S1 – Normal Distribution Summary
S1 – Normal Distribution Summary
 A Continuous random variable (CRV) is a continuous set of
characteristics measured/observed in a trial, which may be subject to
random variation, e.g. length, time and weight.
 The probability density function (pdf) of a CRV is the curve
representing the shape of the distribution.
 The area under the curve gives probabilities. Total area equals 1. Since
the variable is continuous P(X  r) = P(X < r) and P(X  r) = P(X > r).
 The pdf of a Normal distribution is bell shaped and symmetrical about
the mean
 if the CRV X has a normal distribution then X  N(μ, σ2)
 Z  N(0, 1) is called the standard normal distribution
 Table 3 in the formula book and the Appendix of the textbook give
probabilities of P(Z  z), i.e. the left tail
 The table is for positive values of Z only, use symmetry for negatives.
 For other distributions transform the CRV X into Z by linear scaling:
 A Continuous random variable (CRV) is a continuous set of
characteristics measured/observed in a trial, which may be subject to
random variation, e.g. length, time and weight.
 The probability density function (pdf) of a CRV is the curve
representing the shape of the distribution.
 The area under the curve gives probabilities. Total area equals 1. Since
the variable is continuous P(X  r) = P(X < r) and P(X  r) = P(X > r).
 The pdf of a Normal distribution is bell shaped and symmetrical about
the mean
 if the CRV X has a normal distribution then X  N(μ, σ2)
 Z  N(0, 1) is called the standard normal distribution
 Table 3 in the formula book and the Appendix of the textbook give
probabilities of P(Z  z), i.e. the left tail
 The table is for positive values of Z only, use symmetry for negatives.
 For other distributions transform the CRV X into Z by linear scaling:
Z
X 

. On each diagram now have two axes; for X and for Z.
Z
X 

. On each diagram now have two axes; for X and for Z.
 When you know the probability P(Z < z) use Table 4 to find z then
calculate x to which it corresponds using the above formula.
 The table shows probabilities of 0.5 and above, for others use the
symmetry properties of the distribution.
 About 99.7% of data lie within μ  3σ, 95.5% lie within μ  2σ and 68%
of data lie within μ  σ.
 The Central Limit Theorem: For a large sample of n independent
observations of a random variable X, where X has a distribution with a
 When you know the probability P(Z < z) use Table 4 to find z then
calculate x to which it corresponds using the above formula.
 The table shows probabilities of 0.5 and above, for others use the
symmetry properties of the distribution.
 About 99.7% of data lie within μ  3σ, 95.5% lie within μ  2σ and 68%
of data lie within μ  σ.
 The Central Limit Theorem: For a large sample of n independent
observations of a random variable X, where X has a distribution with a
mean of μ and a variance σ2, the distribution of the sample mean X is
mean of μ and a variance σ2, the distribution of the sample mean X is

given approximately by X  N   ,

of X.
 The standard deviation

n
2 
 , irrespective of the distribution
n 

given approximately by X  N   ,

of X.
is called the standard error.
 When X follows a normal distribution the distribution of X is exactly
normal for all values of n. For other distributions n  30 is recommended
to give a good approximation.
 ALWAYS DRAW A DIAGRAM AND SHADE WHAT YOU ARE
INTERESTED IN!
 The standard deviation

n
2 
 , irrespective of the distribution
n 
is called the standard error.
 When X follows a normal distribution the distribution of X is exactly
normal for all values of n. For other distributions n  30 is recommended
to give a good approximation.
 ALWAYS DRAW A DIAGRAM AND SHADE WHAT YOU ARE
INTERESTED IN!