The standard score: the distance from the mean in standard deviations

Science and society, grade 12 (3 units) Lesson 11 of 14Learning

The standard score: the distance from the mean in standard deviations

A mark in maths and a mark in English

Noa got 85 in a maths exam and 80 in an English exam. In which subject did she do better, compared with her year?
In maths the year’s mean is 70 and the standard deviation 10.
In English the mean is 65 and the standard deviation 5.

In maths Noa is 15 points above the mean – one and a half standard deviations: $\displaystyle {\frac{85-70}{10}}={1.5}$ .
In English she is the same 15 points above the mean, but there that is three standard deviations: $\displaystyle {\frac{80-65}{5}}={3}$ .

In English the marks are packed close to the mean, so 80 stands out much more there: Noa did better in English, though the mark is lower.

The standard score

The standard score of a value is its distance from the mean, counted in standard deviations:

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

A value above the mean – a positive standard score; below it – a negative one; the mean itself – 0.
Noa got 60 in maths in the exam before: $\displaystyle {z}={\frac{60-70}{10}}={-1}$ , one standard deviation below the mean.
The standard score does not depend on units: marks, heights, times and weights can all be compared this way.

From a standard score to a value

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

\displaystyle x=\mu+z\sigma

For example, in maths the standard score $\displaystyle {z}={2}$ belongs to the mark $\displaystyle {70+2\cdot10}={90}$ , and the standard score $\displaystyle {z}={-0.5}$ to the mark $\displaystyle {70-0.5\cdot10}={65}$ .

A value from a percent

The 34/13.5/2.5 rule also gives the way back: from a percent to a value.

A programme for gifted pupils takes the top 2.5% in an IQ test, where $\displaystyle {\mu}={100}$ and $\displaystyle {\sigma}={15}$ . From what score are pupils accepted?

The top 2.5% are more than two standard deviations above the mean: $\displaystyle {z}={2}$ .
$\displaystyle {x}={100+2\cdot15}={130}$ : from a score of 130 up.

The percent	The boundary	$\displaystyle z$
The top 2.5%	above $\displaystyle \mu+2\sigma$	$\displaystyle 2$
The top 16%	above $\displaystyle \mu+\sigma$	$\displaystyle 1$
The bottom 16%	below $\displaystyle \mu-\sigma$	$\displaystyle -1$
The bottom 2.5%	below $\displaystyle \mu-2\sigma$	$\displaystyle -2$

Exercise 1

The scores in a university entrance exam are normally distributed, with mean 500 and standard deviation 100.

a

Ron scored 650. Work out his standard score.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

b

Dana’s standard score is $\displaystyle -0.5$ . What is her score?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 2

Dana competes in athletics.
In the long jump she jumped 5.6 metres. The results of girls her age: mean 5 metres, standard deviation 0.3 metres.
In the 100 metres she ran 13.2 seconds. The results of girls her age: mean 14 seconds, standard deviation 0.4 seconds.

a

Work out her standard score in the long jump.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

b

Work out her standard score in the run.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

c

In a run a low time is a good result. In which event does Dana stand out more among girls her age? Explain.

Exercise 3

Birth weight is normally distributed, with mean 3,300 grams and standard deviation 500 grams.
The standard score of a baby’s birth weight is $\displaystyle -1.5$ . How much did the baby weigh?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 4

The weight of apples in an orchard is normally distributed, with mean 180 grams and standard deviation 20 grams.

a

The heaviest 2.5% of the apples are exported. From what weight?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

b

The lightest 16% of the apples go to make juice. Up to what weight?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Formulas: science and society, grade 12 (3 units)

A quadratic model

A quadratic model – $\displaystyle a<0$ : it has a highest point, $\displaystyle a>0$ : a lowest point
$\displaystyle y=ax^{2}+bx+c$ $\displaystyle a\neq0$
The vertex’s $\displaystyle x$ – the axis of symmetry
$\displaystyle x=-\frac{b}{2a}$
The vertex’s $\displaystyle x$ – midway between two values of $\displaystyle x$ with the same $\displaystyle y$
$\displaystyle x=\frac{x_1+x_2}{2}$
The quadratic formula for $\displaystyle {ax^{2}+bx+c}={0}$ – the zeros
$\displaystyle x_{1,2}=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$

Other models

A linear relation – the differences in the table are equal
$\displaystyle y=mx+b$
An exponential relation – the ratio of two neighbouring values in the table is fixed, $\displaystyle b$
$\displaystyle y=a\cdot b^{x}$

The normal distribution

The standard deviation of $\displaystyle n$ data – $\displaystyle \bar{x}$ their mean
$\displaystyle \sigma=\sqrt{\frac{(x_1-\bar{x})^{2}+(x_2-\bar{x})^{2}+\dots+(x_n-\bar{x})^{2}}{n}}$
In a normal distribution – $\displaystyle \mu$ the mean, $\displaystyle \sigma$ the standard deviation: about 68% of the data within one standard deviation of the mean (34% on each side)
$\displaystyle P(\mu-\sigma<x<\mu+\sigma)\approx68\%$
About 95% within two standard deviations of the mean (another 13.5% on each side)
$\displaystyle P(\mu-2\sigma<x<\mu+2\sigma)\approx95\%$
About 2.5% above $\displaystyle \mu+2\sigma$ , and about 2.5% below $\displaystyle \mu-2\sigma$
$\displaystyle P(x>\mu+2\sigma)\approx2.5\%$
How many of $\displaystyle n$ data are $\displaystyle p$ percent of them
$\displaystyle n\cdot p\%$

The standard score

The standard score of the value $\displaystyle x$ – how many standard deviations it is above the mean (or below it, when $\displaystyle z<0$ )
$\displaystyle z=\frac{x-\mu}{\sigma}$
The value whose standard score is $\displaystyle z$
$\displaystyle x=\mu+z\cdot\sigma$