the normal distribution. distribution – any collection of scores, from either a sample or...
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The Normal Distribution
The Normal Distribution
Distribution – any collection of scores, from either a sample or population
Can be displayed in any form, but is usually represented as a histogram
Normal Distribution – specific type of distribution that assumes a characteristic bell shape and is perfectly symmetrical
The Normal Distribution
Can provide us with information on likelihood of obtaining a given score 60 people scored a 6 – 6/350 = .17 = 17% 9 people scored a 1 – 3%
The Normal Distribution
Why is the Normal Distribution so important? Almost all of the statistical tests that we will be
covering (Z-Tests, T-Tests, ANOVA, etc.) throughout the course assume that the population distribution, that our sample is drawn from (but for the variable we are looking at), is normally distributed
Also, many variables that psychologists and health professionals look at are normally distributed
Why this is requires a detailed examination of the derivation of our statistics, that involves way more detail than you need to use the statistic.
The Normal Distribution
Ordinate
Density – what is measured on the ordinate (more on this in Ch. 7)
Abscissa
The Normal Distribution
Mathematically defined as:
Since and e are constants, we only have to determine μ (the population mean) and σ (the population standard deviation) to graph the mathematical function of any variable we are interested in
Don’t worry, understanding this is not necessary to understanding the normal distribution, only a helpful aside for the mathematically inclined
22/2)(2
1)(
XeXf
The Normal Distribution
Using this formula, mathematicians have determined the probabilities of obtaining every score on a “standard normal distribution” (see Table E.10 in your book)To determine these probabilities for the variable you’re interested in we must plug in your variable to the formula Note: This assumes that your variable fits a
normal distribution, if not, your results will be inaccurate
The Normal Distribution
However, this table refers to a Standard Normal Distribution Μ = 0; σ = 1
How do you get your variable to fit?
X
z
The Normal Distribution
Z-Scores Range from +∞ to -∞ Represent the number of standard
deviations your score is from the mean i.e. z = +1 is a score that is 1 standard deviation
above the mean and z = -3 is a score 3 standard deviations below the mean
Now we can begin to use the table to determine the probability that our z score will occur using table E.10
The Normal Distribution
The Normal Distribution
Mean to Z
z
3.503.00
2.502.00
1.501.00
.500.00
-.50-1.00
-1.50-2.00
-2.50-3.00
-3.50-4.00
Normal Distribution
Cutoff at +1.6451200
1000
800
600
400
200
0
The Normal Distribution
The Normal Distribution
Larger Portion
z
3.503.00
2.502.00
1.501.00
.500.00
-.50-1.00
-1.50-2.00
-2.50-3.00
-3.50-4.00
Normal Distribution
Cutoff at +1.6451200
1000
800
600
400
200
0
The Normal Distribution
Smaller Portion
z
3.503.00
2.502.00
1.501.00
.500.00
-.50-1.00
-1.50-2.00
-2.50-3.00
-3.50-4.00
Normal Distribution
Cutoff at +1.6451200
1000
800
600
400
200
0
Reminder: Z-Scores represent # of standard deviations from the mean For this distribution, if μ = 50 and σ = 10,
what score does z = -3 represent? z = +2.5?
z
3.503.00
2.502.00
1.501.00
.500.00
-.50-1.00
-1.50-2.00
-2.50-3.00
-3.50-4.00
Normal Distribution
Cutoff at +1.6451200
1000
800
600
400
200
0
z = -.1 z = 1.645(z = -.1, “Mean to Z”) + (z = 1.645, “Mean to Z”).0398 + .4500 = .4898 = 49%
z = -1.00, “Smaller Portion” = Red + Bluez = -1.645, “Smaller Portion” = Blue(Red + Blue) - Blue = Red
z = -1.645 z = -1.00(z = -1.00, “Smaller Portion”) – (z = -1.645,
“Smaller Portion”).1587 - .0500 = .1087 = 11%
The Normal Distribution
What are the scores that lie in the middle 50% of a distribution of scores with μ = 50 and σ = 10? Look for “Smaller Portion” = .2500
on Table E.10 z = .67 Solve for X using z-score formula Scores = 56.7 and 43.3
The Normal Distribution
3.437.56
507.6
7.650
)10(67.5010
5067.
andX
X
X
X
X
Xz
The Normal Distribution
Other uses for z-scores:1. Converting two variable to a standard metric
You took two exams, you got an 80 in Statistics and a 50 in Biology – you cannot say which one you did better in without knowing about the variability in scores in each If the class average in Stats was a 90 and the s.d. 15,
what would we conclude about your score now? How is it different than just using the score itself?
If the mean in Bio was a 30 and the s.d. was a 5, you did 4 s.d’s above the mean (a z-score of +4) or much better than everyone else
The Normal Distribution
Other uses for z-scores:1. Converting variables to a standard metric
This also allows us to compare two scores on different metrics i.e. two tests scored out of 100 = same metric
one test out of 50 vs. one out of 100 = two different metrics
Is 20/50 better than 40/100? Is it better when compared to the class average?
2. Allows for quick comparisons between a score and the rest of the distribution it is a part of
The Normal Distribution
Standard Scores – scores with a predetermined mean and standard deviation, i.e. a z-scoreWhy convert to standard scores? You can compare performance on two different
tests with two different metrics You can easily compute Percentile ranks but they are population-relative!
Percentile – the point below which a certain percent of scores fall i.e. If you are at the 75th%ile (percentile), then 75%
of the scores are at or below your score
The Normal Distribution
How do you compute %ile? Convert your raw score into a z-score Look at Table E.10, and find the “Smaller Portion” if your
z-score is negative and the “Larger Portion” if it is positive
Multiply by 100
z
3.503.00
2.502.00
1.501.00
.500.00
-.50-1.00
-1.50-2.00
-2.50-3.00
-3.50-4.00
Normal Distribution
Cutoff at +1.6451200
1000
800
600
400
200
0
The Normal Distribution
New Score = New s.d. (z) + New Mean
New IQ Score = 15 (2) + 100 = 130
T-Score – commonly used standardized normal distribution w/ mean = 50 and s.d. = 10
T-Score = 10 (2) + 50 = 70