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# The Normal Curve

Class about the Normal Distribution. Definition and applications. How to use Z-scores to find probabilities.
by

## Susana Muniz

on 22 August 2016

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#### Transcript of The Normal Curve

F distribution
T distribution
Chi-square distribution
Normal distribution

Normal distribution:
a theoretical continuous probability distribution of a random variable

Level of measurement
Number of variables
Purpose
Nominal
Ordinal
Interval
Univariate
Bivariate
Multivariate
Descriptive
Inferential
Statistics
THE NORMAL
DISTRIBUTION
By Susana Muniz

AS 313
T-TH 3:00 - 4:00 pm.

To describe groups of people, countries, cars, and so on...
Frequency Distributions
Measures of Central Tendency
Measures of Dispersion
DESCRIPTIVE STATISTICS
INFERENTIAL STATISTICS
To make guesses/ inferences about a population from sample data
To know how much we can rely on such inferences
POPULATION
sample
Normal distribution:
a theoretical continuous probability distribution of a random variable

Frequency distribution:
an arrangement of data that shows all the possible outcomes of a variable and the number of times each outcome is observed
Probability distribution:
an arrangement of data that shows all the possible outcomes of a variable and their probabilities.

Probability: The likelihood of something to happen. The likelihood of an outcome.
%
%
Probabilities go from 0 to 1.

0
means that the is impossible.
1
means that the outcome is certain.

e.g.

Probability of being struck by a lighting 0.000006
Probability of being in prison (males) 0.005
Probability of being female at born 0.5
Probability of dying for human beings 1

The larger the number, the more probable the outcome, and the smaller, the less likely to happen.

=
# of times something can happen

Total observations of all possible outcomes
=
f

n
Discrete
Continuous
Binomial distribution
Hypergeometric distribution
Poisson distribution

Normal distribution:
a theoretical continuous probability distribution of a random variable

Continuous
SHAPE
Theoretical
A characteristic or feature of a case (e.g. a person) can be:

+
Constant:
only 1 possible outcome. e.g. species: homo sapiens

+
Variable:
2 or more possible outcomes. e.g. gender: male/female

Random variable
: when the possible outcomes happen by chance.
Normal distribution:
a theoretical continuous probability distribution of a random variable

A characteristic or feature of a case (e.g. a person) can be:

+
Constant:
only 1 possible outcome

+
Variable:
2 or more possible outcomes

Random variable
: when the possible outcomes happen by chance.
Features:
1) It is unimodal
2) Its mean, median and mode have the same value
3) It is symmetric about its mean
4) It is non-zero over the entire real line

50%
50%
> 50%
<50%
Non symmetrical
So how do we know if an empirical distribution looks like the normal distribution?

FIRST OF ALL, BY VISUAL INSPECTION!

NORMAL OR NOT?
Skewedness:
a measure of symmetry.

If it is normal, its value is Zero.

The less normal it is, the higher or lower (negative) the value is.
Kurtosis:
a measure of "peakdness"

If it is normal, its value is Zero.

The less normal it is, the higher or lower (negative) the value is.
Skwedness= 0.61
Kurtosis= -0.48

Skwedness= -.14
Kurtosis= 0.9

Skwedness= 5.13
Kurtosis= 27.19

Normal distribution:
a theoretical continuous probability distribution of a random variable.

Features:
1) It is unimodal
2) Its mean, median and mode have the same value
3) It is symmetric about its mean
4) It is non-zero over the entire real line

The standard normal distribution instead of using a scale such as years, income, scores, and so on, it uses STANDARD DEVIATION UNITS or Z SCORES
Z-scores:
it is a value in a distribution expressed as the distance from the mean, in standard deviation units.

Why do we bother in transforming original scores into Z-scores?
Because we want to estimate probabilities!

We want to know how likely or unlikely is something to happen!

% ≈ proportions ≈ probabilities
The areas under the normal curve are constant.

Between the mean (z=0) and +1 standard deviation (z=+1) there will be always 34.13% of the total area under the normal curve.
Below a score
Between two scores
Outside two scores
Thank you.
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Find this presentation at:
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MSc. Susana Muniz
AS 313
T - TH
3:00 to 4:00
0.45
0
0.25
Perfect
Empirical
Distribution

Theoretical Distribution
Perfect
In the normal curve the probabilities of finding values between a specific range of standard deviations are constant.
X Z
It comes from real data. It refers to a real phenomenon
It does not refer to a concrete phenomenon. It is an abstraction. A model.
An empirical distribution will never be normal. It will look like normal, or it will be fairly normally distributed, it will be called normal-like, almost-normal... but it will never be completely perfectly normal
Standard normal distribution
Age 4 11 18 25 32 39 46
(years old)
(points)
Exam Scores 60 90 120 150 180 210 240
Std. Deviations -3 -2 -1 0 1 2 3
X=25, s=7
X=150, s=30
(Z scores)
Z =
X- X
s
Z = 32 - 25
7
= 7
7
Z = 1
Find the value of Z for X=32 years old
Find the value of Z for X =210 points
Z = 210 - 150
30
= 60
30
Z = 2
Z scores
Find the value of Z for X=90, given that X= 150 and s=30
Z =
X- X
s
Z = 90- 150
30

Z = - 60
30
Z = -2
X=90
What is the probability
of X=90 or lower?
Between the mean (z=0) and +1 and -1 standard deviations, there will always be 68.26% of the area.
68.26%
P= 0.3413
P= 0.6826
2.14% + 0.13% = 2.28% or
P= 0.0228
What is the probability of
X= 90 or higher?
13.59+ 34.13 +34.13 +13.59+2.14+0.13 = 97.72% or P=0.9772
also 100 - 2.28= 97.72% or
P= 0.9772
Above a score
Find the probability of finding a value between X=100 and X=130.
X=150 and s=30
Z= X-X
s
Z= 100-150
30
Z= -157
30
Z= -1.67
1)
2) Z= 130-150
30
Z= -20
30
Z= -0.67
Look at
The Table!!!!
Figure A.1 Area between Mean and Z
Z (b)
Area between...
...
1.65 0.4505
1.66 0.4515
1.67 0.4525...

Figure A.1 Area between Mean and Z
Z (b)
Area between...
...
0.65 0.2422
0.66 0.2454
0.67 0.2486...

0.4525 - 0.2486 =
What is the value of X at the 90th percentile?
X=150, s=30
10%
90%
20%
30%
40%
60%
70%
80%
Look at
The Table!!!!
Figure A.1 Area between the mean and Z...

Z (b)
Area between...

1.27 0.3980
1.28 0.3997
1.29 0.4015
1.30 0.4032...
X = X+Z(s)
0.40
Median = 50% percentile
Z?
Z= 1.28
X= 150 + 1.28 (30)

X= 150 + 38.4
X= 188.4
X = X- Z(s)
What is the value of X at the 30th percentile?
0.50
0.30 0.20
Z (b)
Area between

0.51 0.1950
0.52 0.1985
0.53 0.2019
Z= 0.52
X = 150 - 0.52 (30)

X = 150-15.6
X= 134.4
Let's do some SPSS!
--> Explore --> Susana Muniz --> The normal curve
0.2486
Z= -1.67
X= 100
Z= -0.67
X= 130
0.4525
Now what do I do?
0.2486
The normal curve.
Definition
Characteristics.

Z-scores.
What are they?
How to transform X to Z-scores and vice-versa

Probabilities
How to calculate probabilities from Z-scores
How to know X and Z-scores from probabilities
Look at
The Table!!!!
Table. Area under the Normal Curve
FIGURE A.1 Area Between Mean and Z
FIGURE A.2 Area Beyond Z
b b
c c
(b) (c)
Area Area
Between Beyond
Z Meand and Z Z

0.00 0.0000 0.5000
0.01 0.0040 0.4960
0.02 0.0080 0.4920
0.03 0.0120 0.4880
...
Find P for Z=-2.00
P=0.0228
2.00
0.4772
0.0228
0.5000
0.5000
P= 0.9772
0.2039
Empirical
Above the mean
Below the mean
Full transcript