Question

Estimate the standard deviation and the relative standard deviation associated with counts of (a) $100$, (b) $850$, (c) $8.00\mathrm{X}{10}^3$, and (d) $4.00\mathrm{X}{10}^4$.

Short answer

Part (a)${\sigma }_M=\sqrt{100}=10$,${\sigma }_R=10\mathrm{\%}$

Part (b)${\sigma }_M=\sqrt{850}=29.15$,${\sigma }_R=3.43\mathrm{\%}$

Part (c)${\sigma }_M=\sqrt{8.00\times {10}^3}=2828.4$,${\sigma }_R=0.0354\mathrm{\%}$

Part (d)${\sigma }_M=\sqrt{4.00\times {10}^4}=20000$,$\begin{array}{rl}& {\sigma }_R=0.005\mathrm{\%}\end{array}$

Step-by-step solution

Introduction

Standard deviation - The standard deviation, which is determined as a percentage of the mean, is the most commonly used quantitative measure of precision.

Relative standard deviation - Calculating the dispersion of a set of values in relation to the mean, we may analyses the precision in a set of values using relative standard deviation.

Concept

The standard deviation can be calculated using the formula below:

${\sigma }_M=\sqrt{M}$

The formula for calculating the relative standard deviation is:

$\begin{array}{rl}& {\sigma }_R=\frac{{\sigma }_m}{M}\\ & {\sigma }_R=\frac{\sqrt{M}}{M}\end{array}$

Solution to part (a)

The standard deviation formula is as follows:

${\sigma }_M=\sqrt{M}$

Where,

$M=$count's number

${\sigma }_M=$standard deviation

The relative standard deviation formula is as follows:

$\begin{array}{rl}& {\sigma }_R=\frac{{\sigma }_M}{M}\\ & {\sigma }_R=\frac{\sqrt{M}}{M}\end{array}$

Where,

${\sigma }_R=$relative standard deviation

Given is -

Count $=100$

As a result, the standard deviation is

role="math" localid="1646072762981" ${\sigma }_M=\sqrt{100}=10$

Now we'll look at the relative standard deviation-

$\text{Count}=600$

Therefore,

${\sigma }_R=\frac{\sqrt{M}}{M}=\frac{1}{\sqrt{M}}$

Where,

role="math" localid="1646072801916" $\begin{array}{rl}& \sqrt{M}=\sqrt{100}=10\\ & {\sigma }_R=\frac{1}{10}=0.1\\ & {\sigma }_R\mathrm{\%}=\frac{1}{10}\times 100=10\mathrm{\%}\\ & {\sigma }_R=10\mathrm{\%}\end{array}$

Solution to part (b)

The standard deviation is calculated using the formula:

${\sigma }_M=\sqrt{M}$

Where,

$M=$count's number

${\sigma }_M=$standard deviation

The relative standard deviation formula is as follows:

$\begin{array}{rl}& {\sigma }_R=\frac{{\sigma }_M}{M}\\ & {\sigma }_R=\frac{\sqrt{M}}{M}\end{array}$

Where,

${\sigma }_R=$relative standard deviation

Given is -

count $=850$

As a result, the standard deviation is

role="math" localid="1646072862998" ${\sigma }_M=\sqrt{850}=29.15$

Now we'll look at the relative standard deviation-

Count $=850$

Therefore,

${\sigma }_R=\frac{\sqrt{M}}{M}=\frac{1}{\sqrt{M}}$

Where,

role="math" localid="1646072919121" $\begin{array}{rl}& \sqrt{M}=\sqrt{850}=29.15\\ & {\sigma }_R=\frac{1}{29.15}=0.0343\\ & {\sigma }_R\mathrm{\%}=0.0343\times 100=3.43\mathrm{\%}\\ & {\sigma }_R=3.43\mathrm{\%}\end{array}$

Solution to part (c)

The standard deviation formula is as follows:

${\sigma }_M=\sqrt{M}$

Where,

$M=$count's number

localid="1646073100230" ${\sigma }_{\mathrm{M}}=$standard deviation

The relative standard deviation formula is as follows:

$\begin{array}{rl}& {\sigma }_R=\frac{{\sigma }_M}{M}\\ & {\sigma }_R=\frac{\sqrt{M}}{M}\end{array}$

Where,

${\sigma }_R=$relative standard deviation

Given is -

$\text{Count}=8.00\times {10}^3$

As a result, the standard deviation -

localid="1646072949454" ${\sigma }_M=\sqrt{8.00\times {10}^3}=2828.4$

Now we'll look at the relative standard deviation-

Count$=8.00\times {10}^3$

Therefore,

${\sigma }_R=\frac{\sqrt{M}}{M}=\frac{1}{\sqrt{M}}$

Where,

localid="1646072975876" $\begin{array}{rl}& \sqrt{M}=\sqrt{8.00\times {10}^3}=2828.4\\ & {\sigma }_R=\frac{1}{2828.4}=0.000354\\ & {\sigma }_R\mathrm{\%}=0.000354\times 100=0.0354\mathrm{\%}\\ & {\sigma }_R=0.0354\mathrm{\%}\end{array}$

Solution to part (d)

The standard deviation formula is as follows:

${\sigma }_M=\sqrt{M}$

Where,

$M=$count's number

${\sigma }_M=$standard deviation

The relative standard deviation formula is as follows:
$\begin{array}{rl}& {\sigma }_R=\frac{{\sigma }_M}{M}\\ & {\sigma }_R=\frac{\sqrt{M}}{M}\end{array}$

Where,

${\sigma }_R=$relative standard deviation

Given is -

Count$=4.00\times {10}^4$

As a result, the standard deviation -
role="math" localid="1646073007586" ${\sigma }_M=\sqrt{4.00\times {10}^4}=20000$

Now we'll look at the relative standard deviation-

$\text{Count}=4.00\times {10}^4$

Therefore,

${\sigma }_R=\frac{\sqrt{M}}{M}=\frac{1}{\sqrt{M}}$

Where,

role="math" localid="1646073034836" $\begin{array}{rl}& \sqrt{M}=\sqrt{4.00\times {10}^4}=20000\\ & {\sigma }_R=\frac{1}{20000}=0.00005\\ & {\sigma }_R\mathrm{\%}=0.000054\times 100=0.005\mathrm{\%}\\ & {\sigma }_R=0.005\mathrm{\%}\end{array}$