Given the measurements

$\begin{array}{l}\mathbf{x}\mathbf{:}\mathbf{2}\mathbf{.3}\mathbf{,}\mathbf{2}\mathbf{.1}\mathbf{,}\mathbf{1}\mathbf{.8}\mathbf{,}\mathbf{1}\mathbf{.7}\mathbf{,}\mathbf{2}\mathbf{.1}\\ \mathbf{y}\mathbf{:}\mathbf{1}\mathbf{.0}\mathbf{,}\mathbf{1}\mathbf{.1}\mathbf{,}\mathbf{0}\mathbf{.9}\end{array}$

find the mean value and the probable error for$\mathit{x}\mathbf{-}\mathit{y}\mathbf{,}\mathit{x}\mathit{y}\mathbf{,}\frac{\mathbf{x}}{{\mathbf{y}}^{\mathbf{3}}}$

Measurements

Frequency distribution of the number of successful outcomes that can be achieved in a given number of trials, each with an equal chance of success

The mean for x is given below.

$\begin{array}{c}\overline{x}=\frac{1}{n}\sum _{i=1}^n{x}_i\\ =\frac{2.3+2.1+1.8+1.7+2.1}{5}\\ =2\end{array}$

The standard deviation for x is given below.

$\begin{array}{c}{\sigma }_n^2=\frac{{\displaystyle \sum _{i=1}^n{(x-\overline{x})}^2}}{n_a-1}\\ =\frac{2{\left(0.3\right)}^2+2{\left(0.1\right)}^2+{\left(0.2\right)}^2}{4}\\ =0.06\end{array}$

The probable error for x is given below.

$\begin{array}{c}{\sigma }_{ms}=\sqrt{\frac{{\sigma }_z^2}{n}}\\ =\sqrt{\frac{0.06}{5}}\\ =0.11\end{array}$

The mean for y is given below.

$\begin{array}{c}\overline{y}=\frac{1}{n}\sum _{i=1}^n{y}_i\\ =\frac{1+1.1+0.9}{3}\\ =1\end{array}$

The variance for y is given below.

$\begin{array}{c}{\sigma }_n^2=\frac{{\displaystyle \sum _{i=1}^n{(y-\overline{y})}^2}}{n_y-1}\\ =\frac{2{\left(0.1\right)}^2}{2}\\ =0.01\end{array}$

The standard deviation for y is given below.

$\begin{array}{c}{\sigma }_{\text{y}}=\sqrt{\frac{{\sigma }_y^2}{n}}\\ =\sqrt{\frac{0.01}{3}}\\ =0.0577\end{array}$

The probable error for y is given below.

$\begin{array}{c}{r}_y={\sigma }_{y}I\\ =\left(0.6745\right)\left(0.0577\right)\\ =0.039\end{array}$

Let us assume the equations mentioned below.

$\begin{array}{c}E\left(w\right)=w({\mu }_v,{\mu }_{\text{v}})\\ w=x-y\end{array}$

The mean for $x-y$ is given below.

$\begin{array}{c}E\left(w\right)={\mu }_e-{\mu }_v\\ =2-1\\ =1\end{array}$

The standard deviation for $x-y$ is given below.

$\begin{array}{c}{\sigma }_{ma}={\left[\sqrt{{\left(\frac{\partial w}{\partial x}\right)}^2{\sigma }_{mz}^2+{\left(\frac{\partial w}{\partial y}\right)}^2{\sigma }_{m,}^2}\right]}_{(x,y)=(\overline{x},\overline{y})}\\ {\sigma }_{ma}=\sqrt{{\left(1\right)}^2{\left(0.11\right)}^2+{\left(1\right)}^2{\left(0.0577\right)}^2}\\ =0.124\end{array}$

The probable error for $x-y$is given below.

$\begin{array}{c}{r}_{\text{w}}=\left(0.6745\right)\left(0.124\right)\\ =0.083\end{array}$

Let us assume the equations mentioned below.

$\begin{array}{c}E\left(w\right)=w({\mu }_v,{\mu }_{\text{v}})\\ w=xy\end{array}$

The mean for $xy$ is given below.

$\begin{array}{c}E\left(w\right)={\mu }_x{\mu }_y\\ =2\times 1\\ =2\end{array}$

The standard deviation for $xy$is given below.

$\begin{array}{c}{\sigma }_v=\sqrt{{\overline{y}}^2{\sigma }_{me}^2+{\overline{x}}^2{\sigma }_{nvy}^2}\\ =\sqrt{{\left(0.11\right)}^2+2^2{\left(0.0577\right)}^2}\\ =0.16\end{array}$

The probable error for $xy$ is given below.

$\begin{array}{c}{r}_{\text{w}}=0.16\times 0.6745\\ =0.108\end{array}$

Let us assume the equations mentioned below.

$\begin{array}{c}E\left(w\right)=w({\mu }_v,{\mu }_{\text{v}})\\ w=\frac{x}{y^3}\end{array}$

The mean for $\frac{x}{y^3}$ is given below.

$\begin{array}{c}E\left(w\right)=\frac{{\mu }_z}{{\mu }_v^3}\\ =\frac{2}{1}\\ =2\end{array}$

The standard deviation for $\frac{x}{y^3}$ is given below.

$\begin{array}{c}{\sigma }_w=\sqrt{{\left(\frac{1}{y^3}\right)}^2{\sigma }_{me}^2+{(-\frac{3x}{y^4})}^2{\sigma }_m^2}\\ =\sqrt{\left(1\right){\left(0.11\right)}^2+36{\left(0.0577\right)}^2}\\ =0.363\end{array}$

The probable error for$\frac{x}{y^3}$ is given below.

$\begin{array}{c}{r}_v=0.363\times 0.6745\\ =0.245\end{array}$

The values of the function for$x-y$ is given below.

$\begin{array}{c}E\left(w\right)=1\\ {\sigma }_{ma}=0.124\\ r_{\text{w}}=0.083\end{array}$

The values of the function for $xy$ is given below.

$\begin{array}{c}E\left(w\right)=2\\ {\sigma }_v=0.16\\ r_{\text{w}}=0.108\end{array}$

Thvalues of the function for $\frac{x}{y^3}$ is given below.