Normal Distribution. Intuition The sum of two dice The sum of two dice = 7 The total number of...
-
Upload
basil-simpson -
Category
Documents
-
view
218 -
download
0
Transcript of Normal Distribution. Intuition The sum of two dice The sum of two dice = 7 The total number of...
Normal DistributionNormal Distribution
IntuitionIntuition
The sum of two diceThe sum of two dice
= 7= 7
The total number of possibilities is : The total number of possibilities is : 6x6=366x6=36
Simulations using Monte CarloSimulations using Monte Carlo
Number of trials = 10
SumSum
Rela
tive f
requency
(%
)R
ela
tive f
requency
(%
)
Simulations using Monte CarloSimulations using Monte Carlo
Number of trials = 50
SumSum
Rela
tive f
requency
(%
)R
ela
tive f
requency
(%
)
Simulations using Monte CarloSimulations using Monte Carlo
Number of trials = 200
SumSum
Rela
tive f
requency
(%
)R
ela
tive f
requency
(%
)
Simulations using Monte CarloSimulations using Monte Carlo
Number of trials = 1000
SumSum
Rela
tive f
requency
(%
)R
ela
tive f
requency
(%
)
Simulations using Monte CarloSimulations using Monte Carlo
Number of trials = 5000
SumSum
Rela
tive f
requency
(%
)R
ela
tive f
requency
(%
)
Simulations using Monte CarloSimulations using Monte Carlo
Number of trials = 50 000
SumSum
Rela
tive f
requency
(%
)R
ela
tive f
requency
(%
)
Simulations using Monte CarloSimulations using Monte Carlo
Number of trials = 200 000
SumSum
Rela
tive f
requency
(%
)R
ela
tive f
requency
(%
)
Theoretical resultsTheoretical results
SumSum PossibilitiesPossibilities
22 (1,1)(1,1)
33 (1,2)(1,2) (2,1)(2,1)
44 (1,3)(1,3) (2,2)(2,2) (3,1)(3,1)
55 (1,4)(1,4) (2,3)(2,3) (3,2)(3,2) (4,1)(4,1)
66 (1,5)(1,5) (2,4)(2,4) (3,3)(3,3) (4,2)(4,2) (5,1)(5,1)
77 (1,6)(1,6) (2,5)(2,5) (3,4)(3,4) (4,3)(4,3) (5,2)(5,2) (6,1)(6,1)
88 (2,6)(2,6) (3,5)(3,5) (4,4)(4,4) (5,3)(5,3) (6,2)(6,2)
99 (3,6)(3,6) (4,5)(4,5) (5,4)(5,4) (6,3)(6,3)
1010 (4,6)(4,6) (5,5)(5,5) (6,4)(6,4)
1111 (5,6)(5,6) (6,5)(6,5)
1212 (6,6)(6,6)
Theoretical resultsTheoretical results
ProbabilityProbability %% SumSum PossibilitiesPossibilities
1/361/36 2,782,78 22 (1,1)(1,1)
2/362/36 5,565,56 33 (1,2)(1,2) (2,1)(2,1)
3/363/36 8,338,33 44 (1,3)(1,3) (2,2)(2,2) (3,1)(3,1)
4/364/36 11,1111,11 55 (1,4)(1,4) (2,3)(2,3) (3,2)(3,2) (4,1)(4,1)
5/365/36 13,8913,89 66 (1,5)(1,5) (2,4)(2,4) (3,3)(3,3) (4,2)(4,2) (5,1)(5,1)
6/366/36 16,6716,67 77 (1,6)(1,6) (2,5)(2,5) (3,4)(3,4) (4,3)(4,3) (5,2)(5,2) (6,1)(6,1)
5/365/36 13,8913,89 88 (2,6)(2,6) (3,5)(3,5) (4,4)(4,4) (5,3)(5,3) (6,2)(6,2)
4/364/36 11,1111,11 99 (3,6)(3,6) (4,5)(4,5) (5,4)(5,4) (6,3)(6,3)
3/363/36 8,338,33 1010 (4,6)(4,6) (5,5)(5,5) (6,4)(6,4)
2/36 2/36 5,565,56 1111 (5,6)(5,6) (6,5)(6,5)
1/361/36 2,782,78 1212 (6,6)(6,6)
SumSum
=36/36=36/36
SumSum
= 100= 100
Simulations using Monte CarloSimulations using Monte Carlo
Sampling distribution of two dice
SumSum
Rela
tive f
requency
(%
)R
ela
tive f
requency
(%
)
Normal distributionNormal distribution
It is one type of distribution encountered often in the empirical world (heights, weights, abilities, psychological properties, etc.)
Is there a way to express those Is there a way to express those observed data by a mathematical observed data by a mathematical
formula?formula?
Normal distributionNormal distribution
Sampling distribution of two dice
SumSum
Rela
tive f
requency
(%
)R
ela
tive f
requency
(%
)
Normal distributionNormal distributionDefinition: mathematical function that describes phenomena
for a high n.
Properties: - Unimodal and symetric (around the mean)
- Mode = Median = Mean
- Asymptotic of the x-axis
2
2
( )
2
2
1( )
2
x
f x e
= 50 = 2Density function
Normal distributionNormal distributionIn order to know the probability of a score x under the normal curve, we have to compute the area under the curve (integrate) from -∞ to x.
2
2
( )
2
2
1( )
2
xx
g x e
Cumulative density function
x
x
Standard normal distributionStandard normal distribution
There is an infinity of values that can be used for and , parameters. Therefore, a normal distribution will be called standard if = 0 and = 1.
ZZ
PD
FPD
F
Standard normal distributionStandard normal distributionTable: Area under the curve; multiply by 100 for %
How to use the normal distributionHow to use the normal distribution
Solution: Z score transformation (number of standard deviations between x and the mean)
What is the proportion of data that is below a specific score?
Ex.1 : x = 3.70Mean = 2.93SD = 0.33
x xz
s
33.2
33.0
93.27.3
01.99)33.2(Table
Ex.2 : x = 2.50mean = 2.93sd = 0.33
x xz
s
How to use the normal distributionHow to use the normal distribution
What is the proportion of data that is above a specific score?
32.90)30.1(Table We do not have negative z-score values. However, using the symmetrical property, we can use its positive counterpart to find the solution
30.133.0
93.25.3
Ex.3 : x1 = 3.00 ; x2 = 2.85 mean = 2.93sd = 0.33
11
x xz
s
How to use the normal distributionHow to use the normal distribution
What is the proportion of data that is between two specific scores?
21.033.0
93.200.3
32.58)21.0(Table
22
x xz
s
How to use the normal distributionHow to use the normal distribution
24.033.0
93.285.2
48.59)24.0(Table
This is the area above 2.85. What we want is the area below.
52.4048.59100
Now we can substract the area of z2 from z1 => z1-z2
8.1752.4032.58
Ex.4 : x = ?Mean = 2.93SD = 0.33
x xz
ssz x x
x sz x
How to use the normal distributionHow to use the normal distribution
What is the specific score that has 85% of all scores below it?
04.1zTable85
27.393.204.133.0 xszx