The standard score and quality control

Finance and economics, grade 12 (3 units) Lesson 14 of 16Learning

The standard score and quality control

The standard score

The standard score, $\displaystyle z$ , says how many standard deviations a value is from the mean: the difference between the value and the mean, divided by the standard deviation.

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

A value above the mean has a positive standard score; a value below the mean, a negative one.
A value equal to the mean has a standard score of 0.

At a service centre the mean salary is $\displaystyle {\mu}={9000}$ ₪, and the standard deviation $\displaystyle {\sigma}={1000}$ ₪. The standard score of a salary of 10,500 ₪:

z=\frac{10500-9000}{1000}=\frac{1500}{1000}=1.5

A salary of 7500 ₪ is as far from the mean, but below it: $\displaystyle {z}={\frac{7500-9000}{1000}}={-1.5}$ .

Comparing two distributions

A standard score compares values from different distributions: it says where each value stands within its own group.

A chain of shops wants to give a prize to its best salesperson:
Dana sold 62 units in a day, at a branch where the mean is 50 units and the standard deviation 6.
Yossi sold 75 units in a day, at a branch where the mean is 60 units and the standard deviation 10.

The example below works out each one’s standard score:

Dana: $\displaystyle {z}={\frac{62-50}{6}}={\frac{12}{6}}={2}$ .
Yossi: $\displaystyle {z}={\frac{75-60}{10}}={\frac{15}{10}}={1.5}$ .

Yossi sold more units, but Dana is further above her branch’s mean: compared with her branch, she is the better salesperson.

A value from a percent

Sometimes you know the percent and look for the value. The 34/13.5/2.5 rule gives the place on the axis, and from it the value:

\displaystyle x

\displaystyle =

\displaystyle \mu+z\cdot\sigma

At Dana’s branch (mean 50, standard deviation 6) the chain gives a bonus on the 2.5% of the days with the highest sales. From how many units a day?

The top 2.5% lie above $\displaystyle \mu+2\sigma$ , that is $\displaystyle {z}={2}$ .
$\displaystyle {x}={50+2\cdot6}={62}$ units.

And the top 16%? Above $\displaystyle \mu+\sigma$ lie $\displaystyle {13.5\%+2.5\%}={16\%}$ , so from $\displaystyle {50+6}={56}$ units a day.

Quality control

A machine fills one-litre bottles of juice.
The amount in a bottle is spread normally: the mean is 1000 ml, and the standard deviation 4 ml.
A bottle whose amount is further from the mean than two standard deviations leaves the line.

$\displaystyle {\mu-2\sigma}={1000-2\cdot4}={992}$ ml, and $\displaystyle {\mu+2\sigma}={1008}$ ml.
A bottle with less than 992 ml or more than 1008 ml leaves the line.

95% of the bottles are between 992 and 1008 ml, so $\displaystyle {2.5\%+2.5\%}={5\%}$ leave the line.
Out of 1000 bottles: $\displaystyle {0.05\cdot1000}={50}$ bottles or so.

This is how the machine itself is checked too: if 120 bottles leave the line in an hour of 1000 bottles – far more than 50 – the machine has probably gone wrong and needs setting.

Exercise 1

At a high-tech company the monthly salaries are spread normally.
The mean salary is 20,000 ₪, and the standard deviation 4000 ₪.

What is the standard score of a salary of 26,000 ₪?

/ – 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