Reasoning in Psychology Using Statistics 2017

Annoucement Quiz 3 is posted, due Friday, Feb. 24 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. 8th) Annoucement

Outline for next 2 classes Transformations: z-scores Normal Distribution Using Unit Normal Table Combines 2 topics Today Statistical Snowball: For the rest of the course, new concepts build upon old concepts So if you feel like you don’t understand something now, ask now, don’t wait. X X X X Outline for next 2 classes

Characteristics of a standard deviation Change/add/delete a given score, then the standard deviation will change. May change the mean and (if adding or subtracting) the number of scores (n or N) Characteristics of a standard deviation

Characteristics of a standard deviation Change/add/delete a given score, then the standard deviation will change. Add/subtract a constant to each score, then the standard deviation will NOT change. Looking at a numerical example. (subtract 1 from every score) Original sample 2, 4, 6, 8 New sample 1, 3, 5, 7 Original mean 5 New mean 4 2 - 5 = -3 4 - 5 = -1 6 - 5 = +1 8 - 5 = +3 1 - 4 = -3 3 - 4 = -1 5 - 4 = +1 7 - 4 = +3 Original SS 20   Characteristics of a standard deviation

Characteristics of a standard deviation Change/add/delete a given score, then the mean will change. Add/subtract a constant to each score, then the standard deviation will NOT change. Multiply (or divide) each score by a constant, then the standard deviation will change by being multiplied by that constant. 20 21 22 23 24 X 21 - 22 = -1 (-1)2 23 - 22 = +1 (+1)2 s = Characteristics of a standard deviation

Characteristics of a standard deviation Change/add/delete a given score, then the mean will change. Add/subtract a constant to each score, then the standard deviation will NOT change. Multiply (or divide) each score by a constant, then the standard deviation will change by being multiplied by that constant. 42 - 44 = -2 (-2)2 40 42 44 46 48 X 46 - 44 = +2 (+2)2 s = Sold=1.41 Characteristics of a standard deviation

Extreme scores: Range is most affected, IQR is least affected Sample size: Range tends to increase as n increases, IQR & s do not With open-ended distributions, one cannot even compute the Range or σ, so the IQR is the only option Range is unstable when you repeatedly sample from the same population, but the IQR & σ are stable and tend not to fluctuate. When to use which

Location Where is Bone student center? Reference point Direction CVA Rotunda Direction North (and 10o West) Distance Approx. 1625 ft. 1625 ft. Location

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

Locating a score μ Reference point X1 = 162 X1 - 100 = +62 Direction

Locating a score μ Below Above X1 = 162 X1 - 100 = +62 Direction

Locating a score μ Distance Distance X1 = 162 X1 - 100 = +62 X2 = 57

Transforming a score 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

Transforming scores μ X1 - 100 = +1.24 50 If X1 = 162, z = z-score: standardized location of X value within distribution X1 - 100 = +1.24 50 If X1 = 162, z = 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 X2 - 100 = -0.86 50 If X2 = 57, z = Transforming scores

Transforming scores μ μ = 20 σ = 5 X1 - 20 = +1.2 5 If X1 = 26, z = z-score: standardized location of X value within distribution X1 - 20 = +1.2 5 If X1 = 26, z = 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 X2 - 20 = -0.8 5 If X2 = 16, z = Transforming scores

Transforming distributions 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

Properties of z-score distribution Shape: Mean: Standard Deviation: Properties of z-score distribution

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

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

Properties of z-score distribution 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

Properties of z-score distribution 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, μ μ transformation +1 X+1std = 150 50 150 = +1 z is in standard deviation units Properties of z-score distribution

Properties of z-score distribution 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, μ μ transformation -1 X-1std = 50 50 150 +1 X+1std = 150 = +1 = -1 Properties of z-score distribution

Properties of z-score distribution 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 z-score Properties of z-score distribution

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 μ +1 -1 μ 150 50 transformation  z = -0.60 X = 70  X = (-0.60)( 50) + 100 = -30 +100 From z to raw score:

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

SAT examples 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

SAT examples 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? Example 3 z-score of 1.67 (ACT) is higher than z-score of 1.3 (SAT), so report your ACT score. SAT examples

Example with other tests 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 zA = -.5 XB = zB σ + μ XC = zC 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. = (-.5)(5) + 20 = -2.5 + 20 = 17.5 Example with other tests

Population Sample Mean Standard Deviation Z-score X Formula Summary

Wrap up In lab Questions? Using SPSS to convert raw scores into z-scores; copy formulae with absolute reference Questions? Applications: Statistics and the Weather Brian’s Climate Blog NOAA – Jan 2014 Temperature Anomalies Brandon Foltz: Understanding z-scores (~22 mins) StatisticsFun (~4 mins) Chris Thomas. How to use a z-table (~7 mins) Dr. Grande. Z-scores in SPSS (~7 mins) Wrap up