The mean and median of economic data

Finance and economics, grade 10 (3 units) Lesson 13 of 15Learning

The mean and median of economic data

One very high salary

8 people work in a small company.
These are their monthly salaries, in shekels, from the lowest to the highest – the last is the manager’s:
$\displaystyle 6000, 6500, 7000, 7000, 7500, 8000, 9000, 37000$

The mean: $\displaystyle {\frac{88000}{8}}={11000}$ ₪.
The median: there are two values in the middle, the fourth and the fifth, so $\displaystyle {\frac{7000+7500}{2}}={7250}$ ₪.

The manager’s salary is an outlier: it pulls the mean up, and 7 of the 8 workers earn less than the mean. The median hardly moves because of it: had the manager earned 12000 ₪, the median would still be 7250 ₪.

The measure of centre for salaries

The two measures of centre answer two different questions:

The mean – what each worker would get if all the salaries were shared out equally. It fits the question of how much is spent on salaries: $\displaystyle {8\cdot11000}={88000}$ ₪ a month.
The median – half the workers earn less than it or exactly it, and half more. It fits the question of how much a usual worker earns.

Salaries usually include a few very high ones, so the median is lower than the mean. When the news reports “the average wage in the economy”, most workers earn less than it.

Exercise 1

9 people work in a garage.
These are their monthly salaries, in shekels:
$\displaystyle 6000, 6200, 6500, 6500, 7000, 7300, 8000, 9500, 24000$

Work out the mean salary.

/ – 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

Find the median of the salaries.

/ – 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 new worker asks how much people earn at the garage.
Which measure of centre describes it better? Explain.

The mean from a frequency table

A chain of shops asked 20 young workers how much they get an hour:

Hourly wage (₪)	35	40	45	60
Workers	10	6	3	1

In a frequency table you multiply each value by its frequency, add, and divide by the number of values:

\displaystyle \bar{x}=\frac{x_1f_1+x_2f_2+\dots+x_kf_k}{f_1+f_2+\dots+f_k}

\bar{x}=\frac{35\cdot10+40\cdot6+45\cdot3+60\cdot1}{20}=\frac{785}{20}=39.25

The mean: 39.25 ₪ an hour.
The median: 20 values, so it is the mean of the 10th and the 11th. Values 1–10 are 35, and the 11th is 40: $\displaystyle {\frac{35+40}{2}}={37.5}$ ₪.
The mode: 35 ₪ – the wage of half the workers.

Exercise 2

A survey asked 20 young workers their monthly salary:

Salary (₪)	4000	5000	6000	8000
Workers	5	8	4	3

Work out the mean salary.

/ – 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 salaries.

/ – 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

Prices over time

A line chart shows how a price changes from month to month.
This is the price of a kilogram of tomatoes at a greengrocer’s, in the first half of the year:

The highest price was in March, 9 ₪, and the lowest in January and June, 5 ₪.
The biggest rise: from February to March, from 6 ₪ to 9 ₪ – a rise of $\displaystyle {\frac{3}{6}}={50\%}$ .
The mean price over the half year: $\displaystyle {\frac{5+6+9+8+6+5}{6}}={\frac{39}{6}}={6.5}$ ₪.

Exercise 3

This is the price of a litre of petrol at a filling station, in the first half of the year:

In which month was the price highest?

Work out the mean price per litre over the half year.

/ – 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

Ron fills up 40 litres each time.
How many shekels more does a fill-up cost in March than in January?

/ – 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

An axis that does not start at 0

The monthly price of a phone plan at one company, over four years:

This is the same chart, but its axis starts at 49:

Both charts hold the same data: the price rose from 50 ₪ to 54 ₪, a rise of $\displaystyle {\frac{4}{50}}={8\%}$ .
In the second chart the 2024 bar is 5 times as tall as the 2021 bar, because only the part above 49 shows.
Before drawing a conclusion from a chart, check where its axis starts.

Exercise 4

An internet company’s advert says: “Our customers have grown 6 times over!”
Beside it is this chart:

By what percent did the number of customers grow from 2022 to 2024?

/ – 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

Why does the chart make the number look 6 times as big?

Formulas: finance and economics, grade 10 (3 units)

Open as a page

Buying and selling

The unit price – $\displaystyle P$ the price of the pack, $\displaystyle n$ the number of units in it (grams, litres, items)
$\displaystyle u=\frac{P}{n}$
A $\displaystyle p\%$ discount – $\displaystyle a$ the price before the discount, $\displaystyle b$ the price after it
$\displaystyle b$ $\displaystyle =$ $\displaystyle (100\%-p\%)\cdot a$
A price with 18% VAT – $\displaystyle a$ the price without VAT, $\displaystyle b$ the price with VAT
$\displaystyle b$ $\displaystyle =$ $\displaystyle 1.18\cdot a$

A linear model

The total cost – $\displaystyle x$ the number of units, $\displaystyle m$ the cost per unit (the slope), $\displaystyle b$ the fixed cost
$\displaystyle y=mx+b$
The cost per unit from two points $\displaystyle (x_1,y_1)$ , $\displaystyle (x_2,y_2)$ on the line
$\displaystyle m=\frac{y_2-y_1}{x_2-x_1}$
Two plans cost the same – where their two lines meet
$\displaystyle m_1x+b_1$ $\displaystyle =$ $\displaystyle m_2x+b_2$

Profit and loss

The total cost – $\displaystyle F$ the fixed costs, $\displaystyle c$ the cost of one unit, $\displaystyle x$ the number of units
$\displaystyle C=F+c\cdot x$
The revenue – $\displaystyle s$ the selling price of one unit
$\displaystyle R=s\cdot x$
The profit – the revenue minus the cost (a negative profit is a loss)
$\displaystyle P=R-C$
The profit percent – out of the cost
$\displaystyle r\%=\frac{P}{C}$
The break-even point – the revenue equals the cost
$\displaystyle s\cdot x$ $\displaystyle =$ $\displaystyle F+c\cdot x$

Wages

Pay – $\displaystyle w$ the hourly wage, $\displaystyle h$ the number of hours
$\displaystyle S=w\cdot h$
The pay for an overtime hour, in the first two overtime hours of a day
$\displaystyle 1.25\cdot w$
The pay for an overtime hour, from the third overtime hour of a day
$\displaystyle 1.5\cdot w$
Net pay – $\displaystyle G$ the gross pay, $\displaystyle D$ the deductions
$\displaystyle N=G-D$

Measures of centre

The mean of $\displaystyle n$ values
$\displaystyle \bar{x}=\frac{x_1+x_2+\dots+x_n}{n}$
The mean from a frequency table – $\displaystyle f_i$ the frequency of the value $\displaystyle x_i$
$\displaystyle \bar{x}=\frac{x_1f_1+x_2f_2+\dots+x_kf_k}{f_1+f_2+\dots+f_k}$
The place of the median in an ordered list of $\displaystyle n$ values
$\displaystyle \frac{n+1}{2}$