The mode and the range

Data and probability, grade 8 Lesson 5 of 14Learning

The mode and the range

Measures of centre

A measure of centre is one number that stands for a whole list of data: what is typical of it, where its “centre” is.

There are three measures of centre: the mode, the mean and the median.
Each describes the data in its own way, so they sometimes come out different.

The mode

The mode is the value that comes up most often in the data.

We asked 11 pupils how many brothers and sisters they have.
The answers: $\displaystyle 1, 2, 0, 1, 3, 1, 2, 1, 0, 2, 4$

1 comes up four times, 2 three times, 0 twice, 3 and 4 once each.
The mode is 1: it is the most common answer.

The mode of categorical data

Categorical data – a sport, a colour, a town – have a mode too: the answer chosen most often.

We asked the pupils of class 8B which sport they like best:

The mode: football.
The mode is the value – football – not its frequency, 9.

Two modes, or none

When two values come up most often, equally often, the data have two modes.

A die was rolled 8 times: $\displaystyle 2, 5, 3, 5, 6, 2, 1, 4$ .
2 and 5 came up twice each: the modes are 2 and 5.
The ages of 5 cousins: $\displaystyle 3, 7, 9, 12, 15$ .
Each value comes up only once: no value is more common than the others, and there is no mode worth naming.

The mode in a frequency table and a chart

In a frequency table the mode is the value with the largest frequency.

The shoe sizes of 25 pupils:

Size	36	37	38	39	40	41
Frequency	2	5	8	6	3	1

The largest frequency is 8: eight pupils wear size 38.
The mode: size 38.

In a bar chart the mode is the value under the tallest bar:

The range

The range is the difference between the largest value and the smallest value in the data:

\displaystyle R=x_{\max}-x_{\min}

The highest temperatures in Haifa, in one week in spring, in degrees:
$\displaystyle 18, 21, 24, 19, 25, 23, 17$

The largest: 25. The smallest: 17.
The range: $\displaystyle {25-17}={8}$ degrees.

The spread of the data

The range shows how spread out the data are: with a small range the data are close to each other, with a large range some of them are far apart.

The points two basketball players scored in their last five games:

Yoav: $\displaystyle 12, 14, 14, 15, 13$ – the mode 14, the range $\displaystyle {15-12}={3}$ .
Itai: $\displaystyle 4, 14, 14, 25, 8$ – the mode 14, the range $\displaystyle {25-4}={21}$ .

Both have the same mode, but Yoav scores about the same number of points in every game, while Itai’s number changes a lot from game to game.

The range in a frequency table

The range is a difference between values – sizes – not between frequencies.

In a frequency table, look for the largest and the smallest value in the row of values.

In the shoe sizes above: the largest 41, the smallest 36, and the range $\displaystyle {41-36}={5}$ .

Exercise 1

The number of goals a football team scored in 12 games:
$\displaystyle 2, 0, 3, 1, 1, 4, 2, 1, 0, 1, 5, 2$

Find the mode.

/ – 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

Work out the range.

/ – 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

We asked the pupils of class 8C which fruit they like best:

Find the mode.

Roy worked out $\displaystyle {8-3}={5}$ , and said the range is 5.
Do these data have a range? Explain.

Exercise 3

How many books 30 pupils read in the summer holidays:

Books	0	1	2	3	4	6
Frequency	3	7	9	6	3	2

Find the mode.

/ – 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 range.

/ – 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 4

The lowest temperatures in Jerusalem, in one week in winter, in degrees:
$\displaystyle 2, -1, 5, 7, 3, 0, 4$

Work out the range.

/ – 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

Find the mode of each list – whether it has one mode, two, or none.

$\displaystyle 5, 8, 8, 3, 8, 2$
$\displaystyle 4, 9, 4, 9, 1$
$\displaystyle 11, 13, 15, 17$

Exercise 6

Write a list of 6 numbers whose mode is 6 and whose range is 10.

Formulas: data and probability, grade 8

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Relative frequency

The relative frequency of an answer – $\displaystyle f$ its frequency, $\displaystyle n$ the total of all the frequencies
$\displaystyle \frac{f}{n}$

Measures of centre and the range

The mean of $\displaystyle n$ data
$\displaystyle \bar{x}=\frac{x_1+x_2+\dots+x_n}{n}$
The mean from a frequency table – $\displaystyle f_1, \dots, f_k$ the frequencies of the values $\displaystyle x_1, \dots, x_k$
$\displaystyle \bar{x}=\frac{x_1f_1+x_2f_2+\dots+x_kf_k}{f_1+f_2+\dots+f_k}$
The place of the median in an ordered list of $\displaystyle n$ data
$\displaystyle \frac{n+1}{2}$
The range – the largest value minus the smallest
$\displaystyle R=x_{\max}-x_{\min}$

Probability

Equally likely outcomes – $\displaystyle k$ the outcomes in the event $\displaystyle A$ , $\displaystyle n$ all the outcomes
$\displaystyle P(A)=\frac{k}{n}$
An impossible event – 0, a certain event – 1
$\displaystyle 0\le P(A)\le1$
The complement $\displaystyle \overline{A}$ – $\displaystyle A$ does not happen
$\displaystyle P(\overline{A})$ $\displaystyle =$ $\displaystyle 1-P(A)$
How many times an event $\displaystyle A$ is expected to happen in $\displaystyle n$ experiments
$\displaystyle n\cdot P(A)$