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The Standard Normal Distribution

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Title: The Standard Normal Distribution


1
The Standard Normal Distribution
  • PSY440
  • June 3, 2008

2
Outline of Class Period
  • Article Presentation (Kristin M)
  • Recap of two items from last time
  • Using Excel to compute descriptive statistics
  • Using SPSS to generate histograms
  • Standardization (z-transformation) of scores
  • The normal distribution
  • Properties of the normal curve
  • Standard normal distribution the unit normal
    table
  • Intro to probability theory and hypothesis testing

3
Using Excel to Compute Mean SD
  • Step 1 Compute mean of height with formula bar.
  • Step 2 Create deviation scores by creating a
    formula that subtracts the mean from each raw
    score, and apply the formula to all of the cells
    in a blank column next to the column of raw
    scores.
  • Step 3 Square the deviations by creating a
    formula and applying it to the cells in the next
    blank column.
  • Step 4 Use the formula bar to add the squared
    deviations, divide by (n-1) and take the square
    root of the result.
  • Step 5 Check the result by computing the SD with
    the formula bar.

4
Using SPSS to generate histograms
  • Most common answer
  • Most distinctive answer

5
How did this happen?
  • The shape of the histogram will change depending
    on the intervals used on the x axis.
  • For very large samples and truly continuous
    variables, the shape will smooth out, but with
    smaller samples, the shape can change
    considerably if you change the size of the
    intervals.

6
Make sure you are in charge of SPSS and not vice
versa!
  • SPSS has default settings for many of its
    operations that or may not be what you want.
  • You can tell SPSS how many intervals you want in
    your histogram, or how large you want the
    intervals to be.

7
Histogram with 16 intervals
In legacy dialogues, chose interactive and then
choose histogram. (see note) In chart builder,
choose histogram then choose element
properties then click on set parameters
8
The Z transformation
  • If you know the mean and standard deviation
    (sample or population we wont worry about
    which one, since your text book doesnt) of a
    distribution, you can convert a given score into
    a Z score or standard score. This score is
    informative because it tells you where that score
    falls relative to other scores in the
    distribution.

9
Locating a score
  • Where is our raw score within the distribution?
  • The natural choice of reference is the mean
    (since it is usually easy to find).
  • So well subtract the mean from the score (find
    the deviation score).
  • The direction will be given to us by the negative
    or positive sign on the deviation score
  • The distance is the value of the deviation score

10
Locating a score
X1 - 100 62
X1 162
X2 57
X2 - 100 -43
11
Locating a score
X1 - 100 62
X1 162
X2 57
X2 - 100 -43
12
Transforming a score
  • The distance is the value of the deviation score
  • However, this distance is measured with the units
    of measurement of the score.
  • Convert the score to a standard (neutral) score.
    In this case a z-score.

13
Transforming scores
  • A z-score specifies the precise location of each
    X value within a distribution.
  • Direction The sign of the z-score ( or -)
    signifies whether the score is above the mean or
    below the mean.
  • Distance The numerical value of the z-score
    specifies the distance from the mean by counting
    the number of standard deviations between X and ?.

X1 162
X2 57
14
Transforming a distribution
  • We can transform all of the scores in a
    distribution
  • We can transform any all observations to
    z-scores if we know the distribution mean and
    standard deviation.
  • We call this transformed distribution a
    standardized distribution.
  • Standardized distributions are used to make
    dissimilar distributions comparable.
  • e.g., your height and weight
  • One of the most common standardized distributions
    is the Z-distribution.

15
Properties of the z-score distribution
0
16
Properties of the z-score distribution
150
50
0
Xmean 100
1
17
Properties of the z-score distribution
150
50
1
0
Xmean 100
1
X1std 150
-1
18
Properties of the z-score distribution
  • Shape - the shape of the z-score distribution
    will be exactly the same as the original
    distribution of raw scores. Every score stays in
    the exact same position relative to every other
    score in the distribution.
  • Mean - when raw scores are transformed into
    z-scores, the mean will always 0.
  • The standard deviation - when any distribution
    of raw scores is transformed into z-scores the
    standard deviation will always 1.

19
From z to raw score
  • We can also transform a z-score back into a raw
    score if we know the mean and standard deviation
    information of the original distribution.
  • Z (X - ?) --gt (Z)( ?) (X - ?) --gt X (Z)(
    ?) ?
  • ?

X (-0.60)( 50) 100
X 70
20
Lets try it with our data
  • To transform data on height into standard scores,
    use the formula bar in excel to subtract the mean
    and divide by the standard deviation.
  • Can also choose standardize (x,mean,sd)
  • Show with shoe size
  • Observe how height and shoe size can be more
    easily compared with standard (z) scores

21
Z-transformations with SPSS
  • You can also do this in SPSS.
  • Use Analyze . Descriptive Statistics.
    Descriptives .
  • Check the box that says save standardized values
    as variables.

22
The Normal Distribution
  • Normal distribution

23
The Normal Distribution
  • Normal distribution is a commonly found
    distribution that is symmetrical and unimodal.
  • Not all unimodal, symmetrical curves are Normal,
    so be careful with your descriptions
  • It is defined by the following equation
  • The mean, median, and mode are all equal for this
    distribution.

24
The Normal Distribution
  • This equation provides x and y coordinates on the
    graph of the frequency distribution. You can plug
    a given value of x into the formula to find the
    corresponding y coordinate. Since the function
    describes a symmetrical curve, note that the same
    y (height) is given by two values of x
    (representing two scores an equal distance above
    and below the mean)

Y
25
The Normal Distribution
  • As the distance between the observed score (x)
    and the mean increases, the value of the
    expression (i.e., the y coordinate) decreases.
    Thus the frequency of observed scores that are
    very high or very low relative to the mean, is
    low, and as the difference between the observed
    score and the mean gets very large, the frequency
    approaches 0.

Y
26
The Normal Distribution
  • As the distance between the observed score (x)
    and the mean decreases (i.e., as the observed
    value approaches the mean), the value of the
    expression (i.e., the y coordinate) increases.
  • The maximum value of y (i.e., the mode, or the
    peak in the curve) is reached when the observed
    score equals the mean hence mean equals mode.

Y
27
The Normal Distribution
  • The integral of the function gives the area under
    the curve (remember this if you took calculus?)
  • The distribution is asymptotic, meaning that
    there is no closed solution for the integral.
  • It is possible to calculate the proportion of the
    area under the curve represented by a range of x
    values (e.g., for x values between -1 and 1).

Y
28
The Unit Normal Table
  • The normal distribution is often transformed into
    z-scores.

z .00 .01
-3.4 -3.3 0 1.0 3.3 3.4 0.0003 0.0005 0.5000 0.8413 0.9995 0.9997 0.0003 0.0005 0.5040 0.8438 0.9995 0.9997
  • Gives the precise proportion of scores (in
    z-scores) between the mean (Z score of 0) and any
    other Z score in a Normal distribution
  • Contains the proportions in the tail to the left
    of corresponding z-scores of a Normal
    distribution
  • This means that the table lists only positive Z
    scores
  • The .00 column corresponds to column (3) in Table
    B of your textbook.
  • Note that for z0 (i.e., at the mean), the
    proportion of scores to the left is .5 Hence,
    meanmedian.

29
Using the Unit Normal Table
z .00 .01
-3.4 -3.3 0 1.0 3.3 3.4 0.0003 0.0005 0.5000 0.8413 0.9995 0.9997 0.0003 0.0005 0.5040 0.8438 0.9995 0.9997
50-34-14 rule
Similar to the 68-95-99 rule
1
2
-1
-2
0
15.87 (13.59 and 2.28) of the
scores are to the right of the score 100-15.87
84.13 to the left
30
Using the Unit Normal Table
  • Steps for figuring the percentage above or below
    a particular raw or Z score

z .00 .01
-3.4 -3.3 0 1.0 3.3 3.4 0.0003 0.0005 0.5000 0.8413 0.9995 0.9997 0.0003 0.0005 0.5040 0.8438 0.9995 0.9997
  • 1. Convert raw score to Z score (if necessary)
  • 2. Draw normal curve, where the Z score falls on
    it, shade in the area for which you are finding
    the percentage
  • 3. Make rough estimate of shaded areas
    percentage (using 50-34-14 rule)

31
Using the Unit Normal Table
  • Steps for figuring the percentage above or below
    a particular raw or Z score

z .00 .01
-3.4 -3.3 0 1.0 3.3 3.4 0.0003 0.0005 0.5000 0.8413 0.9995 0.9997 0.0003 0.0005 0.5040 0.8438 0.9995 0.9997
  • 4. Find exact percentage using unit normal table
  • 5. If needed, subtract percentage from 100.
  • 6. Check the exact percentage is within the
    range of the estimate from Step 3

32
SAT Example problems
  • The population parameters for the SAT are
  • m 500, s 100, and it is Normally distributed

Suppose that you got a 630 on the SAT. What
percent of the people who take the SAT get your
score or lower?
From the table z(1.3) .9032
33
The Normal Distribution
  • You can go in the other direction too
  • Steps for figuring Z scores and raw scores from
    percentages
  • 1. Draw normal curve, shade in approximate area
    for the percentage (using the 50-34-14 rule)
  • 2. Make rough estimate of the Z score where the
    shaded area starts
  • 3. Find the exact Z score using the unit normal
    table
  • 4. Check that your Z score is similar to the
    rough estimate from Step 2
  • 5. If you want to find a raw score, change it
    from the Z score

34
The Normal Distribution
  • Example What z score is at the 75th percentile
    (at or above 75 of the scores)?
  • 1. Draw normal curve, shade in approximate area
    for the percentage (using the 50-34-14 rule)
  • 2. Make rough estimate of the Z score where the
    shaded area starts (between .5 and 1)
  • 3. Find the exact Z score using the unit normal
    table (a little less than .7)
  • 4. Check that your Z score is similar to the
    rough estimate from Step 2
  • 5. If you want to find a raw score, change it
    from the Z score using mean and standard
    deviation info.

35
The Normal Distribution
  • Finding the proportion of scores falling between
    two observed scores
  • Convert each score to a z score
  • Draw a graph of the normal distribution and shade
    out the area to be identified.
  • Identify the area below the highest z score using
    the unit normal table.
  • Identify the area below the lowest z score using
    the unit normal table.
  • Subtract step 4 from step 3. This is the
    proportion of scores that falls between the two
    observed scores.

36
The Normal Distribution
  • Example What proportion of scores falls between
    the mean and .2 standard deviations above the
    mean?
  • Convert each score to a z score (mean 0, other
    score .2)
  • Draw a graph of the normal distribution and shade
    out the area to be identified.
  • Identify the area below the highest z score using
    the unit normal table
  • For z.2, the proportion to the left .5793
  • Identify the area below the lowest z score using
    the unit normal table.
  • For z0, the proportion to the left .5
  • Subtract step 4 from step 3
  • .5793 - .5 .0793
  • About 8 of the observations fall between the
    mean and .2 SD.

37
The Normal Distribution
  • Example 2 What proportion of scores falls
    between -.2 standard deviations and -.6 standard
    deviations?
  • Convert each score to a z score (-.2 and -.6)
  • Draw a graph of the normal distribution and shade
    out the area to be identified.
  • Identify the area below the highest z score using
    the unit normal table
  • For z-.2, the proportion to the left 1 - .5793
    .4207
  • Identify the area below the lowest z score using
    the unit normal table.
  • For z-.6, the proportion to the left 1 -
    .7257 .2743
  • Subtract step 4 from step 3
  • .4207 - .2743 .1464
  • About 15 of the observations fall between -.2
    and -.6 SD.

38
Hypothesis testing
  • Example Testing the effectiveness of a new
    memory treatment for patients with memory problems
  • Our pharmaceutical company develops a new drug
    treatment that is designed to help patients with
    impaired memories.
  • Before we market the drug we want to see if it
    works.
  • The drug is designed to work on all memory
    patients, but we cant test them all (the
    population).
  • So we decide to use a sample and conduct the
    following experiment.
  • Based on the results from the sample we will make
    conclusions about the population.

39
Hypothesis testing
  • Example Testing the effectiveness of a new
    memory treatment for patients with memory problems

55 errors
60 errors
  • Is the 5 error difference
  • A real difference due to the effect of the
    treatment
  • Or is it just sampling error?

40
Testing Hypotheses
  • Hypothesis testing
  • Procedure for deciding whether the outcome of a
    study (results for a sample) support a particular
    theory (which is thought to apply to a
    population)
  • Core logic of hypothesis testing
  • Considers the probability that the result of a
    study could have come about if the experimental
    procedure had no effect
  • If this probability is low, scenario of no effect
    is rejected and the theory behind the
    experimental procedure is supported

41
Basics of Probability
  • Probability
  • Expected relative frequency of a particular
    outcome
  • Outcome
  • The result of an experiment

42
Flipping a coin example
What are the odds of getting a heads?
n 1 flip
43
Flipping a coin example
What are the odds of getting two heads?
Number of heads
n 2
2
1
1
0
of outcomes 2n
This situation is known as the binomial
44
Flipping a coin example
What are the odds of getting at least one heads?
Number of heads
n 2
2
1
1
0
45
Flipping a coin example
Number of heads
n 3
HHH
3
HHT
2
HTH
2
HTT
1
2
THH
THT
1
TTH
1
TTT
0
23 8 total outcomes
2n
46
Flipping a coin example
Number of heads
Distribution of possible outcomes (n 3 flips)
3
2
X f p
3 1 .125
2 3 .375
1 3 .375
0 1 .125
2
1
2
1
1
0
47
Flipping a coin example
Can make predictions about likelihood of outcomes
based on this distribution.
Distribution of possible outcomes (n 3 flips)
.4
Whats the probability of flipping three heads in
a row?
.3
probability
.2
.1
p 0.125
.125
.125
.375
.375
0
1
2
3
Number of heads
48
Flipping a coin example
Can make predictions about likelihood of outcomes
based on this distribution.
Distribution of possible outcomes (n 3 flips)
.4
Whats the probability of flipping at least two
heads in three tosses?
.3
probability
.2
.1
p 0.375 0.125 0.50
.125
.125
.375
.375
0
1
2
3
Number of heads
49
Flipping a coin example
Can make predictions about likelihood of outcomes
based on this distribution.
Distribution of possible outcomes (n 3 flips)
.4
Whats the probability of flipping all heads or
all tails in three tosses?
.3
probability
.2
.1
p 0.125 0.125 0.25
.125
.125
.375
.375
0
1
2
3
Number of heads
50
Hypothesis testing
Can make predictions about likelihood of outcomes
based on this distribution.
Distribution of possible outcomes (of a
particular sample size, n)
  • In hypothesis testing, we compare our observed
    samples with the distribution of possible samples
    (transformed into standardized distributions)
  • This distribution of possible outcomes is often
    Normally Distributed

51
Inferential statistics
  • Hypothesis testing
  • Core logic of hypothesis testing
  • Considers the probability that the result of a
    study could have come about if the experimental
    procedure had no effect
  • If this probability is low, scenario of no effect
    is rejected and the theory behind the
    experimental procedure is supported
  • A five step program
  • Step 1 State your hypotheses
  • Step 2 Set your decision criteria
  • Step 3 Collect your data
  • Step 4 Compute your test statistics
  • Step 5 Make a decision about your null hypothesis

52
Hypothesis testing
  • Hypothesis testing a five step program
  • Step 1 State your hypotheses as a research
    hypothesis and a null hypothesis about the
    populations
  • Null hypothesis (H0)
  • Research hypothesis (HA)
  • There are no differences between conditions (no
    effect of treatment)
  • Generally, not all groups are equal
  • You arent out to prove the alternative
    hypothesis
  • If you reject the null hypothesis, then youre
    left with support for the alternative(s) (NOT
    proof!)

53
Testing Hypotheses
  • Hypothesis testing a five step program
  • Step 1 State your hypotheses

One -tailed
  • Our theory is that the treatment should improve
    memory (fewer errors).

mTreatment gt mNo Treatment
H0
mTreatment lt mNo Treatment
HA
54
Testing Hypotheses
  • Hypothesis testing a five step program
  • Step 1 State your hypotheses

One -tailed
Two -tailed
  • Our theory is that the treatment should improve
    memory (fewer errors).
  • Our theory is that the treatment has an effect
    on memory.

mTreatment gt mNo Treatment
H0
mTreatment mNo Treatment
H0
mTreatment lt mNo Treatment
HA
mTreatment ? mNo Treatment
HA
55
One-Tailed and Two-Tailed Hypothesis Tests
  • Directional hypotheses
  • One-tailed test
  • Nondirectional hypotheses
  • Two-tailed test

56
Testing Hypotheses
  • Hypothesis testing a five step program
  • Step 1 State your hypotheses
  • Step 2 Set your decision criteria
  • Your alpha (?) level will be your guide for when
    to reject or fail to reject the null hypothesis.
  • Based on the probability of making making an
    certain type of error

57
Testing Hypotheses
  • Hypothesis testing a five step program
  • Step 1 State your hypotheses
  • Step 2 Set your decision criteria
  • Step 3 Collect your data

58
Testing Hypotheses
  • Hypothesis testing a five step program
  • Step 1 State your hypotheses
  • Step 2 Set your decision criteria
  • Step 3 Collect your data
  • Step 4 Compute your test statistics
  • Descriptive statistics (means, standard
    deviations, etc.)
  • Inferential statistics (z-test, t-tests, ANOVAs,
    etc.)

59
Testing Hypotheses
  • Hypothesis testing a five step program
  • Step 1 State your hypotheses
  • Step 2 Set your decision criteria
  • Step 3 Collect your data
  • Step 4 Compute your test statistics
  • Step 5 Make a decision about your null hypothesis
  • Based on the outcomes of the statistical tests
    researchers will either
  • Reject the null hypothesis
  • Fail to reject the null hypothesis
  • This could be correct conclusion or the incorrect
    conclusion

60
Error types
  • Type I error (?) concluding that there is a
    difference between groups (an effect) when
    there really isnt.
  • Sometimes called significance level or alpha
    level
  • We try to minimize this (keep it low)
  • Type II error (?) concluding that there isnt an
    effect, when there really is.
  • Related to the Statistical Power of a test (1-?)

61
Error types
Real world (truth)
H0 is correct
H0 is wrong


Reject H0
Experimenters conclusions
Fail to Reject H0
62
Error types
Real world (truth)
H0 is correct
H0 is wrong


Reject H0
Experimenters conclusions
Fail to Reject H0
63
Error types
Real world (truth)
H0 is correct
H0 is wrong


Type I error
Reject H0
Experimenters conclusions
Fail to Reject H0
Type II error
64
Performing your statistical test
  • What are we doing when we test the hypotheses?

Real world (truth)
H0 is true (no treatment effect)
H0 is false (is a treatment effect)
65
Performing your statistical test
  • What are we doing when we test the hypotheses?
  • Computing a test statistic Generic test

66
Generic statistical test
  • The generic test statistic distribution (think of
    this as the distribution of sample means)
  • To reject the H0, you want a computed test
    statistics that is large
  • Whats large enough?
  • The alpha level gives us the decision criterion

?-level determines where these boundaries go
67
Generic statistical test
  • The generic test statistic distribution (think of
    this as the distribution of sample means)
  • To reject the H0, you want a computed test
    statistics that is large
  • Whats large enough?
  • The alpha level gives us the decision criterion

68
Generic statistical test
  • The alpha level gives us the decision criterion

One -tailed
Two -tailed
69
Generic statistical test
  • The alpha level gives us the decision criterion

One -tailed
Two -tailed
Reject H0
Reject H0
Fail to reject H0
Fail to reject H0
70
Generic statistical test
  • The alpha level gives us the decision criterion

One -tailed
Two -tailed
all of it in one tail
Reject H0
Reject H0
Fail to reject H0
Fail to reject H0
71
Generic statistical test
  • An example One sample z-test
  • Step 1 State your hypotheses
  • We give a n 16 memory patients a memory
    improvement treatment.

mTreatment mpop 60
  • How do they compare to the general population
    of memory patients who have a distribution of
    memory errors that is Normal, m 60, s 8?

mTreatment ? mpop ? 60
72
Generic statistical test
  • An example One sample z-test

H0 mTreatment mpop 60
HA mTreatment ? mpop ? 60
  • We give a n 16 memory patients a memory
    improvement treatment.
  • Step 2 Set your decision criteria

a 0.05
  • How do they compare to the general population
    of memory patients who have a distribution of
    memory errors that is Normal, m 60, s 8?

73
Generic statistical test
  • An example One sample z-test

H0 mTreatment mpop 60
HA mTreatment ? mpop ? 60
  • We give a n 16 memory patients a memory
    improvement treatment.

a 0.05
One -tailed
  • Step 3 Collect your data
  • How do they compare to the general population
    of memory patients who have a distribution of
    memory errors that is Normal, m 60, s 8?

74
Generic statistical test
  • An example One sample z-test

H0 mTreatment mpop 60
HA mTreatment ? mpop ? 60
  • We give a n 16 memory patients a memory
    improvement treatment.

a 0.05
One -tailed
  • Step 4 Compute your test statistics
  • How do they compare to the general population
    of memory patients who have a distribution of
    memory errors that is Normal, m 60, s 8?

-2.5
75
Generic statistical test
  • An example One sample z-test

H0 mTreatment mpop 60
HA mTreatment ? mpop ? 60
  • We give a n 16 memory patients a memory
    improvement treatment.

a 0.05
One -tailed
  • Step 5 Make a decision about your null hypothesis
  • How do they compare to the general population
    of memory patients who have a distribution of
    memory errors that is Normal, m 60, s 8?

Reject H0
76
Generic statistical test
  • An example One sample z-test

H0 mTreatment mpop 60
HA mTreatment ? mpop ? 60
  • We give a n 16 memory patients a memory
    improvement treatment.

a 0.05
One -tailed
  • Step 5 Make a decision about your null hypothesis

- Reject H0
  • How do they compare to the general population
    of memory patients who have a distribution of
    memory errors that is Normal, m 60, s 8?

- Support for our HA, the evidence suggests that
the treatment decreases the number of memory
errors
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