The document discusses standard scores and normal distributions. It defines standard scores as transformed raw scores that allow comparison across different scales by putting them on a common scale. It then focuses on z-scores, which convert values to standardized units relative to the mean and standard deviation. The document also discusses how sample means are distributed normally as sample size increases, with a mean equal to the population mean and standard deviation called the standard error that decreases with larger samples. This allows determining if a sample mean is representative of the population.

1. Week 6
Part A: Standard Scores
1

2. 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

3. Standard Scores
 Standard scores are transformed
raw scores.
 Allow us to determine the exact
position of raw scores in the
distribution.
3

4. 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

5. 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

6. 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

7. z-standard (unit) normal distribution
7

8. Revisiting properties of the normal
distribution
8

9. 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

10. 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

11. 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

12. 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

13. 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

14. Part B: Normal Distribution
14

15. 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

16. 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

17. 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

18. 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

19. 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

20. 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

21. 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

22. 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

23. 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

24. 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

25. 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

26. Other standard scores
Definitions
Z-score T-score IQ GRE
Mean 0 50 100 500
Standard
1 10 15 100
Deviation
26

27. 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

28. 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

29. Part C: Sampling
Distribution of the Means
29

30. 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

31. 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

32. 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

33. Sampling Distribution
 Any distribution that is of sample
statistics and NOT individual
observations/scores.
33

34. 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

35. 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

36. 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

37. Note cards:
• Z score formula
• New score formula
• T score values (sd = 10, mean =
50)
In-Class Activity Creating a Sampling
Distribution!
37