The 68-95-99.7\% rule, also known as the empirical rule, is an estimate of the spread of data which is approximately normally distributed. The majority of scores lie within one standard deviation of the mean, and almost all scores like within three standard deviations of the mean.

Specifically:

The image shows a standard normal distribution with 68% of scores where shaded on both sides.

68\% of scores lie within 1 standard deviation of the mean.

The image shows a standard normal distribution with 95% of scores where shaded on both sides.

95\% of scores lie within 2 standard deviations of the mean.

The image shows a standard normal distribution with 99.7% of scores where shaded on both sides.

99.7\% of scores lie within 3 standard deviations of the mean.

Since the normal distribution is symmetric, we can actually divide these regions up further.

For example, since 95\% of scores lie within 2standard deviations of the mean, then \dfrac{95\%}{2}=47.5\% of scores will lie between the mean and 2 standard deviations above the mean for the normal distribution.

The image shows a standard normal distribution with 47.5% of scores where shaded on the right side.

We could also say that 50\% + \dfrac{68\%}{2}=84\% of scores are less than or equal to 1 standard deviation above the mean.

Standard deviation is a measure of spread that we can apply to everyday contexts. For example, if the mean score in a test was 67 and the standard deviation was 7 marks, then:

A person who received a mark of 74 was one standard deviation above the mean. If the test scores were normally distributed, then they scored better than approximately 84\% of students.
A person who received a score of 53 was two standard deviations below the mean. If the test scores were normally distributed, then they scored better than only 2.5\% of students.

The standard deviation of a data set is a measure of spread, and for normally distributed data we have:

68\% of scores lie within 1 standard deviation of the mean.
95\% of scores lie within 2 standard deviations of the mean.
99.7\% of scores lie within 3 standard deviations of the mean.

Examples

Example 1

The grades in a test are approximately normally distributed. The mean mark is 60 with a standard deviation of 2.

Between which two scores does approximately 68\% of the results lie symmetrically about the mean? Write both scores on the same line, separated by a comma.

Worked Solution

Create a strategy

Use the empirical rule, 68\% of scores from a normal distribution lie one standard deviation above or below the mean.

Apply the idea

One standard deviation below the mean: 60-2=58

One standard deviation above the mean: 60 + 2 = 62

\text{Scores} = 58,\,62

Between which two scores does approximately 95\% of the results lie symmetrically about the mean? Write both scores on the same line, separated by a comma.

Worked Solution

Create a strategy

Use the empirical rule, 95\% of scores from a normal distribution lie two standard deviation above or below the mean.

Apply the idea

One standard deviation below the mean: 60-4=56

One standard deviation above the mean: 60 + 4 = 64

\text{Scores} = 56,\,64

Between which two scores does approximately 99.7\% of the results lie symmetrically about the mean? Write both scores on the same line, separated by a comma.

Worked Solution

Create a strategy

Use the empirical rule, 99.7\% of scores from a normal distribution lie three standard deviation above or below the mean.

Apply the idea

One standard deviation below the mean: 60-6=54

One standard deviation above the mean: 60 + 6 = 66

\text{Scores} = 54,\,66

Example 2

In a normal distribution, what percentage of scores lie between 2 standard deviations below and 3 standard deviations above the mean? Use the empirical rule to find your answer.

Worked Solution

Create a strategy

We need to divide 95\% by 2 and 99.7\% by also 2, and then add them together.

Apply the idea

\displaystyle \text{Percentage}	\displaystyle =	\displaystyle \dfrac{95}{2} + \dfrac{99.7}{2}	Divide the percentages by 2
	\displaystyle =	\displaystyle 47.5 + 49.85	Evaluate the division
	\displaystyle =	\displaystyle 97.35\%	Evaluate the addition

Idea summary

The standard deviation of a data set is a measure of spread, and for normally distributed data we have:

68\% of scores lie within 1 standard deviation of the mean.
95\% of scores lie within 2 standard deviations of the mean.
99.7\% of scores lie within 3 standard deviations of the mean.

Outcomes

U3.AoS1.6

the normal model and the 68–95–99.7% rule, and standardised values (𝑧-scores)

U3.AoS1.18

solve problems using 𝑧-scores and the 68–95–99.7% rule

1.08 The 68-95-99.7% rule

Ideas

The 68-95-99.7% rule

Examples

Example 1

Example 2

Outcomes

U3.AoS1.6

U3.AoS1.18

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