#### 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}$$=$
and $\stackrel{‾}{b}$$=$ .