Converting Z-Scores into Actual Scores Revision Notes for HSC SSCE Mathematics Standard

Converting Z-Scores into Actual Scores

Introduction

After learning how to calculate z-scores, you need to understand how to reverse the process. Converting z-scores back into actual scores is an essential skill that allows you to determine the original data value when you know how many standard deviations it is from the mean.

This conversion is particularly useful when you know someone's standardised score and want to find out what they actually scored on a test, or when comparing performances across different tests or datasets.

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The ability to convert z-scores back to actual scores is a fundamental skill in statistics. It allows you to "undo" the standardisation process and interpret standardised results in terms of the original measurements.

Understanding the conversion formula

To convert a z-score back to an actual score, we need to rearrange the z-score formula to make $x$ the subject. Here's how the algebraic rearrangement works:

Starting with the z-score formula:

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

Multiply both sides by $s$ to eliminate the denominator:

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

Add the mean ( $\bar{x}$ ) to both sides to isolate $x$ :

$x = \bar{x} + z \times s$

This final formula tells us that the actual score equals the mean plus the product of the z-score and the standard deviation. Put simply, you add the mean to the result of multiplying the z-score by the standard deviation.

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The conversion formula $x = \bar{x} + z \times s$ is essentially the opposite of calculating a z-score. You're working backwards from a standardised value to find the original score.

Key formula and definitions

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CONVERTING Z-SCORES INTO ACTUAL SCORES

$x = \bar{x} + z \times s$

Alternatively, you can make $x$ the subject of: $z = \frac{x - \bar{x}}{s}$

Where:

$z$ = z-score or standardised score (the number of standard deviations the score is from the mean)
$x$ = actual score
$\bar{x}$ = mean of the data set
$s$ = standard deviation

Worked example 1: Basic conversion

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Worked Example: Converting a Positive Z-Score

Question: In a normal distribution, the mean was $30$ and the standard deviation was $3.5$ . Blake achieved a standardised score of $2$ . Calculate Blake's actual score using:

a) The formula $x = \bar{x} + z \times s$

b) Making $x$ the subject of $z = \frac{x - \bar{x}}{s}$

Solution:

Method a) Using the direct formula

Write the formula:

$x = \bar{x} + z \times s$

Substitute $\bar{x} = 30$ , $z = 2$ , and $s = 3.5$ into the formula:

$x = 30 + 2 \times 3.5$

Evaluate:

$x = 30 + 7$

$x = 37$

Method b) Rearranging the z-score formula

Write the z-score formula:

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

Substitute $z = 2$ , $\bar{x} = 30$ , and $s = 3.5$ into the formula:

$2 = \frac{x - 30}{3.5}$

Multiply both sides of the equation by $3.5$ :

$2 \times 3.5 = x - 30$

Add $30$ to both sides of the equation (the opposite operation to subtracting $30$ ):

$(2 \times 3.5) + 30 = x - 30 + 30$

Make $x$ the subject and evaluate:

$x = (2 \times 3.5) + 30$

$x = 7 + 30$

$x = 37$

Answer: Blake scored 37.

Worked example 2: Converting a negative z-score

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Worked Example: Converting a Negative Z-Score

Question: A class test (out of $50$ ) has a mean mark of $\bar{x} = 34$ and a standard deviation of $s = 4$ . Mary's standardised test mark was $z = -1.5$ . What was Mary's actual mark?

Solution:

Method a) Using the direct formula

Write the formula and substitute $\bar{x} = 34$ , $z = -1.5$ , $s = 4$ :

$x = \bar{x} + z \times s$

$x = 34 + (-1.5) \times 4$

Evaluate:

$x = 34 + (-6)$

$x = 28$

Method b) Rearranging the z-score formula

Write the z-score formula and substitute $z = -1.5$ , $\bar{x} = 34$ , $s = 4$ :

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

$-1.5 = \frac{x - 34}{4}$

Multiply both sides by $4$ :

$-1.5 \times 4 = x - 34$

Add $34$ to both sides of the equation (the opposite operation to subtracting $34$ ):

$(-1.5 \times 4) + 34 = x - 34 + 34$

Make $x$ the subject and evaluate:

$x = (-1.5 \times 4) + 34$

$x = -6 + 34$

$x = 28$

Answer: Mary scored 28.

Note that Mary's negative z-score indicates her performance was below the mean, which is confirmed by her actual score of $28$ being less than the mean of $34$ .

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When working with negative z-scores, be extra careful with your arithmetic. Remember that multiplying a negative number by a positive number gives a negative result, and adding the mean brings the score back toward the centre of the distribution.

Worked example 3: Interpreting and converting z-scores

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Worked Example: Interpreting and Converting Z-Scores

Question: Jordan achieved a z-score of $2.5$ for a reading test. The state mean for this reading test was $71.5$ and the standard deviation was $6.4$ .

a) Explain the meaning of Jordan's z-score of $2.5$ .

b) What mark did Jordan score in the reading test? Answer correct to the nearest whole number.

Solution:

Part a) Interpretation

A positive z-score indicates the score is above the mean. Remember that a z-score is the number of standard deviations the score is from the mean.

A z-score of $2.5$ is an excellent mark as it is $2.5$ times the standard deviation above the mean. This indicates Jordan performed significantly better than average.

Part b) Calculating the actual score

Method a) Using the direct formula

Write the formula and substitute $\bar{x} = 71.5$ , $z = 2.5$ , $s = 6.4$ :

$x = \bar{x} + z \times s$

$x = 71.5 + 2.5 \times 6.4$

Evaluate:

$x = 71.5 + 16$

$x = 87.5$

$x \approx 88 \text{ (rounded to the nearest whole number)}$

Method b) Rearranging the z-score formula

Write the z-score formula and substitute $z = 2.5$ , $\bar{x} = 71.5$ , $s = 6.4$ :

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

$2.5 = \frac{x - 71.5}{6.4}$

Multiply both sides by $6.4$ :

$2.5 \times 6.4 = x - 71.5$

Add $71.5$ to both sides (the opposite operation to subtracting $71.5$ ):

$(2.5 \times 6.4) + 71.5 = x - 71.5 + 71.5$

Make $x$ the subject and evaluate:

$x = (2.5 \times 6.4) + 71.5$

$x = 16 + 71.5$

$x = 87.5$

$x \approx 88 \text{ (correct to the nearest whole number)}$

Answer: Jordan scored 88.

Exam tips

When tackling z-score conversion problems in exams, keep these practical strategies in mind:

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Essential Exam Strategies:

Choose your method: You can use either the direct formula $x = \bar{x} + z \times s$ or rearrange the z-score formula. The direct formula is usually quicker.
Watch for negative z-scores: When the z-score is negative, the actual score will be below the mean. Take care with your calculations when dealing with negative numbers.
Understand what you're calculating: Remember that you're working backwards from a standardised score to find the original data value.
Check your answer makes sense: If the z-score is positive, your answer should be greater than the mean. If it's negative, your answer should be less than the mean.
Show your working: In exams, always write down the formula, substitute the values, and show your calculations clearly.

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Common Mistake to Avoid:

When dealing with negative z-scores, many students forget that multiplying by a negative number affects the sign of the result. Always double-check your arithmetic when working with negative values!

Remember!

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Key Points to Remember:

The formula for converting z-scores to actual scores is: $x = \bar{x} + z \times s$
This means you add the mean to the product of the z-score and standard deviation
Positive z-scores give actual scores above the mean; negative z-scores give actual scores below the mean
You can solve these problems using either the direct formula or by rearranging the z-score formula algebraically
Always check your answer makes sense in the context of the question
The conversion process is the opposite of calculating a z-score - you're "undoing" the standardisation

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