Chapter 11 Language of Descriptive Statistics

Section 11.1 Terminology and Language

11.1.2 Rounding

The rounding of measurement values is an everyday process.
Info 11.1.3

In principle, there are three ways of rounding:
• Rounding (off) using the $\text{floor}$ function $⌊x⌋$.
• Rounding (up) using the $\text{ceil}$ function $⌈x⌉$.
• Rounding using the $\text{round}$ function (sometimes also called $\text{rnd}$ function).

The $\text{floor}$ function is defined as

$\text{floor}:\mathrm{ }ℝ\to ℝ\mathrm{ }\mathrm{ },\mathrm{ }\mathrm{ }x\mathrm{ }⟼\mathrm{ }\text{floor}\left(x\right)\mathrm{ }=\mathrm{ }⌊x⌋\mathrm{ }=\mathrm{ }max\left\{k\in ℤ : k\le x\right\} .$

If $x\in ℝ$ is a real number, then $\text{floor}\left(x\right)=⌊x⌋$ is the largest integer that is smaller than or equal to $x$. It results from rounding off the value of $x$. If a positive real number $x$ is written as a decimal, then $⌊x⌋$ equals the integer on the left of the decimal point: rounding (off) cuts off the digits on the right of the decimal point. For example $⌊3.142⌋=3$ but $⌊-2.124⌋=-3$. The $\text{floor}$ function is a step function with jumps (in more mathematical terms, jump discontinuities) of height $1$ at all points $x\in ℤ$. The function values at the jumps always lie a step up. They are indicated by the small circles in the figure below, which shows the graph of the $\text{floor}$ function.

Graph of the $\text{floor}$ function

Let a real number $a\ge 0$ be given, written as a decimal number

$a\mathrm{ }=\mathrm{ }{g}_{n} {g}_{n-1} \dots {g}_{1} {g}_{0} . {a}_{1} {a}_{2} {a}_{3} \dots$

This number $a$ can be rounded to $r$ fractional digits ($r\in ℕ{}_{0}$) using the $\text{floor}$ function by

$\stackrel{~}{a}\mathrm{ }=\mathrm{ }\frac{1}{{10}^{r}}·⌊{10}^{r}·a⌋ .$

This process of rounding cuts off the decimal after the $r$th fractional digit. Thus, rounding using the $\text{floor}$ function is in general a rounding off.
Example 11.1.4
Rounding the number ${a}_{1}=2.3727$ to 2 fractional digits using the $\text{floor}$ function results in

${\stackrel{~}{a}}_{1}\mathrm{ }=\mathrm{ }\frac{1}{{10}^{2}}·⌊{10}^{2}·2.3727⌋\mathrm{ }=\mathrm{ }\frac{1}{{10}^{2}}·⌊237.27⌋\mathrm{ }=\mathrm{ }\frac{1}{{10}^{2}}·237\mathrm{ }=\mathrm{ }=2.37 .$

Alternatively, it can be rounded by cutting off the decimal after the second fractional digit (however, this is only possible if the number is given as a decimal which is rarely the case in a computer program).
Rounding the number ${a}_{2}=\sqrt{2}=1.414213562\dots$ to 4 fractional digits using the $\text{floor}$ function results in

${\stackrel{~}{a}}_{2}\mathrm{ }=\mathrm{ }\frac{1}{{10}^{4}}·⌊{10}^{4}·\sqrt{2}⌋\mathrm{ }=\mathrm{ }\frac{1}{{10}^{4}}·⌊14142.1\dots ⌋\mathrm{ }=\mathrm{ }\frac{1}{{10}^{4}}·14142\mathrm{ }=\mathrm{ }1.4142 .$

Rounding the number

${a}_{3}\mathrm{ }=\mathrm{ }\pi \mathrm{ }=\mathrm{ }3,141592654\dots$

to 2 fractional digits using the $\text{floor}$ function results in

${\stackrel{~}{a}}_{3}\mathrm{ }=\mathrm{ }\frac{1}{{10}^{2}}·⌊{10}^{2}·\pi ⌋\mathrm{ }=\mathrm{ }\frac{1}{{10}^{2}}·⌊314.159\dots ⌋\mathrm{ }=\mathrm{ }\frac{1}{{10}^{2}}·314\mathrm{ }=\mathrm{ }3.14 .$

The rounding method using the $\text{floor}$ function is often applied for calculating final grades in certificates ("academic rounding"). If a mathematics student has the individual grades
 Subject Grade Mathematics 1 $1.3$ Mathematics 2 $2.3$ Mathematics 3 $2.0$
then the arithmetic mean of these grades is calculated by

$\frac{1.3+2.3+2.0}{3}\mathrm{ }=\mathrm{ }\frac{5.6}{3}\mathrm{ }=\mathrm{ }1.8\stackrel{‾}{6} .$

Rounding to the first fractional digit using the $\text{floor}$ function would result in the final grade of $\stackrel{~}{a}=1.8$. The rounding methods for calculating final grades always have to be described exactly in the examination regulations.
The counterpart to the $\text{floor}$ function is the $\text{ceil}$ (a.k.a. ceiling) function:
Info 11.1.5

The $\text{ceil}$ function is defined as

$\text{ceil}:\mathrm{ }ℝ\to ℝ\mathrm{ }\mathrm{ },\mathrm{ }\mathrm{ }x\mathrm{ }⟼\mathrm{ }\text{ceil}\left(x\right)\mathrm{ }=\mathrm{ }⌈x⌉\mathrm{ }=\mathrm{ }min\left\{k\in ℤ : k\ge x\right\} .$

If $x\in ℝ$ is a real number, then $\text{ceil}\left(x\right)=⌈x⌉$ is the smallest integer that is greater than or equal to $x$. The $\text{ceil}$ function is a step function with jumps (jump discontinuities) of height $1$ at all points $x\in ℤ$. The function values at the jumps always lie at the bottom. They are indicated by the small circles in the figure below showing the graph of the $\text{ceil}$ function.

Graph of the $\text{ceil}$ function

Let a real number $a\ge 0$ be given as a decimal number

$a\mathrm{ }=\mathrm{ }{g}_{n} {g}_{n-1} \dots {g}_{1} {g}_{0} . {a}_{1} {a}_{2} {a}_{3} \dots$

This number $a$ can be rounded to $r$ fractional digits ($r\in ℕ{}_{0}$) using the $\text{ceil}$ function by

$\stackrel{^}{a}\mathrm{ }=\mathrm{ }\frac{1}{{10}^{r}}·⌈{10}^{r}·a⌉ .$

Rounding using the $\text{ceil}$ function is in general a rounding up to the next decimal digit.
Example 11.1.6
Rounding the number ${a}_{1}=2.3727$ to $2$ fractional digits using the $\text{ceil}$ function results in

${\stackrel{^}{a}}_{1}\mathrm{ }=\mathrm{ }\frac{1}{{10}^{2}}·⌈{10}^{2}·2.3727⌉\mathrm{ }=\mathrm{ }\frac{1}{{10}^{2}}·⌈237.27⌉\mathrm{ }=\mathrm{ }\frac{1}{{10}^{2}}·238\mathrm{ }=\mathrm{ }2.38 .$

Analogously, rounding the number ${a}_{2}=\sqrt{2}=1.414213562\dots$ to $4$ fractional digits using the $\text{ceil}$ function results in

${\stackrel{^}{a}}_{2}\mathrm{ }=\mathrm{ }\frac{1}{{10}^{4}}·⌈{10}^{4}·\sqrt{2}⌉\mathrm{ }=\mathrm{ }\frac{1}{{10}^{4}}·⌈14142.1\dots ⌉\mathrm{ }=\mathrm{ }\frac{1}{{10}^{4}}·14143\mathrm{ }=\mathrm{ }1.4143 .$

Rounding the number ${a}_{3}=\pi =3.141592654\dots$ to $2$ fractional digits using the $\text{ceil}$ function results in

${\stackrel{^}{a}}_{3}\mathrm{ }=\mathrm{ }\frac{1}{{10}^{2}}·⌈{10}^{2}·\pi ⌉\mathrm{ }=\mathrm{ }\frac{1}{{10}^{2}}·⌈314.15\dots ⌉\mathrm{ }=\mathrm{ }\frac{1}{{10}^{2}}·315\mathrm{ }=\mathrm{ }3.15 .$

The rounding method using the $\text{ceil}$ function is often applied, for example, in craftsmen's invoices. A craftsman is mostly paid by the hour. If a repair takes 50 minutes (i.e. $0.8\stackrel{‾}{3}$ hours as a decimal), then a craftsmen will round up and invoice a full working hour. Colloquially, rounding mostly means mathematical rounding:
Info 11.1.7

The $\text{round}$ function (or mathematical rounding) is defined as

$\text{round}:\mathrm{ }ℝ\to ℝ\mathrm{ }\mathrm{ },\mathrm{ }\mathrm{ }x\mathrm{ }⟼\mathrm{ }\text{round}\left(x\right)\mathrm{ }=\mathrm{ }\text{floor}\left(x+\frac{1}{2}\right)\mathrm{ }=\mathrm{ }⌊x+\frac{1}{2}⌋ .$

In contrast to rounding up or rounding off, the maximum change to the number by this rounding is $0.5$.

The $\text{round}$ function is a step function with jumps (jump discontinuities) of height $1$ at all points $x+\frac{1}{2},\mathrm{ }x\in ℤ$. The function values at the jumps always lie a step up. They are indicated by the small circles in the figure below showing the graph of the $\text{round}$ function.

Graph of the $\text{round}$ function

Let a real number $a\ge 0$ be given as a decimal number

$a\mathrm{ }=\mathrm{ }{g}_{n} {g}_{n-1} \dots {g}_{1} {g}_{0} . {a}_{1} {a}_{2} {a}_{3} \dots$

This number $a$ can be rounded to $r$ fractional digits ($r\in ℕ{}_{0}$) using the $\text{round}$ function:

$\stackrel{‾}{a}\mathrm{ }=\mathrm{ }\frac{1}{{10}^{r}}·\text{round}\left({10}^{r}·a\right)\mathrm{ }=\mathrm{ }\frac{1}{{10}^{r}}·⌊{10}^{r}·a+\frac{1}{2}⌋ .$

This rounding method is called mathematical rounding and corresponds to the "normal" rounding process.
Example 11.1.8
The number ${a}_{1}=1.49$ is rounded to one fractional digit using the $\text{round}$ function to

$\begin{array}{ccc}\multicolumn{1}{c}{{\stackrel{‾}{a}}_{1}}& =\hfill & \frac{1}{10}·\mathrm{round}\left(10·1.49\right)\mathrm{ }=\mathrm{ }\frac{1}{10}·⌊10·1.49+0.5⌋\hfill \\ \multicolumn{1}{c}{}& =\hfill & \frac{1}{10}·⌊14.9+0.5⌋\mathrm{ }=\mathrm{ }\frac{1}{10}·⌊15.4⌋\mathrm{ }=\mathrm{ }\frac{1}{10}·15\mathrm{ }=\mathrm{ }1.5 .\hfill \end{array}$

The number ${a}_{2}=1.52$ is rounded to one fractional digit using the $\text{round}$ function to

$\begin{array}{ccc}\multicolumn{1}{c}{{\stackrel{‾}{a}}_{2}}& =\hfill & \frac{1}{10}·\mathrm{round}\left(10·1.52\right)\mathrm{ }=\mathrm{ }\frac{1}{10}·⌊10·1.52+0.5⌋\hfill \\ \multicolumn{1}{c}{}& =\hfill & \frac{1}{10}·⌊15.2+0.5⌋\mathrm{ }=\mathrm{ }\frac{1}{10}·⌊15.7⌋\mathrm{ }=\mathrm{ }\frac{1}{10}·15\mathrm{ }=\mathrm{ }1.5 .\hfill \end{array}$

The number ${a}_{3}=2.3727$ is rounded to two fractional digits using the $\text{round}$ function to

$\begin{array}{ccc}\multicolumn{1}{c}{{\stackrel{‾}{a}}_{3}}& =\hfill & \frac{1}{{10}^{2}}·\mathrm{round}\left({10}^{2}·2.3727\right)\mathrm{ }=\mathrm{ }\frac{1}{100}·⌊100·2.3727+0.5⌋\hfill \\ \multicolumn{1}{c}{}& =\hfill & \frac{1}{100}·⌊237.27+0.5⌋\mathrm{ }=\mathrm{ }\frac{1}{100}·⌊237.77⌋\mathrm{ }=\mathrm{ }\frac{1}{100}·237\mathrm{ }=\mathrm{ }2.37 .\hfill \end{array}$

The number ${a}_{4}=\sqrt{2}=1.414213562\dots$ is rounded to seven fractional digits using the $\text{round}$ function to

$\begin{array}{ccc}\multicolumn{1}{c}{{\stackrel{‾}{a}}_{3}}& =\hfill & \frac{1}{{10}^{7}}·\mathrm{round}\left({10}^{7}·\sqrt{2}\right)\mathrm{ }=\mathrm{ }\frac{1}{{10}^{7}}·⌊{10}^{7}·1.414213562\dots +0.5⌋\hfill \\ \multicolumn{1}{c}{}& =\hfill & \frac{1}{{10}^{7}}·⌊14142135.62\dots +0.5⌋\mathrm{ }=\mathrm{ }\frac{1}{{10}^{7}}·⌊14142136.12\dots ⌋\hfill \\ \multicolumn{1}{c}{}& =\hfill & \frac{1}{{10}^{7}}·14142136\mathrm{ }=\mathrm{ }1.4142136 .\hfill \end{array}$

Exercise 11.1.9
Using the $\text{round}$ function, round the number $\pi =3.141592654\dots$ to four fractional digits: $\stackrel{‾}{\pi }$$=$
.

Exercise 11.1.10
Let the numbers

$a\mathrm{ }=\mathrm{ }\frac{47}{17}\mathrm{ }\mathrm{ }\text{and}\mathrm{ }\mathrm{ }b\mathrm{ }=\mathrm{ }3.7861$

be given.
1. Round each of the numbers $a$ and $b$ to $2$ fractional digits using the $\text{floor}$ function.
The roundings result in $\stackrel{~}{a}$$=$
and $\stackrel{~}{b}$$=$ .
2. Round each of the numbers $a$ and $b$ to $2$ fractional digits using the $\text{ceil}$ function.
The roundings result in $\stackrel{^}{a}$$=$
and $\stackrel{^}{b}$$=$ .
3. Round each of the numbers $a$ and $b$ to $2$ fractional digits using the $\text{round}$ function.
The roundings result in $\stackrel{‾}{a}$$=$
and $\stackrel{‾}{b}$$=$ .