The weighted mean

Science and society, grade 10 (3 units) Lesson 8 of 15Learning

The weighted mean

A final mark by weights

When the values do not all count the same, work out a weighted mean: multiply each value by its weight, add, and divide by the sum of the weights.

\displaystyle \bar{x}=\frac{w_1x_1+w_2x_2+\dots+w_kx_k}{w_1+w_2+\dots+w_k}

Noa’s final mark in a subject is made up of her school mark, weighted 30%, and her matriculation exam mark, weighted 70%.
Her school mark is 90, and her exam mark 80.

\frac{30\cdot90+70\cdot80}{30+70}=\frac{2700+5600}{100}=83

When the weights are percentages adding up to 100%, each mark can be multiplied by its weight as a decimal and the products added: $\displaystyle {0.3\cdot90+0.7\cdot80}={27+56}={83}$ .
The ordinary mean, $\displaystyle {\frac{90+80}{2}}={85}$ , is not the final mark: the exam counts more, and the final mark is nearer to it.

The mean of two groups

The mean of two groups together is a weighted mean of their means, the weights being the number of people in each group.

Two classes in a year group took the same test:
10A has 20 pupils, and their mean is 80.
10B has 30 pupils, and their mean is 70.

\frac{20\cdot80+30\cdot70}{20+30}=\frac{1600+2100}{50}=74

$\displaystyle 20\cdot80$ is the sum of the marks in 10A, and $\displaystyle 30\cdot70$ that in 10B.
The year group’s mean is 74, not 75: it is nearer to the mean of the larger class.

A frequency table as a weighted mean

The mean of a frequency table is a weighted mean: each value’s weight is its frequency.

A falafel stall sold 100 portions in one day, in three sizes:

Price of a portion (₪)	18	22	25
Portions sold	30	50	20

\frac{30\cdot18+50\cdot22+20\cdot25}{100}=\frac{540+1100+500}{100}=21.4

The mean price of a portion sold: 21.4 ₪. It is also the stall’s takings, 2,140 ₪, divided by the number of portions.

The mark needed to reach a mean

When the weighted mean is known and one value is missing, the missing value is the solution of an equation.

Daniel’s school mark is 80 (30%).
He wants a final mark of 87.
What mark does he need in the exam (70%)?

Call the exam mark $\displaystyle x$ :

0.3\cdot80+0.7x=87

$\displaystyle {24+0.7x}={87}$ , so $\displaystyle {0.7x}={63}$ .
$\displaystyle {x}={\frac{63}{0.7}}={90}$ .

Check: $\displaystyle {0.3\cdot80+0.7\cdot90}={24+63}={87}$ . Daniel needs 90 in the exam.

Exercise 1

The final mark is made up of 30% school mark and 70% exam mark.

a

Work out Omer’s final mark: school 85, exam 75.

/ – 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 Maya’s final mark: school 60, exam 95.

/ – 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

The mark in a college course has three parts:
homework – 20%, a mid-term quiz – 30%, a final exam – 50%.
Lior’s marks: homework 90, quiz 70, exam 80.

Work out the course mark.

/ – 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 3

A chain of shops has two branches.
Branch A has 40 employees, and their mean age is 30.
Branch B has 60 employees, and their mean age is 25.

a

Work out the mean age of all the chain’s employees.

/ – 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

Why is the mean not 27.5?

Exercise 4

200 tickets were sold for one showing at a cinema:

Price of a ticket (₪)	40	30	20
Tickets	120	60	20

Work out the mean price of a ticket.

/ – 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 5

The final mark is made up of 30% school mark and 70% exam mark.
Yael’s school mark is 70, and she wants a final mark of 84.

Write an equation and solve it: what mark does Yael need in the exam?

Exercise 6

A class has 30 pupils: 12 boys and 18 girls.
The boys’ mean in a test is 78, and the whole class’s mean is 81.

Write an equation and solve it: what is the girls’ mean?

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

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
$\displaystyle n\cdot P(A)$