Normal Distribution

• Symmetrical (always about the mean)
• Bell shaped curve
• Standard normal
• Area under curve adds up to one

• 68% of data will fall within 1σ of the mean
• 95.4% of data will fall between 2σ of the mean
• 99.7% of data will fall between 3σ of the mean

Rules

P(Z<a) = ϕa

P(Z>a) = 1-ϕa

P(Z<-a) = 1-ϕa

P(Z>-a) = ϕa

P(a<Z<b) = P(Z<b) - P(Z<b)

If the mean is not 0 and/or the standard deviation is not 1 then you will have to standardise first. To do this you have to use the following equation...

(x - μ) ÷ σ = Z

Example Question: P(5<X<6)                                       X ~ N(5.8,0.5²⁾                    

First standardise: P[X< (6-5.8)/0.5] = 0.4                       P[X< (5-5.8)/0.5] = 1.6

P(X<6) - P(X<5)

P(X<0.4) - P(X<1.6)

= ϕ0.4 - (1-ϕ1.6)

= 0.6554 - 0.0548

= 0.6006            

60.06%