The normal distribution

Standardise values using z-scores; interpret and compare z-scores across distributions; apply the empirical (68–95–99.7%) rule to find proportions and expected counts within normally distributed data.

Statistical Analysis · MS-S5

Sub-topics

z-scores

Standardise a data value using

z = \frac{x - \mu}{\sigma}

; interpret a z-score as the number of standard deviations from the mean; find original values, means or standard deviations given a z-score; compare values from different distributions.

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.

Open Math

The normal distribution

Statistical Analysis · MS-S5

Name:

Date:

z-scores

Q1Straightforward

In a class test, scores are normally distributed with a mean of 60 and a standard deviation of 10. A student scores 75. Calculate their z-score.

Q2Straightforward

A data set has a mean of 50 and a standard deviation of 8. A value has a z-score of 1.5. Find the original value

x

Q3Straightforward

Student A scores 72 in an exam where the class mean is 65 and standard deviation is 7. Calculate Student A's z-score, rounded to two decimal places.

Q4Straightforward

Heights in a group have a mean of 168 cm and a standard deviation of 9 cm. A person is 150 cm tall. Calculate their z-score.

Q5Moderate

A quality-control test produces a z-score of 2.4 for a measurement of 88, given that the population mean is 70. Find the standard deviation

\sigma

Q6Moderate

In a distribution with standard deviation 10, the value 95 has a z-score of 1.6. Find the mean

\mu

Q7Moderate

Daily maximum temperatures in a city have a mean of 22°C and a standard deviation of 3°C. Calculate the z-score for a temperature of 28°C.

Q8Moderate

Emma scores 78 in Maths (class mean 66, standard deviation 6) and 84 in English (class mean 72, standard deviation 8). What is Emma's z-score in Maths?

Q9Moderate

A machine produces bolts with a mean diameter of 50 mm and standard deviation of 2 mm. A bolt measures 53.8 mm. Calculate its z-score.

Q10Challenging

A distribution has mean

\mu = 45

and standard deviation

\sigma = 12

. Find the value

x

that corresponds to a z-score of 2.5.

Q11Challenging

A distribution has mean 120 and standard deviation 15. Find the value

x

whose z-score is

-1.8

Q12Challenging

In a normal distribution, the value 62 has a z-score of 0.8, and the value 50 has a z-score of

-1.2

. Find the mean

\mu

of the distribution.

The empirical rule

Q13Straightforward

For a normal distribution, what percentage of values lie within 1 standard deviation of the mean? (Enter your answer as a number.)

Q14Straightforward

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?

Q15Straightforward

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?

Q16Straightforward

In a normal distribution, what percentage of values lie more than 3 standard deviations above the mean?

Q17Moderate

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?

Q18Moderate

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?

Q19Moderate

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?

Q20Moderate

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?

Q21Moderate

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.

Q22Challenging

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.)

Q23Challenging

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?

Q24Challenging

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