Shea Mike

Standard deviation (definitional formula) - Let’s do one This numerator is called “sum of squares” Each of these are deviation scores _ X - µ _ 1 - 5 = - 4 2 - 5 = - 3 3 - 5 = - 2 4 - 5 = - 1 5 - 5 = 0 6 - 5 = 1 7 - 5 = 2 8 - 5 = 3 9 - 5 = 4 (X - µ)2 16 9 4 1 60 Step 1: Find the mean _ X_ 1 2 3 4 5 6 7 8 9 45 ΣX = 45 ΣX / N = 45/9 = 5 Step 2: Subtract the mean from each score Step 3: Square the deviations Step 4: Find standard deviation This is the Variance! a) 60 / 9 = 6.6667 b) square root of 6.6667 = 2.5820 Σ(x - µ) = 0 This is the standard deviation!

Thank you! See you next time!!