Transcript Use z scores.

Describing a
Score’s Position
within a Distribution
Lesson 5
Science & Probability
Learn about populations by studying
samples
 Introduction of error
 Drawing conclusions
 Cannot make states with certainty
 Probability statements
 Use of normal distribution
 Can calculate probability of a result
 Natural variables ≈ normal ~

Probability: Definitions

Probability(P) of an event (A)
 Assuming each outcome equally likely
P(A) = # outcomes Classified as A
total # possible outcomes
P(drawing ♥) =
P(7 of ♥) =
P(15 of ♥) =
P(♥ or ♦ or ♣ or ♠) =
~
Standard Normal Distribution
AKA Unit Normal Distribution
 Parameters
 m = 0, s = 1
 z scores
 Or standard scores
 Distance & direction from m in units
of s ~

Standard Normal Distribution
f
-2
-1
0
Z scores (s)
1
2
Other Standardized Distributions
Many natural variables ≈ normal
 Standardized distributions
 Have defined or set parameters
 IQ: m = 100, s = 15
 ACT: m = 18, s = 6
 SAT: m = 500, s = 100 ~

IQ Scores
m = 100
s = 15
f
z scores
-2
-1
0
1
2
IQ
70
85
100
115
130
The Normal Distribution & Probability
Area under curve = frequency
 Area under curve represents all data
 Proportion (p) including all scores = 1
 p for any area under curve can be
calculated
 Proportion = probability that a
score(s) is in distribution
 Table A.1, pg 797 ~

Probability of obtaining IQ score below the median?
Greater than 115?
Percentile rank of 70?
Use z scores.
f
0.5
0.5
IQ
70
85
100
115
130
Total area under curve = 1.0
Using z scores
AKA standard scores
 distance from mean in units of s
 Uses
 Determining probabilities
 Percentile rank or scores
 Compare scores from different
distributions
 *Technically must use parameters
 text uses sample statistics: X and s ~

z Score Equation
z =
X-m
s
Using z scores
Distance and direction relative to mean
 Standard Normal Distribution
 m = 0, s = 1
 Answer questions by 1st finding z score
 What proportion of population have IQ
scores greater than 115?
 What is the percentile rank for IQ
score of 70?
 What percentage of people have IQ
scores between 70 and 115?

z score for 115?
z
Xi  m
s
z score for 70?
f
IQ
70
85
100
IQ Score
115
130
Handy Numbers

Standard Normal Distribution
 z scores
 Proportions of distribution


i.e., area under curve, table A.1
3 handy proportions
 Same for all normal distributions
 Between z = 0 and ±1
 Between z = 1 and 2 (also -1 & -2)
 Beyond z = ±2 (area in tails) ~
Areas Under Normal Curves
f
.34
.34
.02
.02
.14
-2
.14
-1
0
Z scores (s)
1
2
What % of students scored b/n 18 and 24?
% greater than 30?
% less than 30?
ACT Scores
m = 18
s=6
f
z scores
ACT
-2
6
-1
12
0
1
2
18
24
30
Comparing Scores from
Different Distributions
How to compare ACT to SAT?
 Use z scores
1. Raw ACT score  z score
X m
z
s

2.
Use z score to compute Raw SAT score
X  zs  m
Areas Under Normal Curves
f
.34
.34
.02
.02
.14
-2
.14
-1
0
1
standard deviations
2
Percentile Rank & Percentile
Percentile rank
 % of scores ≤ a particular score (Xi)
th percentile: 84% of IQ scores ≤ 115
 84
 Percentile
 Raw score (Xi) associated with a
particular percentile rank
th percentile
 IQ score of 100 is the 50
 Use z scores & table to determine ~

IQ Scores
f
.34
.34
.02
.02
.14
.14
IQ
70
85
100
115
130
z scores
-2
-1
0
1
2
84th
98th
percentile
rank
2d
16th
50th