Many naturally occurring phenomena can be approximated using a continuous random variable called The Normal Distribution. For example, the heights of females or the weights of Russian ballet dancers.
In a normal distribution, much of the data is gathered around the mean. The distribution has a characteristic 'bell shape' symmetrical about the mean.
Remember we write the symbol, , to stand for the mean. The area of the bell shape = 1.
In any normal distribution, approximately 68% of the data will lie within one standard deviation of the mean. The standard deviation is an important measure of the spread of our data. The greater the standard deviation, the greater our spread of data.
We write the symbol, , to stand for the standard deviation. The standard deviation squared gives us the variance: Var(X) = 2
If X has a normal distribution with mean, , and variance, 2, then we write: X ~ N( , 2)
We will look closely at the normal distribution Z, with mean, = 0, and variance = 1.
Z ~ N(0, 1)
Suppose for this distribution we wanted to calculate the P(Z < 1). Unlike with our discrete random variables we don't have a formula to work this out (there is one but it's way beyond the scope of A-Level maths!). The values of these probabilities have already been calculated and are tabulated in most statistics books.
With Z ~ N(0, 1) the P(Z = z) = (z)
Don't be put off with the Greek letter (phi). (z) just describes the area under the bell from that point!
X ~ N(, 2) and standardising
The only values of the normal distribution that are tabulated are from Z ~ N(0, 1). Not many distributions will have a mean of 0 and a variance of 1 however, so we need to convert any normal distribution of X into the normal distribution of Z!
This is done using the formula: Z = X -
Where is the mean and is the standard deviation.