The Normal Distribution (AQA Level 3 Mathematical Studies (Core Maths): Paper 2A: Statistical Techniques): Revision Note

Properties of the Normal Distribution

The normal distribution is an example of a continuous probability distribution

What is a continuous random variable?

• A continuous random variable (often abbreviated to CRV) is a random variable that can take any value within a range of infinite values

  • Continuous random variables usually measure something

  • For example, height, weight, time, etc

What is a continuous probability distribution?

• A continuous probability distribution is a probability distribution in which the random variable $\mathit{X}$ is continuous

• The probability of $X$ being a particular value is always zero

  • $\mathrm{P}\left(X=k\right)=0$ for any value $k$

  • Instead we define the probability density function $f\left(x\right)$ for a specific value

  • We talk about the probability of $X$ being within a certain range

• A continuous probability distribution can be represented by a continuous graph (the values for $X$ along the horizontal axis and probability density on the vertical axis)

• The area under the graph between the points $x=a$ and $x=b$ is equal to $\mathrm{P}\left(a\le X\le b\right)$

  • The total area under the graph equals 1

• As $\mathrm{P}\left(X=k\right)=0$ for any value $k$, it does not matter if we use strict or weak inequalities

  • $\mathrm{P}\left(X\le k\right)=\mathrm{P}\left(X<k\right)$ for any value $k$

What is a normal distribution?

• A normal distribution is a continuous probability distribution

• The continuous random variable can follow a normal distribution if:

  • The distribution is symmetrical

  • The distribution is bell-shaped

• If X follows a normal distribution then it is denoted $X~\mathrm{N}(\mu ,{\sigma }^2)$

  • μ is the mean

  • σ² is the variance

  • σ  is the standard deviation

• If the mean changes then the graph is translated horizontally

• If the variance changes then the graph is stretched horizontally

  • A small variance leads to a tall curve with a narrow centre

  • A large variance leads to a short curve with a wide centre

What are the important properties of a normal distribution?

• The mean is μ

• The variance is σ²

  • If you need the standard deviation remember to square root this

• The normal distribution is symmetrical about x = μ

  • Mean = Median = Mode = μ

• The normal distribution curve has two points of inflection

  • x = μ ± σ (one standard deviation away from the mean)

• There are some useful approximate results:

  • Approximately two-thirds (68%) of the data lies within one standard deviation of the mean (μ ± σ)

  • Approximately 95% of the data lies within two standard deviations of the mean (μ ± 2σ)

  • Nearly all of the data (99.7%) lies within three standard deviations of the mean (μ ± 3σ)

Standard Normal Distribution

What is the standard normal distribution? 

• The standard normal distribution is a normal distribution where the mean is 0 and the standard deviation is 1

  • It is denoted by Z

  • $Z~\mathrm{N}(0,1)$

  • The area under the curve is 1

Why is the standard normal distribution important?

• Any normal distribution curve can be transformed to the standard normal distribution curve by a horizontal translation and a horizontal stretch

• Therefore we have the relationship:

  • $z=\frac{x-\mu }{\sigma }$

  • Where $X~N(\mu ,{\sigma }^2)$ and $Z~\mathrm{N}(0,1)$

• So for any value x, a z-score (or z-value) can be calculated which measures how many standard deviations x is away from the mean

• Finding the z-value is also known as standardising the variable

• Probabilities are related by:

  • $\mathrm{P}(X<a)=\mathrm{P}\left(Z<\frac{a-\mu }{\sigma }\right)$ 

  • This will be useful when the mean or variance is unknown

• If a value of x is less than the mean then the z-value will be negative

• You can use the notation $\mathrm{\Phi }\left(z\right)$ which just means $\mathrm{P}(Z<z)$