Grouped data

Science and society, grade 10 (3 units) Lesson 9 of 15Learning

Grouped data

Classes

When there are many different values, they are grouped in classes – intervals of equal width – and the data in each class are counted. These are grouped data.

A company asked 40 employees how many minutes their journey to work takes:

Minutes	0–10	10–20	20–30	30–40	40–50
Employees	5	13	12	7	3

The class 10–20 is “from 10 to less than 20”: an employee whose journey takes 20 minutes is counted in the class 20–30.
The table shows the overall picture, but no longer the data themselves: nobody knows how long each employee’s journey takes exactly.

The histogram

A histogram is a bar chart of grouped data: a bar for each class, its height the class’s frequency. The bars touch, since each class goes on where the one before it ends; the classes’ bounds are written under them.

The class with the highest frequency is the modal class: here 10–20 minutes, with 13 employees.

The class middle and an estimate of the mean

From grouped data only an estimate of the mean can be worked out: assume all the data of a class lie at the class middle, and work out a weighted mean.

\displaystyle \bar{x}\approx\frac{m_1f_1+m_2f_2+\dots+m_kf_k}{f_1+f_2+\dots+f_k}

$\displaystyle m_1, \dots, m_k$ – the class middles, $\displaystyle f_1, \dots, f_k$ – their frequencies.

On the way to work: the middle of the class 0–10 is $\displaystyle {\frac{0+10}{2}}={5}$ , and the other classes’ middles are 15, 25, 35 and 45.

\bar{x}\approx\frac{5\cdot5+15\cdot13+25\cdot12+35\cdot7+45\cdot3}{40}=\frac{25+195+300+245+135}{40}=\frac{900}{40}=22.5

The journey to work takes about 22.5 minutes on average. It is an estimate: within each class the real data may be above or below the middle.

The median’s class

From grouped data the median cannot be found exactly, but the median’s class – the class the median is in – can.

There are 40 employees, so the median lies between the 20th and the 21st value of the ordered list.
Count from the first class:

The class 0–10 – 5 employees: values 1–5.
The class 10–20 – 13 employees: values 6–18.
The class 20–30 – 12 employees: values 19–30.

The 20th and 21st values are in the class 20–30, and that is the median’s class. At least half the employees get to work in less than 30 minutes.

Exercise 1

The heights of 50 Year 10 pupils, in cm:

a

What is the modal class?

b

Work out an estimate of the mean height.

/ – 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

What is the median’s class? Explain.

Exercise 2

A mobile company checked how many gigabytes (GB) of data 80 subscribers used in one month:

GB a month	0–10	10–20	20–30	30–40
Subscribers	16	30	24	10

a

Work out an estimate of the mean use.

/ – 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

What is the median’s class?

c

How many subscribers used 20 GB or more?

/ – 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

d

The company is thinking of offering all its subscribers a 20 GB plan a month.
Will the plan be enough for most of them? Explain with your answers.

Exercise 3

The waiting times of 20 patients in an emergency room, in minutes:
$\displaystyle 12, 35, 48, 27, 8, 55, 31, 19, 42, 24$
$\displaystyle 37, 15, 29, 51, 33, 22, 46, 38, 26, 44$

a

Fill in the frequency of each of the classes 0–15, 15–30, 30–45, 45–60.

0–15

/ – 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

15–30

/ – 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

30–45

/ – 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

45–60

/ – 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

From the frequencies, work out an estimate of the mean waiting time.

/ – 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

Work out the exact mean from the data themselves.

/ – 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

d

Why are the estimate and the exact mean different?

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
$\displaystyle n\cdot P(A)$