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

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Simulations using Monte CarloSimulations using Monte Carlo

Number of trials = 50

SumSum

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Simulations using Monte CarloSimulations using Monte Carlo

Number of trials = 200

SumSum

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Simulations using Monte CarloSimulations using Monte Carlo

Number of trials = 1000

SumSum

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Simulations using Monte CarloSimulations using Monte Carlo

Number of trials = 5000

SumSum

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Simulations using Monte CarloSimulations using Monte Carlo

Number of trials = 50 000

SumSum

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Simulations using Monte CarloSimulations using Monte Carlo

Number of trials = 200 000

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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

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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

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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