8.01 Normal distribution

When we have a set of univariate data, it often happens that most of the measurements will be clustered close to the mean value with the density of the observations falling off as distance away from the mean increases. Results of this kind displayed in a histogram show a central peak with columns of decreasing height on each side of the mean.

The normal distribution is a special type of continuous probability distribution. It is often called the "bell curve" because of the shape of the graph. The normal distribution is important in statistics because it can be used to describe many natural variables. For example, for a large population a person's height, arm span, and IQ are each variables which demonstrate an approximate normal distribution.

The bell curve is symmetrical about the mean (which is where the peak of the bell occurs) and it has the property that the mean, mode and median are all equal. 

Probability distributions that display the distinctive bell curve together with its properties are described as being "normally distributed". 

The shape of the normal distribution will depend on its mean ($\mu$μ) and standard deviation $(\sigma)$(σ). The mean is where the graph peaks. To draw a normal distribution graph we only need the mean and standard deviation values.

 

The normal distribution formula

The probability density function describing the normal distribution function is written in terms of the mean $(\mu)$(μ) and standard deviation $(\sigma)$(σ):

$\phi\left(x\right)=\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{\left(x-\mu\right)^2}{2\sigma^2}}$ϕ(x)=1σ√2π​e−(x−μ)22σ2​ 

We won't be using this formula to find the probabilities. Instead, we will primarily be using our available technology on our calculators.

Exploration

Assume that a population is normally distributed with mean $\mu$μ and standard deviation $\sigma$σ. Use the following applet to discover what happens to the graph as we change the mean and standard deviation.

• What do you notice about the shape of the graph when we change the mean? 
• What happens to the shape of the graph as $\sigma$σ gets closer to $0$0?

For example, this distribution curve comes from a data set that has a very small standard deviation, $\sigma=0.2$σ=0.2,  and hence is clustered tightly around the mean:

This normal distribution has a larger standard deviation, $\sigma=0.9$σ=0.9,  and hence is quite spread.

 

 

Practice questions

Question 1

Question 2

 

The empirical rule

The empirical rule, also known as the $68-95-99.7%$68−95−99.7% rule, is a way of estimating the way that normally distributed data spreads out. These numbers correspond to the approximate proportion within one, two, and three standard deviations of the mean.

• Approximately $68%$68% of scores lie within $1$1 standard deviation of the mean:

• Approximately $95%$95% of scores lie within $2$2 standard deviations of the mean:

• Approximately $99.7%$99.7% of scores lie within $3$3 standard deviations of the mean:

A normal distribution is symmetrical, so we can use these basic values to find approximations of other regions. For example, as $95%$95% of scores lie within $2$2 standard deviations of the mean, so $47.5%$47.5% (half of $95%$95%) will lie between the mean and $2$2 standard deviations above the mean:

We can use a similar trick to conclude that $34%$34% (half of $68%$68%) lies between $1$1 standard deviation below the mean, and the mean itself.

If we add these approximations together, we conclude that $81.5%$81.5% (which is $34%+47.5%$34%+47.5%) of scores lie between $1$1 standard deviation below and $2$2 standard deviations above the mean.

Play around with this applet by moving the endpoints of the shaded region. You will see the percentage of scores lying between the endpoints, and can reveal the percentages of each piece with the toggle:

 

Exploration

Standard deviation is a measure of spread that we can apply to everyday contexts. For example, let's say the mean score in a test was $67$67 and the standard deviation was $7$7 marks. This means that:

• a person who was $1$1 standard deviation above the mean would have received a mark of $74$74 (as this is $67+7$67+7).
• a person who was $2$2 standard deviations below the mean would have received a mark of $53$53 (as this is $67-2\times7$67−2×7).

If we're told that the scores were approximately normally distributed, we could go one step further and determine the percentage of students who scored between $53$53 and $74$74.

The number of students that score between $2$2 standard deviations of the mean would be $95%$95%. The normal distribution is symmetrical, so half of $95%$95% of students scored between the mean and two standard deviations below. In other words, $47.5%$47.5% of students scored between $53$53 and $67$67.

Using the same reasoning, we know that half of $68%$68% of students scored between the mean and $1$1 standard deviation above. This means that $34%$34% of students scored between $67$67 and $74$74.

So putting the two percentages together, we can say that $\left(47.5+34\right)%=81.5%$(47.5+34)%=81.5% of students scored between $53$53 and $74$74.

Practice questions

Question 3

question 4

question 5