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The Normal Distribution Data can be "distributed" (spread out) in different ways. It can be spread out more on the left Or more on the right Or it can be all jumbled up The Normal Distribution A normal distribution is a very important statistical data distribution pattern occurring in many natural and social phenomena. The Normal Distribution A normal distribution is characterized by its mean (μ) and variance (σ2) f(X) Changing μ shifts the distribution left or right. σ µ Changing σ increases or decreases the spread. X A normal distribution is characterized by its mean (μ) and variance (σ2) - Changing μ shifts the distribution left or right. - Changing σ increases or decreases the spread. f(X) σ µ X The Normal Distribution - Mean - Variance - Standard deviation - Standard error of the mean 1 n µ = ! xi n i n 1 2 var(x) = ! = (xi " µ ) ! n "1 i n 1 2 std(x) = var(x) = ! = (x " µ ) ! i i n "1 ! SEM = n 2 The Normal Distribution: as mathematical function (pdf) f ( x) = 1 σ 2π Note constants: π=3.14159 e=2.71828 1 x−µ 2 − ( ) ⋅e 2 σ This is a bell shaped curve with different centers and spreads depending onσand onμ The Normal PDF It s a probability density function (PDF), so no matter what the values of σand μ, must integrate to 1! +∞ ∫σ −∞ 1 2π 1 x−µ 2 − ( ) ⋅ e 2 σ dx =1 The beauty of the normal curve: The area between µ-σ and µ+σ is about 68%; the area between µ-2σ and µ+2σ is about 95%; and the area between µ-3σ and µ+3σ is about 99.7%. Almost all values fall within 3 standard deviations. The beauty of the normal curve: The area between µ-σ and µ+σ is about 68%; the area between µ-2σ and µ+2σ is about 95%; and the area between µ-3σ and µ+3σ is about 99.7%. Almost all values fall within 3 standard deviations. The beauty of the normal curve: The area between µ-σ and µ+σ is about 68%; the area between µ-2σ and µ+2σ is about 95%; and the area between µ-3σ and µ+3σ is about 99.7%. Almost all values fall within 3 standard deviations. The Standard Normal Distribution (Z) All normal distributions can be converted into the standard normal curve by subtracting the mean and dividing by the standard deviation: Z= X −µ σ Somebody calculated all the integrals for the standard normal and put them in a table! So we never have to integrate! Even better, computers now do all the integration. The Normal Distribution 1 n µ = ! i xi n n 1 2 != (x ! µ ) " i n !1 i μ-3σμ-2σ μ-σ μ+3σ μ+2σ μ+σ μ Z= -3 -2 -1 0 1 2 3 X −µ Z- Score σ