the standard normal distribution psy440 june 3, 2008

The Standard Normal DistributionThe Standard Normal Distribution

PSY440

June 3, 2008

Outline of Class PeriodOutline 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

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.

Using SPSS to generate histogramsUsing SPSS to generate histograms

Most common answer: Most distinctive answer:

How did this happen?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.

Make sure you are in charge of SPSS and not vice versa!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.

Histogram with 16 intervalsHistogram 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…”

The Z transformationThe Z transformation

If you know the mean and standard deviation (sample or population – we won’t worry about which one, since your text book doesn’t) 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.

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 we’ll subtract the mean from the score (find the deviation score).

€

X − μ

– 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

Locating a score

€

X − μ

€

=100

X1 = 162X2 = 57

X1 - 100 = +62X2 - 100 = -43

Reference point

Direction

Locating a score

€

X − μ

€

=100

X1 = 162X2 = 57

X1 - 100 = +62X2 - 100 = -43

Reference point

BelowAbove

Transforming a score

€

z =X − μ

σ

– 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.

Raw score

Population meanPopulation standard deviation

Transforming scores

€

=100

X1 = 162

X2 = 57

€

σ =50

€

z =X − μ

σ

X1 - 100 = +1.2050

X2 - 100 = -0.86

50

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 .

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.

Properties of the z-score distribution

€

=0

transformation

€

z =X − μ

σ

15050

€

zmean =100 −100

50= 0

€

σ =50

€

=100

Xmean = 100


€

=0

€

σ =50

transformation

€

z =X − μ

σ

15050

Xmean = 100

€

zmean =100 −100

50

€

z+1std =150 −100

50

= 0

= +1

€

=100

X+1std = 150

+1


€

σ =1

€

=0

€

σ =50

transformation

€

z =X − μ

σ

15050

Xmean = 100

X+1std = 150

€

zmean =100 −100

50

€

z+1std =150 −100

50

€

z−1std =50 −100

50

= 0

= +1

= -1

€

=100

X-1std = 50

+1-1


• 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.

15050 €

σ =1

€

=0

+1-1

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 - ) --> (Z)( σ) = (X - ) --> X = (Z)( σ) +

σ

transformation

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X = Zσ + μ

€

σ =50

€

=100

Z = -0.60X = (-0.60)( 50) + 100X = 70

Let’s 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

Z-transformations with SPSSZ-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.”

The Normal Distribution

• 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.

€

1

2πσ 2e−(X −μ )2 / 2σ 2

1 2-1-2 0


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)

€

1

2πσ 2e−(X −μ )2 / 2σ 2

1 2-1-2 0

Y =


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.

€

1

2πσ 2e−(X −μ )2 / 2σ 2

1 2-1-2 0

Y =


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.

€

1

2πσ 2e−(X −μ )2 / 2σ 2

1 2-1-2 0

Y =


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).

€

1

2πσ 2e−(X −μ )2 / 2σ 2

1 2-1-2 0

Y =

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

• 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 z=0 (i.e., at the mean), the proportion of scores to the left is .5 Hence, mean=median.

• The normal distribution is often transformed into z-scores.

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

15.87% (13.59% and 2.28%) of the scores are to the right of the score100%-15.87% = 84.13% to the left

At z = +1:

13.59%2.28%

34.13%

50%-34%-14% rule

1 2-1-2 0

Similar to the 68%-95%-99% rule


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 area’s percentage (using 50%-34%-14% rule)

• 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

• Steps for figuring the percentage above or below a particular raw or Z score:

Suppose that you got a 630 on the SAT. What percent of the people who take the SAT get your score or lower?

SAT Example problems

• The population parameters for the SAT are: = 500, σ = 100, and it is Normally distributed

€

z =X − μ

σ=

630 − 500

100=1.3 From the table:

z(1.3) =.9032 -1-2 1 2

That’s 9.68% above this score

So 90.32% got your score or lower


• 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


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.


Finding the proportion of scores falling between two observed scores1. Convert each score to a z score

2. Draw a graph of the normal distribution and shade out the area to be identified.

3. Identify the area below the highest z score using the unit normal table.

4. Identify the area below the lowest z score using the unit normal table.

5. Subtract step 4 from step 3. This is the proportion of scores that falls between the two observed scores.

1 2-1-2 0


Example: What proportion of scores falls between the mean and .2 standard deviations above the mean?

1. Convert each score to a z score (mean = 0, other score = .2)


3. Identify the area below the highest z score using the unit normal table:For z=.2, the proportion to the left = .5793

4. Identify the area below the lowest z score using the unit normal table.For z=0, the proportion to the left = .5

5. Subtract step 4 from step 3:

.5793 - .5 = .0793

About 8% of the observations fall between the mean and .2 SD.

1 2-1-2 0


Example 2: What proportion of scores falls between -.2 standard deviations and -.6 standard deviations?

1. Convert each score to a z score (-.2 and -.6)


3. Identify the area below the highest z score using the unit normal table:For z=-.2, the proportion to the left = 1 - .5793 = .4207

4. Identify the area below the lowest z score using the unit normal table.For z=-.6, the proportion to the left = 1 - .7257 = .2743

5. Subtract step 4 from step 3:

.4207 - .2743 = .1464

About 15% of the observations fall between -.2 and -.6 SD.

1 2-1-2 0

Hypothesis testingHypothesis 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 can’t 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.


• Example: Testing the effectiveness of a new memory treatment for patients with memory problems

Memory treatment

No Memorytreatment

Memory patients

MemoryTest

MemoryTest

55 errors

60 errors

5 error diff

• Is the 5 error difference: – A “real” difference due to the effect of the treatment

– Or is it just sampling error?

Testing HypothesesTesting 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

Basics of ProbabilityBasics of Probability

• Probability– Expected relative frequency of a particular outcome

• Outcome– The result of an experiment

Probability = Possible successful outcomes

All possible outcomes

Flipping a coin exampleFlipping a coin example

What are the odds of getting a “heads”?

One outcome classified as heads=

1

2= 0.5

Probability = Possible successful outcomes

All possible outcomes

Total of two outcomes

n = 1 flip


What are the odds of getting two “heads”?

Number of heads

2

1

1

0

One 2 “heads” outcome

Four total outcomes

= 0.25

This situation is known as the binomial # of outcomes = 2n

n = 2


What are the odds of getting “at least one heads”?

Number of heads

2

1

1

0

Four total outcomes

= 0.75

Three “at least one heads” outcome

n = 2


HHH

HHT

HTH

HTT

THH

THT

TTH

TTT

Number of heads3

2

1

0

2

2

1

1

2n = 23 = 8 total outcomes

n = 3

Flipping a coin exampleFlipping a coin exampleNumber of heads

3

2

1

0

2

2

1

1

X f p

3 1 .125

2 3 .375

1 3 .375

0 1 .125Number of heads0 1 2 3

.1

.2

.3

.4

prob

abil

ity

.125 .125.375.375

Distribution of possible outcomes(n = 3 flips)


Number of heads0 1 2 3

.1

.2

.3

.4

prob

abil

ity

What’s the probability of flipping three heads in a row?

.125 .125.375.375 p = 0.125


Can make predictions about likelihood of outcomes based on this distribution.



.1

.2

.3

.4

prob

abil

ity

What’s the probability of flipping at least two heads in three tosses?

.125 .125.375.375 p = 0.375 + 0.125 = 0.50





.1

.2

.3

.4

prob

abil

ity

What’s the probability of flipping all heads or all tails in three tosses?

.125 .125.375.375 p = 0.125 + 0.125 = 0.25



Hypothesis testingHypothesis testingCan 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

Inferential statisticsInferential 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

• 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

– 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

This is the one that you test

• Hypothesis testing: a five step program

– You aren’t out to prove the alternative hypothesis

• If you reject the null hypothesis, then you’re left with support for the alternative(s) (NOT proof!)

In our memory example experiment:


Treatment > No Treatment

Treatment < No Treatment

H0:

HA:

– Our theory is that the treatment should improve memory (fewer errors).

– Step 1: State your hypotheses


One -tailed

In our memory example experiment:


Treatment > No Treatment

Treatment < No Treatment

H0:

HA:

– Our theory is that the treatment should improve memory (fewer errors).



Treatment = No Treatment

Treatment ≠ No Treatment

H0:

HA:

– Our theory is that the treatment has an effect on memory.

One -tailed Two -tailedno direction

specifieddirectionspecified

One-Tailed and Two-Tailed Hypothesis TestsOne-Tailed and Two-Tailed Hypothesis Tests

• Directional hypotheses– One-tailed test

• Nondirectional hypotheses– Two-tailed test



– 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




– 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.)





– 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

Error typesError types

• Type I error (): concluding that there is a difference between groups (“an effect”) when there really isn’t. – Sometimes called “significance level” or “alpha level”

– We try to minimize this (keep it low)

• Type II error (): concluding that there isn’t an effect, when there really is.– Related to the Statistical Power of a test (1-)


Real world (‘truth’)

H0 is correct

H0 is wrong

Experimenter’s conclusions

Reject H0

Fail to Reject H0

There really isn’t an effect

There really isan effect



H0 is correct

H0 is wrong


Reject H0

Fail to Reject H0

I conclude that there is an effect

I can’t detect an effect



H0 is correct

H0 is wrong


Reject H0

Fail to Reject H0

Type I error

Type II error

α

β

Performing your statistical testPerforming your statistical test

H0: is true (no treatment effect) H0: is false (is a treatment effect)

Two populations

One population

• What are we doing when we test the hypotheses?


XA

they aren’t the same as those in the population of memory patients

XA

the memory treatment sample are the same as those in the population of memory patients.

Performing your statistical testPerforming your statistical test

• What are we doing when we test the hypotheses?– Computing a test statistic: Generic test

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test statistic =observed difference

difference expected by chance

Could be difference between a sample and a population, or between different samples

Based on standard error or an estimate of the standard error

““Generic” statistical testGeneric” 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– What’s large enough?

• The alpha level gives us the decision criterion

Distribution of the test statistic

-level determines where these boundaries go

““Generic” statistical testGeneric” statistical test

If test statistic is here Reject H0

If test statistic is here Fail to reject H0

Distribution of the test statistic

• 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– What’s large enough?



Reject H0

Fail to reject H0


One -tailedTwo -tailedReject H0

Fail to reject H0

Reject H0

Fail to reject H0

= 0.05

0.025

0.025split up into the two tails


Reject H0

Fail to reject H0



Fail to reject H0

Reject H0

Fail to reject H0

= 0.050.05

all of it in one tail


Reject H0

Fail to reject H0



Fail to reject H0

Reject H0

Fail to reject H0

= 0.05

0.05

all of it in one tail


An example: One sample z-test

Memory example experiment:• We give a n = 16 memory patients a memory improvement treatment.

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

• After the treatment they have an average score of = 55 memory errors.

€

X

• Step 1: State your hypotheses

H0: the memory treatment sample are the same as those in the population of memory patients.

HA: they aren’t the same as those in the population of memory patients

Treatment = pop = 60

Treatment ≠ pop ≠ 60






€

X

• Step 2: Set your decision criteria

H0: Treatment = pop = 60 HA: Treatment ≠ pop ≠ 60

= 0.05One -tailed






€

X


= 0.05One -tailed

• Step 3: Collect your data






€

X


= 0.05One -tailed

• Step 4: Compute your test statistics

€

zX

=X − μ

X

σX

€

=55 − 60

816

⎛ ⎝ ⎜

⎞ ⎠ ⎟

= -2.5






€

X


= 0.05One -tailed

€

zX

= −2.5


-1-2 1 2

5%

Reject H0






€

X


= 0.05One -tailed

€

zX

= −2.5


- Reject H0

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

the standard normal distribution psy440 june 3, 2008

Documents

z score

standard score

given score

standard neutral score

mean sdstep

deviation scores

formula bar

deviation scorethe distance