Points in Distributions

Up to now describing distributions Comparing scores from different

distributions Need to make equivalent comparisons z scores

standard scores Percentile, Percentile rank ~

Standard Scores

Convert raw scores to z scores raw score: value using original scale of

measurement z scores: # of standard deviations score is from

mean e.g., z = 2

= 2 std. deviations from mean z = 0 = mean ~

z Score Equation

z = X -

Areas Under Distributions

Area = frequency Relative area

total area = 1.0

= proportion of individual values in area under curve

Relative area is independent of shape of distribution ~

Total area under curve = 1.0

0.50.5

10 20 30 40 50 60 70 80 90

Using Areas Under Distributions

Given relative frequency, what is value? e.g., the hottest 10% of days the

temperature is above ____? find value of X at border ~

Areas Under Normal Curves

Many variables normal distribution Normal distribution completely

specified by 2 numbers mean & standard deviation

Many other normal distributions have different & ~

Areas Under Normal Curves Unit Normal Distribution

based on z scores

= 0

= 1 e.g., z = -2

relative areas under normal distribution always the same precise areas from Table B.1 ~

Areas Under Normal Curves

+1 +20-1-2

.34

.14

f

standard deviations

.02

.34

.14

.02

Calculating Areas from Tables

Table B.1 (in our text) The Unit Normal Table Proportions of areas under the normal

curve 3 columns

(A) z (B) Proportion in the body (C) Proportion in the tail

Negative z: area same as positive ~

Calculating Areas from Tables Finding proportions

z < 1 = (from B) z > 1: (from C) ~

+1 +20-1-2

f

z

Calculating Areas from Tables Area: 1 < z < 2

find proportion for z = 2; subtract proportion for z = 1 ~

+1 +20-1-2

f

z

Other Standardized Distributions

Normal distributions, but not unit normal distribution

Standardized variables normally distributed specify and inadvance

e.g., IQ test = 100; = 15 ~

Other Standardized Distributions

115 1301008570

f

IQ Scores

= 100 = 15

z scores +1 +20-1-2

Transforming to & from z scores

From z score to standardized score in population

z = X -

Standardized score ---> z score

X = z +

Normal Distributions: Percentiles/Percentile Rank

Unit normal distributions 50th percentile = 0 = z = 1 is 84th percentile

50% + 34% Relationships

z score & standard score linear z score & percentile rank nonlinear ~

Percentiles & Percentile Rank Percentile

score below which a specified percentage of scores in the distribution fall

start with percentage ---> score Percentile rank

Per cent of scores a given score start with score ---> percentage

Score: a value of any variable ~

Percentiles E.g., test scores

30th percentile = (A) 46; (B) 22

90th percentile = (A) 56; (B) 46 ~

A58565454525048464442

B50463230302323222120

Percentile Rank

e.g., Percentile rank for score of 46 (A) 30%; (B) = 90%

Problem: equal differences in % DO NOT reflect equal distance between values ~

A58565454525048464442

B50463230302323222120

115 1301008570

f

IQ Scores

.34

.14

.02

.34

.14

.02

2d 16th 50th 84th 98thpercentile rank

IQ

z scores +1 +20-1-2

Supplementary Material

Determining Probabilities

Must count ALL possible outcomes e.g. of flipping 2 coins

coin A:

coin B: head

head

head

tail

tail

headtail

tail

outcomes

21 3 4

Determining Probabilities

Single fair dieP(1) = P(2) = … = P(6)

Addition rule keyword: OR P(1 or 3) =

Multiplication rule keyword AND P(1 on first roll and 3 on second roll) = dependent events ~

Conditional Probabilities

Put restrictions on range of possible outcomes P(heart) given that card is Red P(Heart | red card) =

P(5 on 2d roll | 5 on 1st roll)? P = 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

Percentile rank percentile Or probability raw score

What is the 43d percentile of IQ scores? 1. Find area in z table 2. Get z score 3. X = z +