The empirical rule
Apply the 68–95–99.7% rule to find the proportion of data within 1, 2 and 3 standard deviations of the mean; determine percentages in tails and between intervals; calculate expected counts in samples; solve problems using the empirical rule and z-scores together.
Worked examples
Applying the 68–95–99.7% rule
Straightforward
Problem
IQ scores are normally distributed with a mean of 100 and a standard deviation of 15. What percentage of people have IQ scores between 70 and 130? What percentage have a score above 130?
1
Determine how many standard deviations the boundaries are from the mean.
2
Apply the empirical rule for 2 standard deviations.
95% of values lie within .
So 95% of people have IQ scores between 70 and 130.
So 95% of people have IQ scores between 70 and 130.
3
Find the percentage above 130.
The remaining lies outside this range.
By symmetry, half of 5% lies above .
Percentage above 130
By symmetry, half of 5% lies above .
Percentage above 130
Answer
95% of people have IQ scores between 70 and 130. About 2.5% have a score above 130.
Finding a percentile using the empirical rule
Moderate
Problem
In a fitness test, scores are normally distributed with a mean of 50 and standard deviation of 8. What score represents the 16th percentile (i.e., 16% of scores fall below this value)?
1
Recall the empirical rule and the symmetry of the normal distribution.
68% of scores lie between and .
By symmetry, 34% lie between and the mean, and 34% lie between the mean and .
By symmetry, 34% lie between and the mean, and 34% lie between the mean and .
2
Find the proportion below .
50% of scores lie below the mean.
Of these, 34% lie between and the mean.
So: lie below .
Of these, 34% lie between and the mean.
So: lie below .
3
The 16th percentile is a score of 42.
Calculate .
The 16th percentile is a score of 42.
Answer
A score of 42 represents the 16th percentile.
Expected counts in a sample
Challenging
Problem
The lifetime of a brand of light bulb is normally distributed with a mean of 1200 hours and a standard deviation of 100 hours. A company orders 5000 bulbs. How many are expected to last between 1000 and 1100 hours?
1
Identify the z-scores for the boundaries.
(i.e., )
(i.e., )
(i.e., )
2
Find the proportion between these boundaries using the empirical rule.
By the empirical rule:
- 95% lie within — so 47.5% lie between and the mean
- 68% lie within — so 34% lie between and the mean
Proportion between and :
- 95% lie within — so 47.5% lie between and the mean
- 68% lie within — so 34% lie between and the mean
Proportion between and :
3
Calculate the expected number of bulbs.
Answer
Approximately 675 bulbs are expected to last between 1000 and 1100 hours.
Practise
Q1·Straightforward
For a normal distribution, what percentage of values lie within 1 standard deviation of the mean? (Enter your answer as a number.)
Explanation
The empirical rule states:
- 68% of values lie within 1 standard deviation of the mean (between and )
- 95% lie within 2 standard deviations
- 99.7% lie within 3 standard deviations
- 68% of values lie within 1 standard deviation of the mean (between and )
- 95% lie within 2 standard deviations
- 99.7% lie within 3 standard deviations
Q2·Straightforward
Test scores are normally distributed with a mean of 70 and a standard deviation of 10. What percentage of scores lie between 60 and 80?
Explanation
and
The interval from to contains 68% of values by the empirical rule.
The interval from to contains 68% of values by the empirical rule.
Q3·Straightforward
SAT-style scores are normally distributed with a mean of 500 and a standard deviation of 100. What percentage of scores lie between 300 and 700?
Explanation
and
The interval contains 95% of values.
The interval contains 95% of values.
Q4·Straightforward
In a normal distribution, what percentage of values lie more than 3 standard deviations above the mean?
Explanation
99.7% of values lie within 3 standard deviations of the mean.
That leaves in both tails combined.
By symmetry, lies above .
That leaves in both tails combined.
By symmetry, lies above .
Q5·Moderate
Adult male heights are normally distributed with a mean of 176 cm and a standard deviation of 7 cm. What percentage of adult males are between 162 cm and 190 cm tall?
Explanation
The interval contains 95% of values by the empirical rule.
Q6·Moderate
Results from a standardised test are normally distributed with a mean of 50 and a standard deviation of 5. What percentage of results are below 45?
Explanation
By the empirical rule, 68% of values lie between and .
The remaining lies outside this range, split equally between the two tails.
Percentage below
Q7·Moderate
A factory produces bolts with lengths normally distributed with a mean of 150 mm and a standard deviation of 15 mm. In a batch of 2000 bolts, approximately how many bolts have lengths between 120 mm and 180 mm?
Explanation
and
By the empirical rule, 95% of values lie within 2 standard deviations.
Expected count bolts
By the empirical rule, 95% of values lie within 2 standard deviations.
Expected count bolts
Q8·Moderate
In a normally distributed data set with mean 60 and standard deviation 12, a score of 84 is how many standard deviations above the mean?
Explanation
A score of 84 is exactly 2 standard deviations above the mean. By the empirical rule, approximately 97.5% of scores fall below this point.
Q9·Moderate
Delivery times for a courier service are normally distributed with a mean of 200 minutes and a standard deviation of 25 minutes. What proportion of deliveries take more than 250 minutes? Express your answer as a decimal.
Explanation
95% of values lie within 2 standard deviations, so 5% lies outside.
By symmetry, 2.5% lies above minutes.
Proportion
Q10·Challenging
Exam scores are normally distributed with a mean of 60 and a standard deviation of 12. What score marks the 84th percentile? (Hint: the 84th percentile is the score below which 84% of results fall.)
Explanation
The empirical rule tells us 68% of values lie between and .
By symmetry, 34% lie between the mean and .
Since 50% of values lie below the mean:
So the 84th percentile
By symmetry, 34% lie between the mean and .
Since 50% of values lie below the mean:
So the 84th percentile
Q11·Challenging
Blood pressure readings in a population are normally distributed with a mean of 120 mmHg and a standard deviation of 15 mmHg. Only 2.5% of the population has a reading above what value?
Explanation
By the empirical rule, 95% of values lie within 2 standard deviations.
lie outside; by symmetry, 2.5% lie above .
lie outside; by symmetry, 2.5% lie above .
Q12·Challenging
Heights in a school cohort are normally distributed with a mean of 170 cm and a standard deviation of 8 cm. The school has 400 students. Approximately how many students are expected to be taller than 186 cm?
Explanation
So 186 cm .
By the empirical rule, 95% of values lie within 2 standard deviations, leaving 5% outside.
By symmetry, 2.5% lie above .
Expected number of students taller than 186 cm:
Open Math
The empirical rule
Statistical Analysis · MS-S5
Name:
Date:
Q1Straightforward
For a normal distribution, what percentage of values lie within 1 standard deviation of the mean? (Enter your answer as a number.)
Q2Straightforward
Test scores are normally distributed with a mean of 70 and a standard deviation of 10. What percentage of scores lie between 60 and 80?
Q3Straightforward
SAT-style scores are normally distributed with a mean of 500 and a standard deviation of 100. What percentage of scores lie between 300 and 700?
Q4Straightforward
In a normal distribution, what percentage of values lie more than 3 standard deviations above the mean?
Q5Moderate
Adult male heights are normally distributed with a mean of 176 cm and a standard deviation of 7 cm. What percentage of adult males are between 162 cm and 190 cm tall?
Q6Moderate
Results from a standardised test are normally distributed with a mean of 50 and a standard deviation of 5. What percentage of results are below 45?
Q7Moderate
A factory produces bolts with lengths normally distributed with a mean of 150 mm and a standard deviation of 15 mm. In a batch of 2000 bolts, approximately how many bolts have lengths between 120 mm and 180 mm?
Q8Moderate
In a normally distributed data set with mean 60 and standard deviation 12, a score of 84 is how many standard deviations above the mean?
Q9Moderate
Delivery times for a courier service are normally distributed with a mean of 200 minutes and a standard deviation of 25 minutes. What proportion of deliveries take more than 250 minutes? Express your answer as a decimal.
Q10Challenging
Exam scores are normally distributed with a mean of 60 and a standard deviation of 12. What score marks the 84th percentile? (Hint: the 84th percentile is the score below which 84% of results fall.)
Q11Challenging
Blood pressure readings in a population are normally distributed with a mean of 120 mmHg and a standard deviation of 15 mmHg. Only 2.5% of the population has a reading above what value?
Q12Challenging
Heights in a school cohort are normally distributed with a mean of 170 cm and a standard deviation of 8 cm. The school has 400 students. Approximately how many students are expected to be taller than 186 cm?
Worked solutions and answers at openmath.au/year-12/standard-2/the-normal-distribution/the-empirical-rule