Z-Scores (HSC SSCE Mathematics Standard): Revision Notes

Worked Example 1: Finding a z-score from given values

Question: For a normal distribution with mean 62 and standard deviation 11, find the z-score for a score of 84 and explain what it means.

Solution:

Write the formula and identify the values:

$z = \frac{x - \bar{x}}{s}$

Where $x = 84$, $\bar{x} = 62$, $s = 11$

Substitute into the formula:

$z = \frac{84 - 62}{11}$

Calculate:

$z = \frac{22}{11} = 2$

Interpretation: The z-score of 2 tells us that 84 is two standard deviations above the mean.

Worked Example 2: Calculating z-score for test marks

Question: Ruby scored 70 on a class test. The class mean was 78 with a standard deviation of 6. What is her z-score? (Answer to one decimal place)

Solution:

Write the formula:

$z = \frac{x - \bar{x}}{s}$

Substitute the values ($x = 70$, $\bar{x} = 78$, $s = 6$):

$z = \frac{70 - 78}{6}$

Calculate:

$z = \frac{-8}{6} = -1.333...$

Round to one decimal place:

$z \approx -1.3$

Answer: Ruby's z-score is $-1.3$, meaning her score was 1.3 standard deviations below the class mean.

Worked Example 3: Multiple z-score calculations

Question: The heights of a group of young women have a mean of 160 cm and a standard deviation of 8 cm. Determine the z-scores for women who are:

• a) 172 cm tall
• b) 150 cm tall
• c) 160 cm tall

Solution:

Part a) For height = 172 cm:

$z = \frac{x - \bar{x}}{s} = \frac{172 - 160}{8}$

$z = \frac{12}{8} = 1.5$

Part b) For height = 150 cm:

$z = \frac{x - \bar{x}}{s} = \frac{150 - 160}{8}$

$z = \frac{-10}{8} = -1.25$

Part c) For height = 160 cm:

$z = \frac{x - \bar{x}}{s} = \frac{160 - 160}{8}$

$z = \frac{0}{8} = 0$

Worked Example 4: Special z-score values

Question: What is the z-score for a score that is:

• a) Equal to the mean?
• b) One standard deviation above the mean?
• c) One standard deviation below the mean?

Solution:

Part a) When the score equals the mean:

The score is $x = \bar{x}$

Substitute into the formula:

$z = \frac{x - \bar{x}}{s} = \frac{\bar{x} - \bar{x}}{s}$

$z = \frac{0}{s} = 0$

Part b) When the score is one standard deviation above the mean:

The score is $x = \bar{x} + s$

Substitute into the formula:

$z = \frac{x - \bar{x}}{s} = \frac{(\bar{x} + s) - \bar{x}}{s}$

$z = \frac{s}{s} = 1$

Part c) When the score is one standard deviation below the mean:

The score is $x = \bar{x} - s$

Substitute into the formula:

$z = \frac{x - \bar{x}}{s} = \frac{(\bar{x} - s) - \bar{x}}{s}$

$z = \frac{-s}{s} = -1$