Normal Distributions

normal distribution sketch graph

conditions that need to be met

• data is continuous

• distribution is bell-shaped + symmetrical

• mode + median + mean are approximatly equal

what two things does a normal distibution have

• mean (μ)

• standard deviation (σ)

what do different values of μ + σ give

different normal distributions

3 main important percentages

• 68%

• 95%

• virtually all (99.8%) observations

where does 68% of the observation lies within

_+ one standard deviation of the mean

where does 95% of the observation lies within

_+ two standard deviation of the mean

where does virtually all (99.8%) of the observation lies within

_+ three standard deviation of the mean

graph showing previous statments

between μ and μ + 2σ

lies 47.5% if the area lies on right side of mean because of symmetry

between μ and μ - 2σ

lies 47.5% if the area lies on left side of mean because of symmetry

between μ and μ + 3σ

lies 49.9% if the area lies on right side of mean because of symmetry

between μ and μ - 2σ

lies 49.9% if the area lies on left side of mean because of symmetry

what is the variance

measure of how spread out the data is

what is the variance

(standard deviation)²

notation

N(μ, σ²)

what is μ

mean

what is σ²

variance

number of s.d from mean

value - mean

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standard deviation

when question says ‘between 2σ - 3σ’ what do you do for the percentages

• do 49.9% - 47.5% = 2.4%

• or 99.8% - 95% = 4.8% but then 4.8/2 + = 2.4%

• (because thats how you find out only one side)

when question says ‘above 2σ’ what do you do for the percentages

• 50% - 47% = 2.5%

• or 100% - 95% = 5 and then 5/2 = 2.5%

• (because you need to find out 1 side)

example question