Points in Distributions n Up to now describing distributions n Comparing scores from different distributions l Need to make equivalent comparisons l z scores standard scores l Percentile, Percentile rank ~

Standard Scores n Convert raw scores to z scores n raw score: value using original scale of measurement n z scores: # of standard deviations score is from mean l e.g., z = 2 = 2 std. deviations from mean l z = 0 = mean ~

z Score Equation z = X -  

Areas Under Distributions n Area = frequency n Relative area l total area = 1.0 = proportion of individual values in area under curve l Relative area is independent of shape of distribution ~

Total area under curve =

Using Areas Under Distributions n Given relative frequency, what is value? l e.g., the hottest 10% of days the temperature is above ____? l find value of X at border ~

Areas Under Normal Curves Many variables  normal distribution l Normal distribution completely specified by 2 numbers l mean & standard deviation n Many other normal distributions have different  &  ~

Areas Under Normal Curves n Unit Normal Distribution l based on z scores  = 0  = 1 l e.g., z = -2 n relative areas under normal distribution always the same l precise areas from Table B.1 ~

Areas Under Normal Curves f standard deviations

Calculating Areas from Tables n Table B.1 (in our text) l The Unit Normal Table l Proportions of areas under the normal curve n 3 columns l (A) z l (B) Proportion in the body l (C) Proportion in the tail n Negative z: area same as positive ~

Calculating Areas from Tables n Finding proportions l z < 1 = (from B) l z > 1: (from C) ~ f z

Calculating Areas from Tables n Area: 1 < z < 2 l find proportion for z = 2; l subtract proportion for z = 1 ~ f z

Other Standardized Distributions n Normal distributions, l but not unit normal distribution n Standardized variables l normally distributed specify  and  in  advance n e.g., IQ test  = 100;  = 15 ~

Other Standardized Distributions f IQ Scores  = 100  = 15 z scores

Transforming to & from z scores n From z score to standardized score in population z = X -   n Standardized score ---> z score X = z  + 

Normal Distributions: Percentiles/Percentile Rank n Unit normal distributions 50th percentile = 0 =  l z = 1 is 84th percentile 50% + 34% n Relationships l z score & standard score linear l z score & percentile rank nonlinear ~

Percentiles & Percentile Rank n Percentile l score below which a specified percentage of scores in the distribution fall l start with percentage ---> score n Percentile rank Per cent of scores  a given score l start with score ---> percentage n Score: a value of any variable ~

Percentiles n E.g., test scores l 30 th percentile = (A) 46; (B) 22 l 90 th percentile = (A) 56; (B) 46 ~ A B

Percentile Rank n e.g., Percentile rank for score of 46 l (A) 30%; (B) = 90% n Problem: equal differences in % DO NOT reflect equal distance between values ~ A B

f IQ Scores d 16 th 50 th 84 th 98 th percentile rank IQ z scores

Supplementary Material

Determining Probabilities n Must count ALL possible outcomes n e.g. of flipping 2 coins coin A: coin B: head tail headtail outcomes 2134

Determining Probabilities n Single fair die P(1) = P(2) = … = P(6) n Addition rule l keyword: OR l P(1 or 3) = n Multiplication rule l keyword AND l P(1 on first roll and 3 on second roll) = l dependent events ~

Conditional Probabilities n Put restrictions on range of possible outcomes l P(heart) given that card is Red l P(Heart | red card) = n P(5 on 2d roll | 5 on 1st roll)? l P = l 1st & 2d roll independent events ~

Know/want Diagram Raw Score (X) z score area under distribution z = X -   X = z  +  Table: column B or C Table: z - column A

Percentage  raw score n Percentile rank  percentile l Or probability  raw score n What is the 43d percentile of IQ scores? l 1. Find area in z table l 2. Get z score 3. X = z  + 