The normal distribution
Explore the normal distribution as a continuous probability distribution; standardise values using z-scores; apply the empirical (68–95–99.7%) rule to find probabilities and proportions; compare values across different normal distributions.
Worked examples
Standardising and interpreting a z-score
Straightforward
Problem
, so and . (a) Find the -score for . (b) Interpret what this z-score means.
1
Part (a): Calculate the z-score using .
2
Part (b): Interpret the z-score, and show how to reverse the process.
A z-score of 2.5 means that is 2.5 standard deviations above the mean. You can also reverse the process: if someone tells you , the corresponding raw value is:
Answer
, meaning is 2.5 standard deviations above the mean.
Applying the empirical rule (combining regions)
Moderate
Problem
, so and . Using the empirical rule, find .
1
Identify how many standard deviations each endpoint is from the mean.
2
Apply the empirical rule.
The empirical rule states that in any normal distribution:
So:
3
Consider the asymmetric case , where , by splitting at the mean.
Answer
.
Comparing values across two distributions
Challenging
Problem
Alice sat two exams: English, where her score was 74 against a class mean of 65 and class SD of 12; and History, where her score was 88 against a class mean of 80 and class SD of 6. In which exam did she perform better relative to her class?
1
Calculate both z-scores.
2
Compare the z-scores to interpret relative performance.
Alice's History z-score () is higher than her English z-score (), so she performed better relative to her class in History, even though her raw score in English is closer to the mean in number. The z-score strips away the effect of different means and spreads, making scores from different distributions comparable.
Answer
Alice performed better relative to her class in History (z ≈ 1.33) than in English (z = 0.75).
Practise
Q1·Straightforward
, so and .
Find the -score corresponding to .
Find the -score corresponding to .
Explanation
A value of 45 is exactly one standard deviation above the mean.
Q2·Straightforward
, so and .
A value in this distribution has -score . Find the value of .
A value in this distribution has -score . Find the value of .
Explanation
A -score of means the value is two standard deviations below the mean.
Q3·Straightforward
, so and .
Using the empirical rule, find .
Using the empirical rule, find .
Explanation
and .
By the empirical rule, approximately 68% of values in a normal distribution lie within one standard deviation of the mean:
By the empirical rule, approximately 68% of values in a normal distribution lie within one standard deviation of the mean:
Q4·Straightforward
The lifespans of a species of insect are normally distributed with mean 24 days and standard deviation 4 days. Using the empirical rule, what proportion have lifespans between 16 and 32 days?
Explanation
and .
By the empirical rule, approximately 95% of values lie within two standard deviations of the mean:
By the empirical rule, approximately 95% of values lie within two standard deviations of the mean:
Q5·Moderate
Scores on an aptitude test are modelled by , so and .
Using the empirical rule, find .
Using the empirical rule, find .
Explanation
.
By the empirical rule, 68% of values lie in , so lies outside this interval. By symmetry, half is in the upper tail:
By the empirical rule, 68% of values lie in , so lies outside this interval. By symmetry, half is in the upper tail:
Q6·Moderate
, so and .
Using the empirical rule, find .
Using the empirical rule, find .
Explanation
and .
By the empirical rule, .
By the symmetry of the normal curve:
By the empirical rule, .
By the symmetry of the normal curve:
Q7·Moderate
is normally distributed with mean . Using the empirical rule, 99.7% of values fall between 65 and 95. Find the standard deviation .
Explanation
Since 99.7% lie between 65 and 95, and the normal distribution is symmetric about :
Check: ✓
Check: ✓
Q8·Moderate
In a Mathematics exam, a student scored 78 in a class where and . Find the student's -score for Mathematics.
Explanation
Q9·Moderate
The same student also sat a Science exam, scoring 85 in a class where and . Find the student's -score for Science. Round to 3 decimal places if needed.
Explanation
Comparing the two z-scores: , so the student performed better in Mathematics relative to their class.
Q10·Moderate
The heights of plants in a large population are normally distributed with mean 45 cm and standard deviation 6 cm. In a sample of 2000 plants, how many would you expect to have heights between 33 cm and 57 cm? (Use the empirical rule.)
Explanation
and .
By the empirical rule, 95% of values lie within two standard deviations of the mean.
By the empirical rule, 95% of values lie within two standard deviations of the mean.
Q11·Challenging
, so and .
Using the empirical rule, find .
Using the empirical rule, find .
Explanation
and .
Split at the mean and use symmetry:
Split at the mean and use symmetry:
Q12·Challenging
A normally distributed random variable has standard deviation . A value of corresponds to a -score of . Find the mean .
Explanation
Q13·Challenging
, so and .
Using the empirical rule, find .
Using the empirical rule, find .
Explanation
.
By the empirical rule, 95% of values lie in , so lies outside this interval. By symmetry, half is in the lower tail:
By the empirical rule, 95% of values lie in , so lies outside this interval. By symmetry, half is in the lower tail:
Open Math
The normal distribution
Statistics and Probability · MAV-12-07
Name:
Date:
Q1Straightforward
, so and .
Find the -score corresponding to .
Find the -score corresponding to .
Q2Straightforward
, so and .
A value in this distribution has -score . Find the value of .
A value in this distribution has -score . Find the value of .
Q3Straightforward
, so and .
Using the empirical rule, find .
Using the empirical rule, find .
Q4Straightforward
The lifespans of a species of insect are normally distributed with mean 24 days and standard deviation 4 days. Using the empirical rule, what proportion have lifespans between 16 and 32 days?
Q5Moderate
Scores on an aptitude test are modelled by , so and .
Using the empirical rule, find .
Using the empirical rule, find .
Q6Moderate
, so and .
Using the empirical rule, find .
Using the empirical rule, find .
Q7Moderate
is normally distributed with mean . Using the empirical rule, 99.7% of values fall between 65 and 95. Find the standard deviation .
Q8Moderate
In a Mathematics exam, a student scored 78 in a class where and . Find the student's -score for Mathematics.
Q9Moderate
The same student also sat a Science exam, scoring 85 in a class where and . Find the student's -score for Science. Round to 3 decimal places if needed.
Q10Moderate
The heights of plants in a large population are normally distributed with mean 45 cm and standard deviation 6 cm. In a sample of 2000 plants, how many would you expect to have heights between 33 cm and 57 cm? (Use the empirical rule.)
Q11Challenging
, so and .
Using the empirical rule, find .
Using the empirical rule, find .
Q12Challenging
A normally distributed random variable has standard deviation . A value of corresponds to a -score of . Find the mean .
Q13Challenging
, so and .
Using the empirical rule, find .
Using the empirical rule, find .
Worked solutions and answers at openmath.au/year-12/advanced/random-variables/the-normal-distribution