(10) Normal Distribution

Continuous random variable

Can take one of infinitely many values

Area under continuous probability distribution

1

Characteristics of normal distribution

• Bell-shaped curve with asymptotes at each end

• Symmetrical about the mean (mean = median = mode)

• Total area under curve = 1

• Points of inflection at µ ± σ

%s of data within standard deviations

68% ± 1σ

95% ± 2σ

99.7% ± 3σ

(± σ of the mean)

Standard normal distribution Z

µ = 0

σ = 1

Z

x-µ / σ

When can a normal distribution be used to approximate a binomial distribution

• n is large

• p is close to 0.5

How to approximate binomial dist. using normal

µ = np

σ = (np(1-p))^1/2

When are continuity corrections needed

When using normal to approximate binomial

Sample mean normal distribution

X̄~N(µ, σ²/n)

Standardised sample mean normal distribution

Z~N(0,1)

Z = X̄ - µ / (σ/(n)^1/2)

Continuity corrections

P(X≤k) = P(Y<k+0.5)

P(X<k) = P(Y<k−0.5)

P(X≥k) = P(Y>k−0.5)

P(X>k) = P(Y>k+0.5)

P(X=k) = P(k−0.5<Y<k+0.5)