Normal distribution

Symbol for mean

µ

µ

Mean

Variance

σ²

Relationship between variance and standard deviation

Variance = standard deviation squared

σ

Standard deviatiom

What percentage of normally distributed data lies within 1 standard deviation of the mean

68%

What percentage of data lies within 2 standard deviations of the mean

95%

What percentage of the data lies within 3 standard deviations of the been

≈100

The normal distribution is written as

X ~ N(µ,σ²)

What does the area under the normal distribution curve between an interval tell us

The probability of getting a value within the interval

One standard deviation

68% and the dist

Standardisation formula

Z = (X - µ)/σ

Standardise P(X>5) where X ~ N(4, 2²)

P(Z > (5-4)/2) = P(Z > 1/2)

Steps to fining the variance or mean given the values of probabilities Eg X ~ N(µ,σ²) where P(X < a) = y and P(X < b) = w

1) standardise the probabilities to Z~N(0,1²) using the formula to get P(Z < (a-µ)/σ) = y and P(Z < (b-µ)/σ) = w

2) find the value of (a-µ)/σ and (b-µ)/σ using the calculator with the given probabilities of y and w and the known variance and mean of standard normal distribution

3) form 2 simultaneous equations and solve them to find the standard deviation and mean

x~N(n,p)

find P(x=y)

0

as X is a continuous random variable and is assigned to a range of values as it could take any one of an infinate number of values on a given interval

in normal distrabution what can be said about the mean median and mode

they are all equal

where are the points of inflection on a normal distribution curve

one standard deviation away from the mean

what can X~B(n,p) as n becomes large be approximated to

X~N(np, np(1-p))