Edexcel IAL - Statistics 1- 6.1 Normal Distribution, Mean and Variance- Study notes  - New syllabus

The Normal Distribution

The normal distribution is a continuous probability distribution that is widely used to model real-life data such as heights, weights, measurement errors, and examination marks.

A random variable that follows a normal distribution is written as  

\( X \sim N(\mu, \sigma^2) \)

where:

\( \mu \) is the mean

\( \sigma^2 \) is the variance

\( \sigma \) is the standard deviation

Shape and Symmetry

The normal distribution has the following key features:

• It is symmetric about the mean \( \mu \)
• The mean, median, and mode are equal
• The curve is bell-shaped

Because of the symmetry, probabilities on either side of the mean are equal.

Mean and Variance

For a normally distributed random variable \( X \sim N(\mu, \sigma^2) \):

\( E(X) = \mu \)

\( \mathrm{Var}(X) = \sigma^2 \)

These results may be used directly without proof.

Standardisation

Probabilities for the normal distribution are found by converting values of \( X \) to a standard normal variable \( Z \), where

\( Z = \dfrac{X – \mu}{\sigma} \)

The standard normal distribution is written as

\( Z \sim N(0,1) \)

Use of Normal Distribution Tables

Tables give values of the cumulative distribution function

\( P(Z \leq z) \)

To find probabilities involving \( X \):

• Convert values of \( X \) to \( Z \) using standardisation
• Use the table to find probabilities for \( Z \)
• Use symmetry when appropriate

Interpolation is not required in this syllabus.

Z- table value:

Using Symmetry

For the standard normal distribution:

\( P(Z \leq 0) = 0.5 \)

\( P(Z \leq -z) = 1 – P(Z \leq z) \)

These properties are frequently used to simplify calculations.

Solving Problems Involving Equations

Some questions may require:

• Finding unknown values of \( \mu \) or \( \sigma \)
• Solving simultaneous equations involving probabilities

These problems rely on correct use of standardisation and normal tables, not on derivation of formulas.