Measures of centre and choosing one

Science and society, grade 10 (3 units) Lesson 7 of 15Learning

Measures of centre and choosing one

Pay in a small company

When the data hold a few very large values, the mean is pulled towards them, and the median describes the ordinary value better.

A small high-tech company has 9 employees.
Each one’s monthly pay, in thousands of shekels, before tax:

The mean: $\displaystyle {\frac{9+10+10+11+12+12+12+14+45}{9}}={\frac{135}{9}}={15}$ thousand ₪.
The median: the fifth value in the ordered list – 12 thousand ₪.
The mode: 12 thousand ₪.

Eight of the nine employees earn less than the mean. The manager’s pay, 45 thousand ₪, is an outlier: it pulls the mean up, and does not affect the median.

What an outlier does to each measure

A change in one value always changes the mean; it changes the median and the mode only if that value is in the middle or is the most common one.

The manager’s pay rose from 45 to 72 thousand ₪, and everyone else’s stayed the same:

The sum grew by 27, to 162. The new mean: $\displaystyle {\frac{162}{9}}={18}$ thousand ₪ – up by $\displaystyle {\frac{27}{9}}={3}$ .
The median stayed 12: the fifth value did not change.
The mode stayed 12.

By the mean, “pay in the company rose by 20%”. But no employee other than the manager got a rise.

Choosing the measure that fits

The measure of centre is chosen by the question it is to answer:

The median – when there are outliers, and the ordinary value is wanted: pay, house prices, waiting times. That is why the news often reports “the median wage”.
The mean – when the total matters: the mean times the number of data is the sum. A budget for all the employees’ pay, the water a town needs, a team’s points.
The mode – when the most common value is wanted, also in data that are not numbers: the shoe size to order most of, the bus line most pupils take.

The same data in two quotes

Two measures of the same data can tell different stories, both of them true. So on reading “the average wage” or “the typical wage”, ask: which measure is it?

In the company of the example:

The management says: “Our mean pay is 15 thousand ₪” – true.
The employees say: “Most of us earn 12 thousand ₪ or less” – also true.

Exercise 1

The prices of seven homes sold on one street last year, in millions of shekels:
$\displaystyle 1.8, 2.0, 2.1, 2.1, 2.3, 2.5, 9.6$
The last home is a large villa.

Work out the mean price.

/ – 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

What is the median?

/ – 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

A family wants to know roughly what an ordinary home on the street costs. Which measure fits? Explain.

Exercise 2

The times of 8 runners in a 5 km run, in minutes:
$\displaystyle 22, 24, 25, 25, 26, 27, 28, 31$
The mean is 26 minutes, and the median 25.5 minutes.
A ninth runner walked most of the way, and finished in 62 minutes.

Work out the mean of the nine runners.

/ – 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

Find the median of the nine runners.

/ – 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

Which measure changed more? Explain why.

Exercise 3

Which measure of centre fits each case? Explain each answer.

A shoe shop decides which size to order the most pairs of.
The water authority plans how much water all the town’s households need together.
A news story on rents in a city with a few luxury towers.

Exercise 4

The monthly pay of the 10 workers in a small factory, in thousands of shekels:
$\displaystyle 7, 7, 7, 8, 8, 9, 10, 12, 15, 37$

Work out the mean pay.

/ – 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

Work out the median pay.

/ – 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

The management wants to show that pay at the factory is high, and the workers’ committee that it is low.
Which measure will each of them quote? What will each say?

Exercise 5

At the factory of Exercise 4, every worker got a rise of a thousand ₪ a month.

What happened to the mean, the median and the mode? Explain without working out all the data again.

Formulas: science and society, grade 10 (3 units)

Open as a page

Information and representations

$\displaystyle p$ percent of $\displaystyle n$
$\displaystyle p\%\cdot n$ $\displaystyle =$ $\displaystyle \frac{p}{100}\cdot n$
A rate per 1,000 people – $\displaystyle k$ the number of cases, $\displaystyle N$ the number of people
$\displaystyle \frac{k}{N}\cdot1000$
The average rate of change between the points $\displaystyle (x_1,y_1)$ and $\displaystyle (x_2,y_2)$
$\displaystyle \frac{y_2-y_1}{x_2-x_1}$
A linear relation – $\displaystyle m$ the change in $\displaystyle y$ when $\displaystyle x$ grows by 1, $\displaystyle b$ the value of $\displaystyle y$ when $\displaystyle {x}={0}$
$\displaystyle y=mx+b$

Measures of centre

The mean of $\displaystyle n$ data
$\displaystyle \bar{x}=\frac{x_1+x_2+\dots+x_n}{n}$
A weighted mean – $\displaystyle w_1, \dots, w_k$ the weights of the values $\displaystyle x_1, \dots, x_k$
$\displaystyle \bar{x}=\frac{w_1x_1+w_2x_2+\dots+w_kx_k}{w_1+w_2+\dots+w_k}$
The mean of grouped data – $\displaystyle m_1, \dots, m_k$ the class middles, $\displaystyle f_1, \dots, f_k$ their frequencies
$\displaystyle \bar{x}\approx\frac{m_1f_1+m_2f_2+\dots+m_kf_k}{f_1+f_2+\dots+f_k}$
The place of the median in an ordered list of $\displaystyle n$ data
$\displaystyle \frac{n+1}{2}$

Probability

Equally likely outcomes – $\displaystyle k$ the outcomes in the event $\displaystyle A$ , $\displaystyle n$ all the outcomes
$\displaystyle P(A)=\frac{k}{n}$
The complement $\displaystyle \overline{A}$ – $\displaystyle A$ does not happen
$\displaystyle P(\overline{A})$ $\displaystyle =$ $\displaystyle 1-P(A)$
A path of a tree diagram – the product of the probabilities $\displaystyle p_1$ , $\displaystyle p_2$ on its branches
$\displaystyle P=p_1\cdot p_2$
An event of several paths – the sum of their probabilities
$\displaystyle P(A)=P_1+P_2+\ldots$
Estimating a probability from relative frequency – $\displaystyle A$ happened $\displaystyle f$ times in $\displaystyle n$ experiments
$\displaystyle P(A)\approx\frac{f}{n}$
How many times an event $\displaystyle A$ is expected to happen in $\displaystyle n$ experiments
$\displaystyle n\cdot P(A)$