2. Objectives
Introduce the Gaussian Distribution
Properties of the Standard Gaussian
Distribution
Introduce the Central Limit Theorem
Use Gaussian Distribution in an
inferential fashion
3. Theoretical Distribution
Empirical distributions
based on data
Theoretical distribution
based on mathematics
derived from model or estimated from data
4. Gaussian Distribution
Why are Gaussian distributions so important?
Many dependent variables are commonly
assumed to be normally distributed in the
population
If a variable is approximately normally
distributed we can make inferences about
values of that variable
Example: Sampling distribution of the mean
5. So what?
Remember the Binomial distribution
With a few trials we were able to calculate
possible outcomes and the probabilities of those
outcomes
Now try it for a continuous distribution with an infinite
number of possible outcomes. Yikes!
The Gaussian distribution and its properties are well
known, and if our variable of interest is normally
distributed, we can apply what we know about the
Gaussian distribution to our situation, and find the
probabilities associated with particular outcomes
6. Gaussian Distribution
Symmetrical, bell-shaped curve
Also known as normal distribution
Point of inflection = 1 standard deviation
from mean
Mathematical formula
f(X)=
1
σ2π
(e)
−
(X−µ)2
2σ2
7. Since we know the shape of the curve, we can
calculate the area under the curve
The percentage of that area can be used to
determine the probability that a given value
could be pulled from a given distribution
The area under the curve tells us about the probability-
in other words we can obtain a p-value for our result
(data) by treating it as a normally distributed data
set.
8. Key Areas under the Curve
For Gaussian
distributions
+ 1 SD ~ 68%
+ 2 SD ~ 95%
+ 3 SD ~ 99.9%
10. Problem:
Each Gaussian distribution with its own
values of m and s would need its own
calculation of the area under various points
on the curve
11. Gaussian Probability Distributions
Standard Normal Distribution – N(0,1)
We agree to use the
standard Gaussian
distribution
Bell shaped
µ=0
σ=1
Note: not all bell shaped
distributions are normal
distributions
12. Gaussian Probability Distribution
Can take on an
infinite number of
possible values.
The probability of
any one of those
values occurring is
essentially zero.
Curve has area or
probability = 1
13. Gaussian Distribution
The standard Gaussian distribution will
allow us to make claims about the
probabilities of values related to our own
data
How do we apply the standard Gaussian
distribution to our data?
14. Z-score
If we know the population mean and
population standard deviation, for any
value of X we can compute a z-score by
subtracting the population mean and
dividing the result by the population
standard deviation
z=
X−µ
σ
15. Important z-score info
Z-score tells us how far above or below the mean a value is
in terms of standard deviations
It is a linear transformation of the original scores
Multiplication (or division) of and/or addition to (or
subtraction from) X by a constant
Relationship of the observations to each other remains
the same
Z = (X-m)/s
then
X = sZ + m
[equation of the general form Y = mX+c]
16. Probabilities and z scores: z tables
Total area = 1
Only have a probability from width
For an infinite number of z scores each point has a
probability of 0 (for the single point)
Typically negative values are not reported
Symmetrical, therefore area below negative value =
Area above its positive value
Always helps to draw a sketch!
17. Probabilities are depicted by areas under the curve
Total area under the curve is
1
The area in red is equal to
p(z > 1)
The area in blue is equal to
p(-1< z <0)
Since the properties of the
normal distribution are
known, areas can be looked
up on tables or calculated on
computer.
18. Strategies for finding probabilities for the
standard normal random variable.
Draw a picture of standard normal
distribution depicting the area of
interest.
Re-express the area in terms of
shapes like the one on top of the
Standard Normal Table
Look up the areas using the table.
Do the necessary addition and
subtraction.
19. Suppose Z has standard normal
Guassian Find p(0<Z<1.23)
23. Example
Data come from distribution: m = 10, s =
3
What proportion fall beyond X=13?
Z = (13-10)/3 = 1
=normsdist(1) or table ⇒ 0.1587
15.9% fall above 13
24. Example: IQ
A common example is IQ
IQ scores are theoretically normally
distributed.
Mean of 100
Standard deviation of 15
25. IQ’s are normally distributed with mean 100 and standard
deviation 15. Find the probability that a randomly selected
person has an IQ between 100 and 115
(100 115)
(100 100 100 115 100)
100 100 100 115 100
(
15 15 15
(0 1) .3413
P X
P X
X
P
P Z
< < =
− < − < − =
− − −
< < =
< < =
26. Say we have GRE scores are normally distributed with mean 500 and
standard deviation 100. Find the probability that a randomly selected
GRE score is greater than 620.
We want to know what’s the probability of
getting a score 620 or beyond.
p(z > 1.2)
Result: The probability of randomly getting a
score of 620 is ~.12
620 500
1.2
100
z
−
= =
27. Work time...
What is the area for scores less than z = -1.5?
What is the area between z =1 and 1.5?
What z score cuts off the highest 30% of the
distribution?
What two z scores enclose the middle 50% of the
distribution?
If 500 scores are normally distributed with mean = 50
and SD = 10, and an investigator throws out the 20
most extreme scores, what are the highest and lowest
scores that are retained?
28. Standard Scores
Z is not the only transformation of scores to
be used
First convert whatever score you have to a z
score.
New score – new s.d.(z) + new mean
Example- T scores = mean of 50 s.d. 10
Then T = 10(z) + 50.
Examples of standard scores: IQ, GRE, SAT
29. Wrap up
Assuming our data is normally distributed
allows for us to use the properties of the normal
distribution to assess the likelihood of some
outcome
This gives us a means by which to determine
whether we might think one hypothesis is more
plausible than another (even if we don’t get a
direct likelihood of either hypothesis)
