The Normal Distribution Chapter Review

A normal distribution is symmetrical about the:

• mean

• standard deviation

• maximum

• z score

Answer explanation

A normal distribution is defined by the mean and standard deviation.

The ages of the citizens who live in Bishop are normally distributed. The mean age is 53 years and the standard deviation is 14. What percentage of the citizens is younger than 67?

• 34%

• 68%

• 84%

• 95%

Answer explanation

The score 67 is one standard deviation to the right of the mean. So the percentage can be calculated by adding 50% of your data (left half of the mean) with 34% (half of 68%), this gives 84%.

Zoe scored 76 in a science test. The science test had a mean of 60 and a standard deviation of 8. A recent history test had a mean of 56 and a standard deviation of 12. What mark in the history test would have been equivalent to Zoe’s science mark?

• 68

• 72

• 76

• 80

Answer explanation

In this question, you had to firstly calculate Zoe's z-score using the information for science, then using the z-score, apply it to the formula $x=\overline{x}+s\times z$ to work out Zoe's equivalent score in history.

What is the z-score if $x=20,\ \overline{x}=16\ and\ s=4\ ?$  

• -2

• -1

• 1

• 2

Answer explanation

You can either use the z-score formula, or visualise how the information would look on a normal distribution graph (i.e. 16 is in the middle, and the standard deviation is 4, so 20 is one standard deviation to the right of the mean, therefore z = 1.

A student’s mark on a test is 50. The mean mark for their class is 55 and the standard deviation is 2.5. Their standard score is:

• -2.0

• -1.5

• 2.0

• 2.5

Answer explanation

The 'standard score' or 'standardized score' refers to the 'z-score'. Remember to try and list out the variables and what they equal to in the question.

Aaron’s class achieved the following results in a topic test: 44, 91, 78, 55, 65, 49, 61, 55, 74, 46, 73, 63, 57 and 29. Using the population standard deviation ( $\sigma$ ) , how many scores have a z-score greater than 2?

• 0

• 1

• 2

• 3

Answer explanation

When you are given a set of scores or data values and asked to find either the mean and/or standard deviation, type these into the calculator. STAT MODE - 1- VAR. Then SHIFT 1, 4.

Susan scored 80 in a maths test. The test mean was 70 and the standard deviation was 5. This test is being rescaled to a mean of 60 and a standard deviation of 8 without changing Susan’s z-score. What is Susan’s new score?

• 64%

• 70%

• 76%

• 80%

Answer explanation

Read these questions carefully, they are quite common and mistakes are made very easily. Here you need to calculate the z-score using the first set of information, then re-use the same z-score to find Susan's new 'actual' score.

Benjamin achieved a z-score of 2.5 on his last assessment task. The class mean was 58% and standard deviation 6%. What was Benjamin’s mark?

• 60.5%

• 61%

• 67%

• 73%

Answer explanation

A simple question where you just need to apply the 'actual' score formula.

What percentage of scores in a normal distribution has a z-score between 0 and 3?

• 0.15%

• 0.30%

• 49.85%

• 97%

Answer explanation

Think about the percentage within 3 standard deviations of the mean (i.e. z = -3 to z = 3),

i.e. 99.7%, then half it to get the percentage between 0 and 3, i.e. 49.85%

The results of a literacy exam were normally distributed. Samuel gained a z-score of −1. What percentage of students scored better than Samuel?

• 16%

• 34%

• 68%

• 84%

Answer explanation

Read this question carefully! It wants you to find the percentage of students who did better than Samuel, i.e. what percentage of scores are greater than z = -1? 50% + 34% = 84%.