points in distributions n up to now describing distributions n comparing scores from different...
TRANSCRIPT
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 +