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Normal Distribution Properties

Neil Trivedi

Teacher

Neil Trivedi

Contents

The Normal Distribution

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Properties of a Normal Distribution

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Percentage Rules

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Using Calculators

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The Normal Distribution

Normal distributions are a selection of bell-shaped curves that are all symmetrical with a larger concentration of values at the centre.

The height and spread of the curves are determined by the mean and standard deviation. Remember, the standard deviation tells us how close our values are to the mean, so the larger the standard deviation the "wider" the curve as the points are generally further away from the mean.

A graph compares several normal distribution curves, showing how changing the mean 𝜇 shifts the curve horizontally and changing the standard deviation 𝜎 alters its spread and height.

If we have an event and its outcomes follow that of a normal distribution with mean and standard deviation then we would say:


Note: represents the outcome of an event and represents the probability of that happening.

You do not need to know or work with the actual formula of the normal distribution.

Properties of a Normal Distribution

a) Normal distributions are for continuous data only.

b) Just like all probabilities in a discrete distribution add up to , the area under a normal distribution curve is also .

c) The probability of getting anything above or below the mean is due to symmetry.

d) The probability of say is zero as the chance of obtaining an exact value with continuous variables is impossible this is because has no area and hence no probability.

e) The probability of finding and are the same probability.

f) To find the probability of you just find the area between the two values.

A normal distribution curve is shown with a horizontal arrow pointing to the right, indicating a positive shift in the mean 𝜇 along the 𝑥–axis.

Percentage Rules

A normal distribution curve is shown with the region between 𝜇−𝜎 and 𝜇+𝜎 shaded, representing the probability within one standard deviation of the mean.

You also need to be aware of the , , rule meaning:

of data is within one standard deviation of mean

of data is within two standard deviations of mean

of data is within three standard deviations of mean

It is generally accepted that all values lie within standard deviations from the mean

No matter how simple the problem is, ALWAYS sketch. It helps visualising the area under the curve we’re concerned about.

Example 1:

The height, in cm, for a given population is, by definition, normally distributed using

Find:

a)

Single Step: Noticing is the mean, we can plot the graph and apply normal distribution properties.

A normal distribution curve is shown with a rightward arrow and shaded region, indicating a shift to the right and the probability area toward the upper tail of the distribution.

We know that it is the area above the mean, and hence the probability is .


b)

Single Step: Noticing and are standard deviations away from the mean and , we can plot the graph and apply normal distribution properties.

A normal distribution curve is shown with most of the central region shaded and a right-pointing arrow, representing the probability 𝑃(𝑋<𝑥) up to a value on the right of the mean.

We know that the data is within two standard deviations of the mean, and hence the probability is .


c)

Single Step: Noticing is standard deviations away from the mean , we can plot the graph and apply normal distribution properties.

A normal distribution curve is shown with a right-pointing arrow and a small shaded region in the upper tail beyond 𝑥≈1.96, representing a right-tail probability.

We know that the probability of data lying within three standard deviations of the mean is . Therefore, the probability of data lying outside three standard deviations is: .

represents only one tail (one side of the distribution). Due to symmetry, we divide the probability found by to obtain:

No answer provided.

Using Calculators

Depending on which calculator you have, you will have to use different ways to find the probability of a normal distribution . In general, follow these steps:

• Press MODE.

• Find the DISTRIBUTION setting

• Choose Normal CD (Cumulative Distribution)

• Input the upper value, lower value, mean and standard deviation

If the lower value is effectively , just type in (just enough that it is more than standard deviations away from the mean), and if it is the case for the upper value , we can do the same.

Example 2:

A basketball player in Japan is considered “exceptionally tall” if their height exceeds cm. The height, in cm, of male basketball players in Japan follows a normal distribution .

a) The probability of a basketball player chosen at random being “exceptionally tall”.

Single Step: Sketch the diagram to find .

A normal distribution curve is shown with a small shaded region in the far right tail, representing the probability 𝑃(𝑋>𝑥) for a large value of 𝑥.

The mean is , the standard deviation is , and we take the lower value to be , and upper value to be . Inputting into our calculator gives

dp


b) A sample of basketball players is randomly collected. What is the probability that at most, of them are considered as “exceptionally tall”.

Step 1: Recognise the distribution.

Let denote the number of “exceptionally tall” players out of .

We are using binomial distribution as there are basketball players, a fixed probability of success ( calculated in part a) and each trial is independent (one player's height does not affect the next person's height.)

We know that . The probability of success is found in part a).

Step 2: Find the required probability.

Using the Binomial CD mode of our calculator, we get

dp

No answer provided.
The Inverse and Cumulative Normal Distribution