Standard Scores
In order to compare scores from
different distributions and in
different units of measurement, we
need a common scale or common
unit of measurement.
Convert scores from each
distribution to standard scores on a
common scale.
2
Standard Scores
Standard scores are transformed
raw scores.
Allow us to determine the exact
position of raw scores in the
distribution.
3
Z-Scores
The most common or the basis of all
standard scores is a z-score.
Can be used as descriptive statistics
and as inferential statistics.
Descriptive: describes exactly where each
individual is located.
Inferential: determines whether a specific
sample is representative of its population
or is extreme and unrepresentative.
4
How does it tell us this?
If our observation X is from a
population with mean μ and standard
deviation σ, then
If the observation X is from a sample
with mean and standard deviation s,
then
5
z-standard (unit) normal distribution
The mathematics of z-score
transformation converts every
observation in a distribution to its z-
score.
With this transformation, the mean
of the new (z) distribution becomes 0
and the standard deviation becomes
1.
The transformed distribution is called
z-standard (unit) normal distribution.
6
z-standard (unit) normal distribution
7
Revisiting properties of the normal
distribution
8
Interpretation of z-scores
A z-score shows the distance of a
score from the mean in terms of
standard deviation.
A z-score of .05 means that a score is
half a deviation above the mean.
A z-score of -.05 means that a score
is .05 standard deviations below the
mean.
9
What does a z-score tell us?
Answers the question:
“How many standard deviations away
from the mean is this observation in
a normal distribution.”
10
When is the Z score useful?
The z score transformation is useful
when we seek to compare the
relative standings of observations
from distributions with different
means and/or different standard
deviations.
11
For example:
Last semester, Matt scored 70 in
Ms. Lauren’s math class. The
average score of the class was 60
and the standard deviation was 15.
This year, Matt is in Ms. Molly’s
class. He scored 88. The mean
score was 90 and the standard
deviation was 4.
In which class did Matt perform
better?
12
In notation:
Write down the information
given and the information you
need:
X1= 70 X2= 88
μ1 = 60 μ2 = 90
σ1 =15 σ2 = 4
13
Part B: Normal Distribution
14
Normal Distribution
The normal distribution is not a
single distribution but a family of
distributions, each which is
determined by its mean and
standard deviation.
Properties:
Unimodal
Continuous
Asymptotic
Theoretical!
15
z-standard (unit) normal distribution
When all scores are converted to z-
scores and plotted, they form a z-
distribution.
A z-score distribution is a normal
distribution with the fixed mean of 0
and the standard deviation of 1.
Any set of scores can be
transformed into z-scores and
plotted on a this distribution.
16
Area under the normal curve
In every normal distribution, the
distance between the mean and a
given Z score cuts off a fixed
proportion of the total area under
the curve.
Statisticians have provided us with
tables indicating the value of these
proportions for each possible Z
score.
17
See page 634 of text:
Mean to Z is the percentage
between the mean and z score or
sd.
‘Area beyond Z’ represents the
smaller portion.
18
Smaller Portion
Larger portion
Smaller portion
So, if we have a Z of +1, the smaller portion would be to
the right. The larger portion would be to the left.
19
Smaller Portion
Larger portion
Smaller portion
So, if we have a Z of -2, the smaller portion would be to
the left. The larger portion would be to the right.
20
Probability versus Percentile
Probability of an event is the
proportion of times the event would
happened if we could repeat the
operation a great many times.
Always between 0 (never happen) and
1 (always happen).
Percentile is the point which a
specified observations falls.
21
Practice interpreting table:
What is the probability of selecting
a score that falls beyond 1 Z?
What is the probability of selecting
a score that fall below – 2Z?
What is the percentile rank of
someone who has a Z score of 2?
What is the percentile rank of
someone who has a Z score of 1?
22
New score = σ(z)+μ
How can we convert a particular
score to a distribution with a
different mean and standard
deviation?
Compute a “new score.”
23
Example of newscore[1]:
There are several IQ tests, each
consisting of different number of
items. Yet, all the IQ test results are
reported on the same scale with the
mean of 100 and a deviation of 15.
How can we convert any score from
any test to this common scale that
everybody can understand?
24
Example of newscore[2]:
1. Convert the given X into a Z-score.
2. Using the new score formula, we
obtain:
New score = σnew distribution (z) + μnew distribution
New IQ score = 15 (z) + 100
New distribution New distribution
standard deviation mean
25
Other standard scores
Definitions
Z-score T-score IQ GRE
Mean 0 50 100 500
Standard
1 10 15 100
Deviation
26
Gre 200 300 400 500 600 700 800
If you scored a 600, how many standard deviations
away from the mean would you be? What percentage of
people did worse than you?
If you scored a 300?
27
T Score
What if we want to convert Matt’s
Z-score of .67 to a T score.
New score = σ (z) + μ
Tscore = 10 (.67) + 50 = 56.70
Standard deviation Matt’s raw score of Mean of T
of T distribution 70 converted to a distribution
Z-score 28
Part C: Sampling
Distribution of the Means
29
Is the Sample Representative of the
Population?
Often make conclusions/inferences
about the population from the
sample under study.
How do we know if a sample is
representative of the population
when every sample is different?
How can we transform a population
distribution of individuals to a
population distribution of sample
means?
30
Sampling error
Every sample is different from the
population.
Sampling error is the
discrepancy/error between the
sample and the population.
Random sampling is used to minimize
this error so that it occurs randomly
31
Distribution of Sample Means
Randomly group people into similar
sized samples.
Calculate the sample means.
Place them into a distribution.
Result in a normal curve which is
the distribution of sample means.
32
Sampling Distribution
Any distribution that is of sample
statistics and NOT individual
observations/scores.
33
Properties of the distribution of
sample means
Approaches a normal distribution as
sample size increases.
Mean of the distribution is equal to
the population mean of individuals.
Standard deviation of this
distribution is called the standard
error of the sample mean.
34
Standard Error of the Sample Mean
Standard error (σx) = σ
√n
Measures the standard distance
between the sample mean and the
population mean.
A measure of how good an estimate
will have for population mean.
As sample size increases, the
standard error decreases.
35
Probability and the Distribution of
Sample Means: What does it tell us?
What is the probability of obtaining
a specific sample mean from the
population of samples?
36
Note cards:
• Z score formula
• New score formula
• T score values (sd = 10, mean =
50)
In-Class Activity Creating a Sampling
Distribution!
37
Be the first to comment