Introduction to Statistics for the Social Sciences SBS
Introduction to Statistics for the Social Sciences SBS 200 - Lecture Section 001, Fall 2017 Room 150 Harvill Building 10: 00 - 10: 50 Mondays, Wednesdays & Fridays. http: //www. youtube. com/watch? v=o. SQJP 40 Pc. GI
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Schedule of readings Before next exam (September 22 nd) Please read chapters 1 - 5 in Open. Stax textbook Please read Appendix D online On syllabus this is referred to as online readings 1 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
Remember bring your writing assignment forms notebook and clickers to each lecture
t o n e v a h u o y Even if tered your s i g e r t l l i t ye s n a c u o y clicker ipate partic
Everyone will want to be enrolled in one of the lab sessions Labs continue this week
Describing Data Visually 8 8 9 10 11 11 12 12 13 13 13 14 14 14 15 15 16 16 16 17 17 17 18 18 18 19 19 20 20 21 21 22 23 24 24 25 25 25 27 28 29 This is a dot plot Re w e i v
Overview Frequency distributions 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 The normal curve
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 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 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 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
Variability Some distributions are more variable than others 5’ 5’ 6” 6’ 6’ 6” 7’ What might this be? Let’s say this is our distribution of heights of men on U of A baseball team Mean is 6 feet tall 5’ 5’ 6” 6’ 6’ 6” 7’ What might this be?
Dispersion: Variability Some distributions are more variable than others The larger the variability the wider the curve tends to be A 5’ 5’ 6” 6’ 6’ 6” 7’ B 5’ 5’ 6” C The smaller the variability the narrower the curve tends to be Range: The difference between the largest and smallest observations Range for distribution A? 5’ 5’ 6” 6’ 6’ 6” 7’ Range for distribution B? Range for distribution C?
Wildcats Basketball team: Tallest player = 85” (same as 7’ 1”) (Deandre Ayton) Shortest player = 71” (same as 5’ 11”) (Parker Jackson-Cartwritght and Tyler Trillo) Fun fact: Mean is 78 Range: The difference between the largest and smallest scores 85” – 71” = 14” xmax - xmin = Range is 14”
Baseball Wildcats Baseball team: Tallest player = 78” (same as 6’ 6”) (Tylor Megill) Shortest players = 69” (same as 5’ 9”) (Cal, Louis, Cesar, and Cory) Fun fact: Mean is 72 Range: The difference between the largest and smallest score 78” – 69” = 9” xmax - xmin = Range Please note: No reference is made to numbers between the min and max Range is 9” (78” – 69” )
Deviation scores Diallo is 0” Let’s build it up again… U of A Baseball team Diallo is 6’ 0” Diallo’s deviation score is 0 6’ 0” – 6’ 0” = 0 Diallo 5’ 8” 5’ 10” 6’ 2” 6’ 4”
Deviation scores Diallo is 0” Preston is 2” Let’s build it up again… U of A Baseball team Diallo is 6’ 0” Diallo’s deviation score is 0 Preston is 6’ 2” Preston’s deviation score is 2” 6’ 2” – 6’ 0” = 2 5’ 8” 5’ 10” 6’ 2” 6’ 4”
Deviation scores Diallo is 0” Preston is 2” Mike is -4” Hunter is -2 Let’s build it up again… U of A Baseball team Diallo is 6’ 0” Diallo’s deviation score is 0 Hunter Preston is 6’ 2” Preston’s deviation score is 2” Mike is 5’ 8” Mike’s deviation score is -4” 5’ 8” 5’ 10” 6’ 2” 6’ 4” 5’ 8” – 6’ 0” = -4 Hunter is 5’ 10” Hunter’s deviation score is -2” 5’ 10” – 6’ 0” = -2
Deviation scores Diallo is 0” Preston is 2” Mike is -4” Hunter is -2 Shea is 4 David is 0” Let’s build it up again… U of A Baseball team David Shea Diallo’s deviation score is 0 Preston’s deviation score is 2” Mike’s deviation score is -4” Hunter’s deviation score is -2” Shea is 6’ 4” 5’ 8” 5’ 10” 6’ 2” 6’ 4” Shea’s deviation score is 4” 6’ 4” – 6’ 0” = 4 David is 6’ 0” David’s deviation score is 0 6’ 0” – 6’ 0” = 0
Deviation scores Diallo is 0” Preston is 2” Mike is -4” Hunter is -2 Shea is 4 David is 0” Let’s build it up again… U of A Baseball team David Shea Diallo’s deviation score is 0 Preston’s deviation score is 2” Mike’s deviation score is -4” Hunter’s deviation score is -2” Shea’s deviation score is 4” David’s deviation score is 0” 5’ 8” 5’ 10” 6’ 2” 6’ 4”
Deviation scores Diallo is 0” Preston is 2” Mike is -4” Hunter is -2 Shea is 4 David is 0” Let’s build it up again… U of A Baseball team 5’ 8” 5’ 10” 6’ 2” 6’ 4”
Deviation scores: The amount by which each observation deviates from the mean (above or below) (x - µ) 5’ 8” 5’ 10” 6’ 2” 6’ 4” Deviation scores Diallo is 0” Preston is 2” Mike is -4” Hunter is -2 Shea is 4 David is 0”
Deviation scores: The amount by which each observation deviates from the mean (above or below) (x - µ) Standard deviation: The average amount by which observations deviate on either side of their mean Rememb er It’s rela tive to the m e n 5’ 8” a 5’ 10” Based o differe n from t nce he mea 6’ 0” n 6’ 2” 6’ 4” Deviation scores Diallo is 0” Preston is 2” Mike is -4” Hunter is -2 Shea is 4 David is 0”
Deviation scores: The amount by which observations deviate on either side of their mean (x - µ) Deviation scores (x - µ) Diallo is 0” Preston is 2” Mike is -4” Hunter is -2 Shea is 4 David is 0” How far away is each score from the mean? Mean Diallo Mike Preston Shea Hunter How do we find each deviation score? Preston Hunte r (x - µ) Diallo Mik e Preston Find distance of each person from the mean (subtract their score from mean) Deviation score (x - µ) = ? 5’ 8” 5’ 9” 5’ 10’ 5’ 11” 6’ 0” 6’ 1” 6’ 2” 6’ 3” 6’ 4” - 6’ 0” = - 4” 6’ 0” = - 3” 6’ 0” = - 2” 6’ 0” = - 1” 6’ 0 = 0 6’ 0” = + 1” 6’ 0” = + 2” 6’ 0” = + 3” 6’ 0” = + 4” Diallo
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” Mike is -4” Hunter is -2 Shea is 4 David is 0” How far away is each score from the mean? Mean Diallo Mike Preston Shea How do we find the average height? Σx = average height N How do we find the average spread? Σ(x - µ) = average deviation N Add up Deviation scores Σ (x - µ) = ? 5’ 8” 5’ 9” 5’ 10’ 5’ 11” 6’ 0” 6’ 1” 6’ 2” 6’ 3” 6’ 4” - 6’ 0” = - 4” 6’ 0” = - 3” 6’ 0” = - 2” 6’ 0” = - 1” 6’ 0 = 0 6’ 0” = + 1” 6’ 0” = + 2” 6’ 0” = + 3” 6’ 0” = + 4” Σ(x - x) = 0 Σ(x - µ) = 0
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” Mike is -4” Hunter is -2 Shea is 4 David is 0” How far away is each score from the mean? Mean Diallo Mike Preston Shea Σ (x - µ) = ? Square the deviations 2 Σ(x - µ) N 2 Σ(x - x) 2 Σ(x - µ) 5’ 8” 5’ 9” 5’ 10’ 5’ 11” 6’ 0” 6’ 1” 6’ 2” 6’ 3” 6’ 4” - 6’ 0” = - 4” 6’ 0” = - 3” 6’ 0” = - 2” 6’ 0” = - 1” 6’ 0 = 0 6’ 0” = + 1” 6’ 0” = + 2” 6’ 0” = + 3” 6’ 0” = + 4” Σ(x - x) = 0 Σ(x - µ) = 0 Big proble m
Standard deviation: The average amount by which observations deviate on either side of their mean How far away is each score from the mean? Mean Deviation scores (x - µ) Diallo is 0” Preston is 2” Mike is -4” Hunter is -2 Shea is 4 David is 0” Step 1 Find the mean Diallo Step 2 Mike Find each deviation scoreΣ (x - µ) = ? Preston Shea Step 4 Divide by n and take square root 2 Σ(x - µ) N 5’ 8” - 6’ 0” = - 4” 5’ 9” - 6’ 0” = - 3” 5’ 10’ - 6’ 0” = - 2” 5’ 11” Step - 6’ 0” = 3 - 1” Square 6’ 0”each - 6’ 0 deviation = 0 6’ 1” add - 6’ 0”them = + 1” up And 6’ 2” - 6’ 0” = + 2” 6’ 3” - 6’ 0” = + 3” 6’ 4” - 6’ 0” = + 4” 2 Σ(x - x) 2 Σ(x - µ) Σ(x - x) = 0 Σ(x - µ) = 0 score
Writing Assignment: Let’s try two problems Step 1 Find the mean Step 2 Find each deviation score Step 3 Square each deviation score and add them up Step 4 Divide by n and take square root
Standard deviation (definitional formula) This numerator is called “sum of squares” - Let’s get you started Step 1: Find the mean Step 2: Subtract the mean from each score Step 3: Square the deviations Step 4: Find standard deviation You finish it! Each of these are deviation scores _ X_ _ X - µ _ 1 2 3 4 5 6 7 8 9 45 1 -5=-4 (X - µ)2 16
Standard deviation (definitional formula) - Let’s do one This numerator is called “sum of squares” Step 1: Find the mean 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 Each of these are deviation scores 2 _ X_ _ X - µ _ 1 2 3 4 5 6 7 8 9 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 45 0 b) square root of 6. 6667 = 2. 5820 This is the standard deviation! (X - µ) 16 9 4 1 0 1 4 9 16 60 Σ(x - µ) = 0
Standard deviation - Let’s get you started How many kids? Step 1: Find the mean Step 2: Subtract the mean from each score (deviations) Step 3: Square the deviations Step 4: Add up the squared deviations Step 5: Find standard deviation You finish it! _ X_ 3 2 3 1 2 4 8 2 1 4 Σx = 30 X- µ_ 3 -3= 0 (X - µ )2 0 Σ(x - µ) = 0 Σ(x - µ)2 = 38
Standard deviation - Let’s do one Definitional formula How many kids? Step 1: Find the mean _ X_ = 30 3 2 3 1 2 4 8 2 1 4 = 30/10 = 3 Step 2: Subtract the mean from each score (deviations) Step 3: Square the deviations Step 4: Add up the squared deviations Step 5: Find standard deviation a) 38 / 10 = 3. 8 Σx = 30 This is the Variance! b) square root of 3. 8 = 1. 95 X- µ_ 3 -3= 0 2 - 3 = -1 3 -3= 0 1 - 3 = -2 2 - 3 = -1 4 -3= 1 8 -3= 5 2 - 3 = -1 1 - 3 = -2 4 -3= 1 (X - µ )2 0 1 0 4 1 1 25 1 4 1 Σ(x - µ) = 0 Σ(x - µ)2 = 38 This is the standard deviation!
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