Reasoning in Psychology Using Statistics Psychology 138 2015

Reasoning in Psychology Using Statistics Psychology 138 2015

• Quiz 3 is posted, due Friday, Feb. 20 at 11: 59 pm – Covers • Tables and graphs • Measures of center • Measures of variability – You may want to have a calculator handy • Exam 2 is two weeks from today (Wed. Mar. 4 th) Annoucement Reasoning in Psychology Using Statistics

• Transformations: z-scores • Normal Distribution • Using Unit Normal Table – Combines 2 topics Outline for 2 classes Reasoning in Psychology Using Statistics Today

• Where is Bone student center? – Reference point – CVA Rotunda – Direction – North (and 10 o West) – Distance – Approx. 1625 ft. Location Reasoning in Psychology Using Statistics 1625 ft.

• Where is a score within distribution? – Reference point – Obvious choice is mean – Direction – Negative or positive sign on deviation score – Distance – Value of deviation score Locating a score Reasoning in Psychology Using Statistics μ Subtract mean from score (deviation score).

Reference point μ X 1 = 162 X 2 = 57 X 1 - 100 = +62 X 2 - 100 = -43 Locating a score Reasoning in Psychology Using Statistics Direction

Below μ Above X 1 = 162 X 2 = 57 X 1 - 100 = +62 X 2 - 100 = -43 Locating a score Reasoning in Psychology Using Statistics Direction

μ Distance X 1 = 162 X 2 = 57 X 1 - 100 = +62 X 2 - 100 = -43 Locating a score Reasoning in Psychology Using Statistics Distance

• Direction and Distance • Deviation score is valuable, • BUT measured in units of measurement of score • AND lacks information about average deviation • SO, convert raw score (X) to standard score (z). Raw score Population mean Population standard deviation Transforming a score Reasoning in Psychology Using Statistics

μ z-score: standardized location of X value within distribution If X 1 = 162, z = X 1 - 100 = +1. 24 50 If X 2 = 57, z = X 2 - 100 = -0. 86 50 • Direction. Sign of z-score (+ or -): whether score is above or below mean • Distance. Value of z-score: distance from mean in standard deviation units Transforming scores Reasoning in Psychology Using Statistics

μ = 20 σ=5 μ z-score: standardized location of X value within distribution If X 1 = 26, z = X 1 - 20 = +1. 2 5 If X 2 = 16, z = X 2 - 20 = -0. 8 5 • Direction. Sign of z-score (+ or -): whether score is above or below mean • Distance. Value of z-score: distance from mean in standard deviation units Transforming scores Reasoning in Psychology Using Statistics

• Can transform all of scores in distribution – Called a standardized distribution • Has known properties (e. g. , mean & stdev) • Used to make dissimilar distributions comparable – Comparing your height and weight – Combining GPA and GRE scores – z-distribution • One of most common standardized distributions • Can transform all observations to z-scores if know distribution mean & standard deviation Transforming distributions Reasoning in Psychology Using Statistics

• Shape: • Mean: • Standard Deviation: Properties of z-score distribution Reasoning in Psychology Using Statistics

• Shape: Same as original distribution of raw scores. Every score stays in same position relative to every other score. transformation 50 μ 150 original μZ z-score • Note: this is true for other shaped distributions too: – e. g. , skewed, mulitmodal, etc. Properties of z-score distribution Reasoning in Psychology Using Statistics

• Shape: Same as original distribution of raw scores. Every score stays in same position relative to every other score • Mean o o If X = μ, z = ? Meanz always = 0 transformation 50 Xmean = 100 μ 150 μZ = 0 =0 Properties of z-score distribution Reasoning in Psychology Using Statistics

• Shape: Same as original distribution of raw scores. Every score stays in same position relative to every other score • Mean: always = 0 • Standard Deviation: Properties of z-score distribution Reasoning in Psychology Using Statistics

• Shape: Same as original distribution of raw scores. Every score stays in same position relative to every other score • Mean: always = 0 • Standard Deviation: For z, transformation 50 X+1 std = 150 μ +1 = +1 z is in standard deviation units Properties of z-score distribution Reasoning in Psychology Using Statistics

• Shape: Same as original distribution of raw scores. Every score stays in same position relative to every other score • Mean: always = 0 • Standard Deviation: For z, transformation 50 X+1 std = 150 X-1 std = 50 μ 150 -1 μ +1 = -1 Properties of z-score distribution Reasoning in Psychology Using Statistics

• Shape: Same as original distribution of raw scores. Every score stays in same position relative to every other score • Mean: always = 0 • Standard Deviation: always = 1, so it defines units of zscore Properties of z-score distribution Reasoning in Psychology Using Statistics

• If know z-score and mean & standard deviation of original distribution, can find raw score (X) – have 3 values, solve for 1 unknown (z)( σ) = (X - μ) X = (z)( σ) + μ transformation 50 μ X = 70 150 μ +1 X = (-0. 60)( 50) + 100 z = -0. 60 = -30 +100 From z to raw score: Reasoning in Psychology Using Statistics -1

• Population parameters of SAT: μ= 500, σ= 100 Example 1 A student got 580 on the SAT. What is her z-score? Another student got 420. What is her z-score? SAT examples Reasoning in Psychology Using Statistics

• Population parameters of SAT: μ= 500, σ= 100 Example 2 Student said she got 1. 5 SD above mean on SAT. What is her raw score? X = z σ + μ = (1. 5)(100) + 500 = 150 + 500 = 650 • Standardized tests often convert scores to: μ = 500, σ = 100 (SAT, GRE) μ = 50, σ = 10 (Big 5 personality traits) SAT examples Reasoning in Psychology Using Statistics

• SAT: μ = 500, σ = 100 • ACT: μ = 21, σ = 3 Example 3 Suppose you got 630 on SAT & 26 on ACT. Which score should you report on your application? z-score of 1. 67 (ACT) is higher than z-score of 1. 3 (SAT), so report your ACT score. SAT examples Reasoning in Psychology Using Statistics

Example 4 On Aptitude test A, a student scores 58, which is. 5 SD below the mean. What would his predicted score be on other aptitude tests (B & C) that are highly correlated with the first one? Test B: μ = 20, σ = 5 XB < or > 20? How much: 1? 2. 5? 5? 10? Test C: μ = 100, σ = 20 XC < or > 100? How much: 20? 10? If XA = -. 5 SD, then z. A = -. 5 XB = z. B σ + μ= (-. 5)(5) + 20 = -2. 5 + 20 = 17. 5 XC = z. C σ + μ = (-. 5)(20) + 100 = -10 + 100 = 90 Find out later that this is true only if perfectly correlated; if less so, then XB and XC closer to mean. Example with other tests Reasoning in Psychology Using Statistics

Population Mean Standard Deviation Z-score Formula Summary Reasoning in Psychology Using Statistics Sample

• In lab – Using SPSS to convert raw scores into z-scores; copy formulas with absolute reference • Questions? Wrap up Reasoning in Psychology Using Statistics