Transcript I. What are z scores?

Outline
I. What are z-scores?
II.Locating scores in a distribution
A. Computing a z-score from a raw
score
B. Computing a raw score from a zscore
C. Using z-scores to standardize
distributions
III. Comparing scores from
different distributions
I. What are z scores?
You scored 76
How well did you perform?
 serves as reference point:
Are you above or below average?
 serves as yardstick:
How much are you above or below?
•
•
•
•
•
Convert raw score to a z-score
z-score describes a score relative to  & 
Two useful purposes:
Tell exact location of score in a distribution
Compare scores across different
distributions
II. Locating Scores in a
distribution
z
X

Deviation from  in SD units
Relative status, location, of a raw
score (X)
z-score has 2 parts:
1. Sign tells you above (+) or
below (-) 
2. Value tells magnitude of
distance in SD units
A. Converting a raw score
(X) to a z-score:
X
z

Example:
Spelling bee:  = 8
Garth X=6
Peggy X=11


=2
z=
z=
B. Converting a z-score to
a raw score:
X    z
Example:
Spelling bee:  = 8
Hellen z = .5 
Andy z = 0 
=2
X=
X=
raw score = mean + deviation
C. Using z-scores to
Standardize a Distribution
Convert each raw score to a z-score
What is the shape of the new dist’n?
Same as it was before!
Does NOT alter shape of dist’n!
Re-labeling values, but order stays
the same!
What is the mean?
=0
Convenient reference point!
What is the standard deviation?
=1
z always tells you # of SD units from
!
An entire population of scores is transformed into z-scores. The
transformation does not change the shape of the population but the
mean is transformed into a value of 0 and the standard deviation is
transformed to a value of 1.
Example:
Student
X
Garth
6
Peggy
11
Andy
8
Hellen
9
Humphry
5
Vivian
9
X-
z
N=6
N=6
=8
= 0
=2
=1
So, a distribution of z-scores
always has:
=0
=1
A standardized distribution
helps us compare scores from
different distributions
III. Comparing Scores
From Different Dist’s
Example:
Jim in class A scored 18
Mary in class B scored 75
Who performed better?
Need a “common metric”
Express each score relative to it’s
own  & 
Transform raw scores to z-scores
Standardize the distributions
 they will now have same  & 
Example:
Class A: Jim scored 18
 = 10
=5
1810
z
1.6
5
Class B: Mary scored 75
 = 50
 = 25
7550
z
25
1
Who performed better? Jim!
Two z-scores can always be
compared
Outline: Probability and
The Normal Curve
I. Probability
A. Probability and inferential statistics
B. What is probability?
II. The Normal Curve
A. Probability and the Normal Curve
B. Properties of the Standard Normal
Curve
C. The Unit Normal Table
III.
Solving Problems with the Normal
Curve
A. Problem Type 1
B. Problem Type 2
C. Cautions
I.
Probability
A. Probability & Inferential Statistics
Transition to inferential statistics
Why is probability so important?
Links samples and populations
Example 1:
The jar is a “population”
One marble is a “sample”
How likely to get BLACK?
But, isn’t the goal of inferential stats the
opposite?
Example 2:
Choose 10 marbles, blindfolded
“Sample” has 8 BLACK & 2 WHITE
Which jar did marbles come from?
This is inferential statistics!
“Judgments under uncertainty”
B. What is probability?
Likelihood of an “event” occurring
Can range from 0 (never) to 1.0 (always)
Defined in terms of a fraction, proportion, or
percentage
p(A) =
Number of outcomes classified as A
Total number of possible outcomes
Example #1:
Toss a coin, what is probability of heads?
p(Heads) = 1/2
1 = one way to get heads
2 = two possible outcomes (heads or
tails)
½ =
.50
=
50%
Example #2:
Select a card from a deck of 52
cards
What is probability of selecting a king?
p(King) = 4/52
4 = four ways to get a king
52 = 52 possible outcomes
4/52
=
.077
=
7.7%
Compute probability from
a frequency distribution
f
p
N
X
f
9
1
8
3
7
4
6
2
Σf = N = 10
What is the probability of selecting a
score with x = 8?
p(x = 8) = f/N = 3/10 = .30 = 30%
What is the probability of selecting a
score with x < 8?
p(x < 8) = f/N = 6/10 = .60 = 60%
II. The Normal Curve
A. Probability and the Normal
Curve
– Special statistical tool called the
Normal Curve
– Theoretical curve defined by
mathematical formula
– Known proportions/areas under
the curve
– Used to solve problems when we
don’t know the population
B. Properties of the
Standard Normal Curve
• Theoretical, idealized curve
• Based on mathematical formula
• Bell-shaped, symmetrical, unimodal
• μ = Md = Mo
• 50% of scores above m, 50% below
• Standardized: μ = 0, σ = 1
• A probability distribution, tails not anchored to axis
• Total area under the curve will sum to 1.0
• Exact percentiles associated with each z-score
• Area under curve provided in Unit Normal Table
• Can be applied to any normal distribution once the
distribution is standardized (converted to z-scores)
Why is the normal curve
so important?
(1) Many variables normally
distributed in population
(2) Can use normal curve to solve
many problems
Two types of problems:
(1) What proportion of dist’n falls
above, below, or between
particular z-scores?
(2) What z-score is associated with
particular proportions/probabilities
under the curve?
C. The Unit Normal
Table (UNT)
A = z-scores
B = Proportion in body (larger portion)
C = Proportion in tail (smaller portion)
• Curve is symmetrical, only + z
scores shown
• Columns B & C always sum to 1.0
• Proportions/probabilities are always
positive
z-score cuts curve into
two portions (B & C)
Let’s Practice!
Tip: Always sketch a
curve first!
Examples 1:
What proportion of distribution falls
above z = 1.5?
p (z > 1.5)
What proportion falls below z = -.5?
p (z < -0.5)
Examples 2:
What z-score separates the
lower 75% from the upper
25%? (same as 75th
percentile)
What z-scores separate the
middle 60% of the distribution
from the rest of the
distribution?
II. Solving Problems
with the Normal Curve
Hints and Tips
Two types of problems:
(1) Finding proportion associated with X or z
(2) Finding X or z associated with proportion
Problem Type #1 Steps to Follow:
(a) Sketch curve
(b) Convert raw score to z-score
(c) Look up proportion for this z-score
(d) Sometimes add/subtract proportions
Problem type #2 Steps to Follow:
(a) Sketch curve
(b) Look up z-score associated with proportion
(c) Convert z-score back to a raw score (X)
Always sketch a normal curve first!
A. Problem Type 1: Finding
Area Under the Curve
Problem #1:
Exam:
μ = 60
σ = 10
What percentage will score below 70?
(1)
Sketch a normal curve
(2)
Convert raw score to z-score
z=
(3)
Plan your strategy
(4)
Refer to Unit Normal Table
Problem #2:
Exam μ = 60 σ = 10
What is percentile rank of student
who scored 55?
(1) Sketch a normal curve
(2)
(3)
Convert raw score to z-score
z=
Plan your strategy
(4)
Refer to UNT
Problem #3:
Exam
μ = 60
σ = 10
What proportion of people will
score between 60 and 80?
(1) Sketch a normal curve
(2) Convert raw score to z-score
z=
(3)
Plan your strategy
(4)
Refer to UNT
Problem #4:
Exam:
μ = 60
σ = 10
What proportion of people will
score between 50 and 80?
(1) Sketch a normal curve
(2) Convert raw scores to zscores
z1 =
z2 =
(3) Plan your strategy
(4) Refer to UNT
B. Problem Type 2: Finding a
Score Associated
with a Proportion or Percentile
Problem #5:
Standardized Exam:
μ = 60 σ = 10
Assign A+ to the 95th percentile
What is cut-off score for earning an
A+?
(1) Sketch curve
(2) Plan your solution
(3) Refer to UNT
z=
(4) Convert z-score back to raw
score:
x=+zσ
x=
Problem #6:
Exam:
μ = 60
σ = 10
Assign F to 15th percentile (and
below)
What is cut-off score for earning an
F?
(1) Sketch curve
(2) Plan your solution
(3) Refer to UNT
z=
(4) Convert z-score back to raw
score:
x=+zσ
x=
C. Cautions
In order to use the UNT to solve problems,
you must:
• have known μ and σ
• assume your variable is normally distributed
Why?
• If you don’t know μ & σ, can’t compute a zscore
• If variable is not normally distributed,
percentages given by UNT won’t apply!
• z-scores can be negative but proportions/
percentiles cannot!
• Pay close attention to the words…
– Above, Below, Within, Beyond
The Distribution of
Sample Means
Inferential statistics:
Generalize from a sample to a
population
Statistics vs. Parameters
Why?
Population not often possible
Limitation:
Sample won’t precisely reflect
population
Samples from same population
vary
“sampling variability”
Sampling error = discrepancy
between sample statistic and
population parameter
The Distribution of
Sample Means
• Extend z-scores and normal curve
to SAMPLE MEANS rather than
individual scores
• How well will a sample describe a
population?
• What is probability of selecting a
sample that has a certain mean?
• Sample size will be critical
– Larger samples are more
representative
– Larger samples = smaller error
The Distribution of
Sample Means
Population of 4 scores: 2 4 6 8 
=5
4 random samples (n = 2):
X 4 X
1
2
6
X
3
5
X
4
3
X is rarely exactly 
Most
X
a little bigger or smaller than 
Most
X
will cluster around 
Extreme low or high values of X are relatively rare
With larger n, X s will cluster closer to µ (the DSM will have
smaller error, smaller variance)
We don’t actually compute a DSM!
A Distribution of Sample
Means
X=4
X=5 X=6
The distribution of sample means for n = 2. This
distribution shows the 16 sample means obtained by
taking all possible random samples of size n=2 that can
be drawn from the population of 4 scores. The known
population mean from which these samples were drawn
is µ = 5.
The Distribution of
Sample Means
A distribution of sample means ( X)
All possible random samples of size n
A distribution of a statistic (not raw scores)
“Sampling Distribution” of X
Probability of getting an X, given known  and 
Important properties
(1)
Mean
(2)
Standard Deviation
(3)
Shape
Properties of the DSM
Mean?
uX  u
Called expected value of
X
Standard Deviation?
Any X can be viewed as a deviation from 

= Standard Error of the Mean
x
 σ/
x
Variability of
X
n
around 
Special type of standard deviation, type of “error”
Average amount by which X deviates from 
Less error = better, more reliable,
estimate of population parameter

x
influenced by two things:
(1) Sample size (n)
Larger n = smaller standard errors
x 
Note: when n = 1 
 as “starting point” for
x
x gets smaller as n increases
 


(2) Variability in population ()
Larger  = larger standard errors
Note:
 
x
m
The distribution of sample means for random samples of size (a) n = 1, (b) n
= 4, and (c) n = 100 obtained from a normal population with µ = 80 and σ =
20. Notice that the size of the standard error decreases as the sample size
increases.
Shape of the DSM?
Central Limit Theorem = DSM will
approach a normal dist’n as n
approaches infinity
Very important!
True even when raw scores NOT normal!
True regardless of  or 
What about sample size?
(1) If raw scores ARE normal, any n
will do
(2)If raw scores NOT normal, n
must be “sufficiently large”
For most distributions  n  30
Why are Sampling
Distributions important?
• Tells us probability of getting X, given  & 
• Distribution of a STATISTIC rather than raw
scores
• Theoretical probability distribution
• Critical for inferential statistics!
• Allows us to estimate likelihood of making an
error when generalizing from sample to popl’n
• Standard error = variability due to chance
• Allows us to estimate population parameters
• Allows us to compare differences between
sample means – due to chance or to
experimental treatment?
• Sampling distribution is the most
fundamental concept underlying all statistical
tests
Working with the
Distribution of Sample
Means
• If we assume DSM is normal
• If we know  & 
• We can use Normal Curve & Unit
Normal Table!
z
X 
x
Example #1:
 = 80  = 12
What is probability of getting X  86
if n = 9?
Example #1b:
 = 80  = 12
What if we change n =36
What is probability of getting X  86
Example #2:
 = 80
 = 12
What X marks the point beyond
which sample means are likely to
occur only 5% of the time? (n = 9)