The normal distribution is a continuous probability distribution that can be presented on a graph. Continuous probability distributions represent continuous random variables, which can take one or more values. Some examples of continuous variables that can be displayed on a normal distribution graph are:
Height
Weight
Measurement errors
The normal distribution curve
The bell-shaped curve on the normal distribution graph is symmetrical to the mean, which can be represented by the symbol \(\mu\). The area under the curve is equal to 1. Below is an example of what the normal distribution graph looks like:
Normal distribution graph
The standard deviation tells you how spread out the data are. When this is calculated from the curve above, it can tell you certain things about the data:
68% of the data fall within one standard deviation from the mean, making the probability likely.
95% of the data fall within two standard deviations of the mean, making the probability very likely.
Nearly all the data, 99.7% of the data fall within three standard deviations of the mean, making the probability almost certain.
Standard deviation is a measurement of how spread out the data is, and it can be notated with the symbol \(\sigma\).
What is the normal distribution formula?
When looking at normal distribution, the notation can be written as:
\[X \sim N(\mu, \sigma^2)\]
\(\mu\) = the mean \(\sigma^2\) = the population variance
When you are given this notation, it gives you the information needed to create a normal distribution curve.
The distribution of X is modelled as \(X \sim N(23, 0.25^2)\). Sketch the distribution of X.
To do this, you can start by identifying the mean and the population variance:
\[\mu = 23 \qquad \sigma^2 = 0.25\]
You know that on a normal distribution graph, the curve is symmetrical about the mean, which allows you to draw the bell shape:
Bell curve
The lengths of wingspans are normally distributed with a mean of 22 cm and a standard deviation of 0.4. Sketch the distribution of X.
To do this, you know that the bell-shaped curve will be symmetrical about the mean; therefore, you can sketch the graph as follows:
Worked example
Normal distribution can help you to find probabilities, and to do this you can use the normal cumulative function on your calculator.
What is the standard normal distribution?
Standard normal distribution is a way of standardising normal distribution. It has a mean, \(\mu\), of 0 and a standard deviation, \(\sigma\), of 1. The notation for the standard normal distribution can be written as:
\[Z \sim N(0,1^2)\]
Normal distribution - Key takeaways
- Normal distribution is a continuous probability distribution that can be presented on a graph.
- Normal distribution graphs show continuous random variables such as height, weight and measurement errors.
- You can use your calculator to find probabilities from a normal distribution graph.
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