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Reasoning in Psychology Using Statistics Psychology 138 2015 • Exam 2 in lecture and lab on Wednesday • Be prepared to do calculations (including square roots) on calculator Announcements Reasoning in Psychology Using Statistics • Mathematical cautions – – – • Different scales: convert to z-scores Restriction of range (e.g., age & height) Outliers (especially in small samples) Interpretive caution – Causal claims Cautions with Correlations Reasoning in Psychology Using Statistics • Change all scores to z-scores Both variables on same scale Correlation stays the same What happens to means? Y 28 28 r= = = 0.898 60.8 *16 31.2 Convert X and Y to z-scores zy 1.5 1.0 0.5 0 -0.5 -1.0 -1.5 6 5 4 3 2 1 1 2 3.6 3 4 5 6 X -1.5 -1 -.5 0 .5 1 1.5 zx Pearson’s r, z transformation Reasoning in Psychology Using Statistics • Total data for positive correlation between SAT and GPA. • What correlation between SAT and GPA in only those with admitted and studied (400 < SAT < 700)? • Get r = 0 Restriction of range Reasoning in Psychology Using Statistics • One extreme score can change correlation (especially in small sample). • On left, 5 observations, high X associated with high Y: good predictability. Outliers Reasoning in Psychology Using Statistics • On right, same 5 observations plus 1 other, high X associated with high or low Y: poor predictability. • We’d like to say: – • X causes Y To be able to do this: 1. The causal variable must come first 2. There must be co-variation between the two variables 3. Need to eliminate plausible alternative explanations • Correlation procedures address point 2, but say nothing about points 1 and 3. • Careful: Do not make casual claims based on correlations Causal claims Reasoning in Psychology Using Statistics • Directionality Problem – Happy people sleep well • Or is it that sleeping well makes you happy? Causal claims Reasoning in Psychology Using Statistics • Third Variable Problem: – Happy people sleep well – Or does sleeping well make you happy? – OR something else makes people happy and sleep well! • Regular exercise • Minimal use of drugs & alcohol • Being a conscientious person • Being a good relationship • Etc. Causal claims Reasoning in Psychology Using Statistics Statistical procedures to help organize, summarize & simplify large sets of data 1. One variable (frequency distribution) • Display results in a frequency distribution table & histogram (or bar chart if categorical variable). Make a deviations table to get measures of central tendency (mode, median, mean) & variability (range, standard deviation, variance). • 2. Two variables (bivariate distribution) • • 3. Display results: Make a scatterplot. Make a bivariate deviations or z-table table to get Pearson’s r. Z-scores & normal distribution Review for Exam 2: Descriptive statistics Reasoning in Psychology Using Statistics • Are hours sleeping related to GPA? – You conduct a survey. • Your sample of 10 gives these results for average hours per night sleeping: 7, 6, 7, 8, 8, 7, 9, 5, 9, 6 • You also have respondents give their overall GPA: 2.4, 3.9, 3.5, 2.8, 3.0, 2.1, 3.9, 2.9, 3.6, 2.7 – We will focus on sleep results first and then both variables together. • What kind of scales are they? • To find standard deviation, will we use formula for population or sample? Example Reasoning in Psychology Using Statistics Hrs. sleep n=10 7,6,7,8,8 7,9,5,9,6 X 9 8 f p % cf c% 7 6 5 ∑ 10 1.0 100 Step 1: Frequency distribution & histogram Reasoning in Psychology Using Statistics Hrs. sleep n=10 7,6,7,8,8 7,9,5,9,6 ∑ X 9 8 f 2 2 7 6 5 3 2 1 10 p % cf c% 1.0 100 Will enter first two columns as X and Y axes for frequency distribution Step 1: Frequency distribution & histogram Reasoning in Psychology Using Statistics Hrs. sleep n=10 p = f/n ∑ X 9 8 f 2 2 p .2 .2 % 20 20 7 6 5 3 2 1 .3 .2 .1 30 20 10 10 cf c% 1.0 100 Step 1: Frequency distribution & histogram Reasoning in Psychology Using Statistics ∑ X 9 8 f 2 2 p .2 .2 % 20 20 7 6 5 3 2 1 .3 .2 .1 30 20 10 10 cf c% 1 10 1.0 100 Step 1: Frequency distribution & histogram Reasoning in Psychology Using Statistics ∑ X 9 8 f 2 2 p .2 .2 % 20 20 7 6 5 3 2 1 .3 .2 .1 30 20 10 10 cf c% 3 1 30 10 1.0 100 Step 1: Frequency distribution & histogram Reasoning in Psychology Using Statistics ∑ X 9 8 f 2 2 p .2 .2 % 20 20 cf c% 7 6 5 3 2 1 .3 .2 .1 30 20 10 6 3 1 60 30 10 10 1.0 100 Step 1: Frequency distribution & histogram Reasoning in Psychology Using Statistics ∑ X 9 8 f 2 2 p .2 .2 % 20 20 7 6 5 3 2 1 .3 .2 .1 30 20 10 10 cf c% 8 6 3 1 80 60 30 10 1.0 100 Step 1: Frequency distribution & histogram Reasoning in Psychology Using Statistics ∑ X 9 8 f 2 2 p .2 .2 % 20 20 cf 10 8 c% 100 80 7 6 5 3 2 1 .3 .2 .1 30 20 10 6 3 1 60 30 10 10 1.0 100 Step 1: Frequency distribution & histogram Reasoning in Psychology Using Statistics Hrs. sleep X 9 8 f 2 2 7 6 5 3 2 1 F R E Q U E N C Y 6 5 4 3 2 1 5 6 7 8 9 SCORE Step 1: Frequency distribution & histogram Reasoning in Psychology Using Statistics • Suppose that you combine two groups together. – How do you compute the new group mean? Group 1 X1 =110 110 110 110 110 110 110 110 Group 2 X 2 =140 140 140 140 New Group X1n1 + X 2 n 2 XN = n1 + n 2 (110 * 7) + (140 * 3) = = 119 7+ 3 A weighted mean Reasoning in Psychology Using Statistics • Suppose that you combine two groups together. – How do you compute the new group mean? Group 1 X1 =110 110 110 110 110 110 110 110 Group 2 X 2 =140 140 140 140 New Group X1n1 + X 2 n 2 XN = n1 + n 2 (110 * 7) + (140 * 3) = = 119 7+ 3 A weighted mean Reasoning in Psychology Using Statistics Be careful computing the mean of this distribution, remember there are groups here X 9 f 2 8 7 6 2 3 2 5 1 9 9 8 8 7 7 7 6 6 5 • The mean • The standard deviation – Change/add/delete a given score, – Change/add/delete a given score, then the mean will change. then the mean will change. – Add/subtract a constant to each score, then the mean will change by adding(subtracting) that constant. – Add/subtract a constant to each score, then the standard deviation will NOT change. – Multiply (or divide) each score by a constant, then the mean will change by being multiplied by that constant. – Multiply (or divide) each score by a constant, then the standard deviation will change by being multiplied by that constant. Characteristics of a mean & standard deviation Reasoning in Psychology Using Statistics Hrs. sleep n = 10 X 9 9 (X - X) (X - X)2 Create table, sorted in descending order 8 8 7 7 7 6 6 5 Step 2: Deviations table Reasoning in Psychology Using Statistics X Hrs. sleep n = 10 (X - X) (X - X)2 9 9 8 8 7 7 7 6 6 5 ∑ 72 Step 2: Deviations table Reasoning in Psychology Using Statistics Mode = 7 (filled in) Median = 7 (arrow) Mean = (∑X)/n = 72/10 = 7.2 Range = 5 to 9 X Hrs. sleep n = 10 ∑ (X - X) (X - X)2 9 1.8 9 1.8 8 .8 8 .8 7 -.2 7 -.2 7 -.2 6 -1.2 6 -1.2 5 -2.2 72 X 7.2 = 9-7.2 0 Step 2: Deviations table Reasoning in Psychology Using Statistics Mode = 7 Median = 7 Mean = (∑X)/n = 72/10 = 7.2 Range = 5 to 9 X Hrs. sleep n = 10 ∑ (X - X) (X - X)2 9 1.8 3.24 = 1.82 9 1.8 3.24 8 .8 .64 8 .8 .64 7 -.2 .04 7 -.2 .04 7 -.2 .04 6 -1.2 1.44 6 -1.2 1.44 5 -2.2 4.84 72 X 7.2 0 15.6 = SS Step 2: Deviations table Reasoning in Psychology Using Statistics Mode = 7 Median = 7 Mean = ∑X/n = 72/10 = 7.2 Range = 5 to 9 SD for sample s= å( X - X ) 2 n -1 = √15.6/9 = √1.73 = 1.32 Person A B C D E F G H I J Hrs. 7 6 7 8 8 7 9 5 9 6 GPA 2.4 3.9 3.5 2.8 3.0 2.1 3.9 2.9 3.6 2.7 G P A 4.0 3.5 3.0 2.5 2.0 1.5 1.0 5 6 7 8 Hours of sleep Step 3: Scatterplot Reasoning in Psychology Using Statistics 9 Person A B C D E F G H I J Hrs. 7 6 7 8 8 7 9 5 9 6 GPA 2.4 3.9 3.5 2.8 3.0 2.1 3.9 2.9 3.6 2.7 What does shape of envelope indicate about correlation? low positive correlation G P A 4.0 B 3.5 3.0 G C H 2.5 I DE J 2.0 A F 1.5 1.0 5 6 7 8 Hours of sleep Step 3: Scatterplot Reasoning in Psychology Using Statistics 9 Person Hrs. GPA A 7 2.4 B 6 3.9 C 7 3.5 D 8 2.8 E 8 3.0 F 7 2.1 G 9 3.9 H 5 2.9 I 9 3.6 J 6 2.7 K 5 1.0 What does shape of envelope indicate about correlation? moderate positive correlation G P A 4.0 B 3.5 3.0 G C H 2.5 I DE J 2.0 A F 1.5 1.0 K 5 6 7 8 Hours of sleep Step 3: Scatterplot, Effect of outlier Reasoning in Psychology Using Statistics 9 Person Hrs. GPA A 7 2.4 B 6 3.9 C 7 3.5 D 8 2.8 E 8 3.0 F 7 2.1 G 9 3.9 H 5 2.9 I 9 3.6 J 6 2.7 K 9 1.0 What does shape of envelope indicate about correlation? low negative correlation G P A 4.0 B 3.5 3.0 G C H 2.5 I DE J 2.0 A F 1.5 1.0 K 5 6 7 8 Hours of sleep Step 3: Scatterplot, Effect of outlier Reasoning in Psychology Using Statistics 9 (X - X) X (X - X)2 Y (Y - Y ) (Y - Y )2 (X - X)(Y - Y ) 9 1.8 3.24 3.9 0.82 0.67 1.476 9 1.8 3.24 3.6 0.52 0.27 0.936 8 0.8 0.64 3.0 -0.08 0.01 -0.064 8 0.8 0.64 2.8 -0.28 0.86 -0.224 7 -0.2 0.40 3.5 0.42 0.18 -0.084 7 -0.2 0.04 2.4 -0.68 0.46 0.136 7 -0.2 0.04 2.1 -0.98 0.96 0.196 6 -1.2 1.44 3.9 0.82 0.67 -0.984 6 -1.2 1.44 2.7 -0.38 0.14 0.456 5 -2.2 4.84 2.9 -0.18 0.03 0.396 0.0 15.6 30.8 0.0 3.47 2.24 SSX 3.08 SSY SP Sum 72 Mean 7.2 Step 4: Bivariate Deviations Table Reasoning in Psychology Using Statistics n=10 Note signs! +r or – r? SP r= SSX SSY X= sX = åX n å( X - X ) n -1 2 = XY co-deviations ___2.24___ = _2.24_ = _2.24_ = .304 = √ 15.6 * 3.47 √54.132 7.357 X deviations, Y deviations 72 = 7.2 10 15.6 = = 1.73 = 1.32 9 Y= åY n = 308 = 3.08 10 SSY 3.08 sY = = = .58 n -1 9 Pearson’s r & summary statistics Reasoning in Psychology Using Statistics SRA (Scientific Reasoning Assessment) (fictional) • Based on normative data: Normal, μ = 50.0, σ = 10.0 • Preparing for your analyses – Write down what you know – Make a sketch of the distribution (make a note: population or sample) 40 μ 60 An example Reasoning in Psychology Using Statistics – – – – Determine the shape What is best measure of center? What is best measure of variability? Mark the mean (center) and standard deviation on your sketch SRA (Scientific Reasoning Assessment) (fictional) • Based on normative data: Normal distr., = 50.0, σ = 10.0 • Question 1 • If George got a 35 on the SRA, what is his percentile rank? Unit Normal Table z= 0.0668 40 -1.0 m 60 1.0 X -m s = 35 - 50 -15 = = -1.5 10 10 • Since a normal distribution, can use Unit Normal Table to infer percentile. That’s 6.68% at or below this score (definition of percentile) z-scores & Normal Distribution Reasoning in Psychology Using Statistics SRA (Scientific Reasoning Assessment) (fictional) • Based on normative data: Normal distr., μ = 50.0, σ = 10.0 • Question 2 Unit Normal Table • What proportion of people get between a 40 and 60 on the SRA? 0.1587 0.1587 40 -1.0 m 60 1.0 That’s about 32% outside these two scores That leaves 68% between these two scores X - m = 40 - 50 = -10 = -1.0 z= 10 10 s = 60 - 50 10 = = 1.0 10 10 • Since a normal distribution, can use Unit Normal Table to infer percentile. z-scores & Normal Distribution Reasoning in Psychology Using Statistics SRA (Scientific Reasoning Assessment) (fictional) • Based on normative data: Normal distr., μ = 50.0, σ = 10.0 z= X -m s X = Zs + m transformation • Question 3a • Suppose that Chandra took a different reasoning assessment (the RSE: Based on normative data, Normal distr., μ= 100, σ = 15). She received a 130 on the RSE. Assuming that they are highly positively correlated, what is the equivalent score on the SRA? z-scores & Normal Distribution Reasoning in Psychology Using Statistics SRA (Scientific Reasoning Assessment) (fictional) • Based on normative data: Normal distr., μ = 50.0, σ = 10.0 X - m (for RSE) z= s 130 - 100 30 = = = 2.0 15 15 X = Zs + m (for SRA) transformation X = (2.0)10 + 50 X = 70 • Question 3a • Suppose that Chandra took a different reasoning assessment ( RSE: Based on normative data, Normal distr., μ= 100, σ = 15). She received a 130 on the RSE. Assuming that they are highly positively correlated, what is the equivalent score on the SRA? • Now know that predict equivalent only if rRSE,SRA = 1.0. z-scores & Normal Distribution Reasoning in Psychology Using Statistics SRA (Scientific Reasoning Assessment) (fictional) • Based on normative data: Normal distr., μ = 50.0, σ = 10.0 X - m (for RSE) z= s 130 - 100 30 = = = 2.0 15 15 X = Zs + m (for SRA) transformation X = (2.0)10 + 50 X = 70 • Question 3c • Suppose that Chandra took a different reasoning assessment (the RSE: Based on normative data, Normal distr., μ= 100, σ = 15). She received a 130 on the RSE. Assuming that they are perfectly positively correlated, what is the equivalent score on the SRA? • What percent of those taking either test will score below Chandra? Know z = 2 From Unit Normal Table, p(z ≥ 2) = .0228 p(z < 2) = 1 - .0228 = .9772 = 98% z-scores & Normal Distribution Reasoning in Psychology Using Statistics SRA (Scientific Reasoning Assessment) (fictional) • Based on normative data: Normal distr., μ = 50.0, σ = 10.0 X - m (for RSE) = 2.0 z= s zy = r zx so zSRA = .8 * 2 = 1.6 X = Zs + m (for SRA) transformation X = (1.6)10 + 50 X = 66 Reasoning in Psychology Using Statistics • Question 3b • Suppose that Chandra took a different reasoning assessment ( RSE: Based on normative data, Normal distr., μ= 100, σ = 15). She received a 130 on the RSE. Assuming that they are highly positively correlated, what is the equivalent score on the SRA? • If rRSE,SRA = .8, what is our best estimate of her actual score? • More on this later in course z-scores, Normal Distribution, & Correlation • In lab: continue to review, including SPSS • Questions? Wrap up Reasoning in Psychology Using Statistics