Normal Distribution. Intuition The sum of two dice The sum of two dice = 7 The total number of possibilities is : 6x6=36.

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Presentation transcript:

Normal Distribution

Intuition The sum of two dice The sum of two dice = 7 The total number of possibilities is : 6x6=36

Simulations using Monte Carlo Number of trials = 10 Sum Relative frequency (%)

Simulations using Monte Carlo Number of trials = 50 Sum Relative frequency (%)

Simulations using Monte Carlo Number of trials = 200 Sum Relative frequency (%)

Simulations using Monte Carlo Number of trials = 1000 Sum Relative frequency (%)

Simulations using Monte Carlo Number of trials = 5000 Sum Relative frequency (%)

Simulations using Monte Carlo Number of trials = Sum Relative frequency (%)

Simulations using Monte Carlo Number of trials = Sum Relative frequency (%)

Theoretical results SumPossibilities 2(1,1) 3(1,2)(2,1) 4(1,3)(2,2)(3,1) 5(1,4)(2,3)(3,2)(4,1) 6(1,5)(2,4)(3,3)(4,2)(5,1) 7(1,6)(2,5)(3,4)(4,3)(5,2)(6,1) 8(2,6)(3,5)(4,4)(5,3)(6,2) 9(3,6)(4,5)(5,4)(6,3) 10(4,6)(5,5)(6,4) 11(5,6)(6,5) 12(6,6)

Probability%SumPossibilities 1/362,782(1,1) 2/365,563(1,2)(2,1) 3/368,334(1,3)(2,2)(3,1) 4/3611,115(1,4)(2,3)(3,2)(4,1) 5/3613,896(1,5)(2,4)(3,3)(4,2)(5,1) 6/3616,677(1,6)(2,5)(3,4)(4,3)(5,2)(6,1) 5/3613,898(2,6)(3,5)(4,4)(5,3)(6,2) 4/3611,119(3,6)(4,5)(5,4)(6,3) 3/368,3310(4,6)(5,5)(6,4) 2/365,5611(5,6)(6,5) 1/362,7812(6,6) Sum=36/36Sum = 100

Simulations using Monte Carlo Sampling distribution of two dice Sum Relative frequency (%)

Normal 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 observed data by a mathematical formula?

Normal distribution Sampling distribution of two dice Sum Relative frequency (%)

Normal distribution Definition: mathematical function that describes phenomena for a high n. Properties: - Unimodal and symetric (around the mean) - Mode = Median = Mean - Asymptotic of the x-axis  = 50  = 2 Density function

Normal distribution In 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. Cumulative density function x x

Standard 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. Z PDF

Standard normal distribution Table: Area under the curve; multiply by 100 for %

How 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 = 3.70 Mean = 2.93 SD = 0.33

Ex.2 : x = 2.50 mean = 2.93 sd = 0.33 How to use the normal distribution What is the proportion of data that is above a specific score? We do not have negative z-score values. However, using the symmetrical property, we can use its positive counterpart to find the solution

Ex.3 : x 1 = 3.00 ; x 2 = 2.85 mean = 2.93 sd = 0.33 How to use the normal distribution What is the proportion of data that is between two specific scores?

How to use the normal distribution This is the area above What we want is the area below. Now we can substract the area of z2 from z1 => z1-z2

Ex.4 : x = ? Mean = 2.93 SD = 0.33 How to use the normal distribution What is the specific score that has 85% of all scores below it?