The standard score: a value's relative position

Dana got 80 in a history test and 85 in a maths test. In which test did she do better, compared with her year group?
In history the year group’s mean is 70, and the standard deviation 5.
In maths the mean is 75, and the standard deviation 10.

• In history Dana is 10 points above the mean, which is two standard deviations: $\displaystyle {\frac{80-70}{5}}={2}$.
• In maths she is the same 10 points above the mean, but there that is one standard deviation: $\displaystyle {\frac{85-75}{10}}={1}$.

In history the marks are packed more closely round the mean, so 80 stands out more there: compared with her year group, Dana did better in history, although her mark there is lower.

The standard score $\displaystyle z$ of a value $\displaystyle x$ is its distance from the mean, in standard deviations:

• A value above the mean – a positive standard score; below the mean – a negative one; the mean itself – 0.
• A standard score gives a value’s relative position in its own group, so it lets us compare values from different groups – different means, different standard deviations, even different units.
• A standard score can be worked out in any set of numerical data, whatever the shape of its distribution.

When the standard score is known, the value is the mean plus $\displaystyle z$ standard deviations:

In Dana’s year group’s maths test ($\displaystyle {\bar{x}}={75}$, $\displaystyle {S}={10}$):

• the standard score $\displaystyle {z}={1.5}$ belongs to the mark $\displaystyle {75+1.5\cdot10}={90}$;
• the standard score $\displaystyle {z}={-0.8}$ belongs to the mark $\displaystyle {75-0.8\cdot10}={67}$.

In a test:
The mark 90 has the standard score 2.
The mark 60 has the standard score $\displaystyle -1$.
What are the mean and the standard deviation?

By $\displaystyle {x}={\bar{x}+z\cdot S}$ we get a system of two equations:

• Subtracting the second equation from the first: $\displaystyle {3S}={30}$, so $\displaystyle {S}={10}$.
• Substituting in the second: $\displaystyle {\bar{x}-10}={60}$, so $\displaystyle {\bar{x}}={70}$.

The marks in a maths exam:
At the first sitting the mean was 76.6, and the standard deviation 10.
At the second sitting the mean was 80, and the standard deviation 5.
Rami sat the exam at both sittings and got 82 at each.

Work out Rami’s standard score at the first sitting.

Work out his standard score at the second sitting.

At which sitting did Rami do better compared with the others who sat the exam? Explain.

In a race the mean time is 64 minutes, and the standard deviation 12 minutes.

The standard score of Omer’s time is 1.5. In how many minutes did he finish the race?

The standard score of Shira’s time is $\displaystyle -0.75$. In how many minutes did she finish?

Ayala and Pnina sat a final test.
Ayala’s standard score is 1.25, and her mark 92.
Pnina’s standard score is $\displaystyle -0.5$, and her mark 64.

Is Pnina’s mark above the mean or below it? Explain.

Work out the mean and the standard deviation of the marks in the test. Call the mean $\displaystyle m$ and the standard deviation $\displaystyle s$.

In a set of data:
The standard score of the value 70 is $\displaystyle z_1$.
The standard score of the value 74 is $\displaystyle z_2$.
It is known that $\displaystyle {z_1}={3z_2}$.
Work out the mean. Call it $\displaystyle m$, and write an equation with no letter but $\displaystyle m$.