Given the measurements

$\begin{array}{l}\mathbf{x}\mathbf{:}\mathbf{5}\mathbf{.7}\mathbf{,}\mathbf{4}\mathbf{.5}\mathbf{,}\mathbf{4}\mathbf{.8}\mathbf{,}\mathbf{5}\mathbf{.1}\mathbf{,}\mathbf{4}\mathbf{.9}\\ \mathbf{y}\mathbf{:}\mathbf{61}\mathbf{.5}\mathbf{,}\mathbf{60}\mathbf{.1}\mathbf{,}\mathbf{59}\mathbf{.7}\mathbf{,}\mathbf{60}\mathbf{.3}\mathbf{,}\mathbf{58}\mathbf{.4}\end{array}$

find the mean value and the probable error for$\mathit{x}\mathbf{+}\mathit{y}\mathbf{,}\frac{\mathbf{y}}{\mathbf{x}}\mathbf{,}{\mathit{x}}^{\mathbf{2}}$

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.

$\begin{array}{c}\overline{x}=\frac{1}{n}\sum _{i=1}^n{x}_i\\ =\frac{5.7+4.5+4.8+5.1+4.9}{5}\\ =5\end{array}$

The variance for x is given below.

$\begin{array}{c}{\sigma }_x^2=\frac{{\Sigma }_{i=1}^n{(x-\overline{x})}^2}{n_x-1}\\ =\frac{2{\left(0.1\right)}^2+{\left(0.7\right)}^2+{\left(0.2\right)}^2+{\left(0.5\right)}^2}{4}\\ =0.02\end{array}$

The standard deviation for x is given below.

$\begin{array}{c}{\sigma }_{ma}=\sqrt{\frac{{\sigma }_z^2}{n}}I\\ =\sqrt{\frac{0.02}{5}}\end{array}$

The probable error for x is given below.

$\begin{array}{c}{r}_x={\sigma }_{mx}I\\ =\left(0.6745\right)\left(0.02\right)\\ =0.135\end{array}$

The mean for y is given below.

$\begin{array}{c}\overline{y}=\frac{1}{n}\sum _{i=1}^n{y}_i\\ =\frac{61.5+60.1+59.7+60.3+58.4}{5}\\ =60\end{array}$

The variance for y is given below.

$\begin{array}{c}{\sigma }_z^2=\frac{{\Sigma }_{i=1}^n{(y-\overline{y})}^2}{n_y-1}\\ =\frac{{\left(1.5\right)}^2+{\left(0.1\right)}^2+2{\left(0.3\right)}^2+{\left(1.6\right)}^2}{4}\\ =1.25\end{array}$

The standard deviation for y is given below.

$\begin{array}{c}{\sigma }_{my}=\sqrt{\frac{{\sigma }_y^2}{n}}I\\ =\sqrt{\frac{1.25}{5}}\\ =0.5\end{array}$

The probable error for y is given below.

$\begin{array}{c}{r}_y={\sigma }_{y}I\\ =\left(0.6745\right)\left(0.5\right)\\ =0.337\end{array}$

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.

$\begin{array}{c}\mathbf{E}\left(w\right)={\mu }_v+{\mu }_y\\ =5+60\\ =65\end{array}$

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

$\begin{array}{c}\sigma ={\left[\sqrt{{\left(\frac{\partial w}{\partial x}\right)}^2{\sigma }_{mx}^2+{\left(\frac{\partial w}{\partial y}\right)}^2{\sigma }_{my}^2}\right]}_{(x,y)=(x,y)}\\ \sigma =\sqrt{{\left(1\right)}^2{\left(0.2\right)}^2+{\left(1\right)}^2{\left(0.5\right)}^2}\\ =0.538\end{array}$

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

$\begin{array}{c}{r}_w=\left(0.6745\right)\left(0.538\right)\\ =0.363\end{array}$

Let us assume the equations mentioned below.

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

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

$\begin{array}{c}E\left(w\right)=\frac{{\mu }_y}{{\mu }_z}\\ =\frac{60}{5}\\ =12\end{array}$

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

$\begin{array}{c}{\sigma }_w=\sqrt{{\left(\frac{1}{\overline{x}}\right)}^2{\sigma }_{ma}^2+{(-\frac{\overline{y}}{\overline{x}})}^2{\sigma }_{me}^2}\\ =\sqrt{{\left(\frac{1}{5}\right)}^2{\left(0.5\right)}^2+{\left(\frac{60}{5^2}\right)}^2{\left(0.2\right)}^2}\\ =0.49\end{array}$

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

$\begin{array}{c}{r}_w=0.49\times 0.6745\\ =0.33\end{array}$

Let us assume the equations mentioned below.

$\begin{array}{c}E\left(w\right)=w({\mu }_v,{\mu }_{\text{v}})\\ w=x^2\end{array}$

The mean for $x^2$is given below.

$\begin{array}{c}E\left(w\right)={\mu }^2\\ =5^2\\ =25\end{array}$

The standard deviation for $x^2$is given below.

$\begin{array}{c}{\sigma }_w=\sqrt{{\left(2x\right)}^2{\sigma }_m^2}\\ =\sqrt{{(2\times 5)}^2{\left(0.2\right)}^2}\\ =2\end{array}$

The probable error for$x^2$ is given below.

$\begin{array}{c}{r}_v=2\times 0.6745\\ =1.35\end{array}$

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

$\begin{array}{c}\mathbf{E}\left(w\right)=65\\ \sigma =0.538\\ r_w=0.363\end{array}$

The values of the function for$\frac{y}{x}$ is given below.

$\begin{array}{c}E\left(w\right)=12\\ {\sigma }_w=0.49\\ r_w=0.33\end{array}$

The values of the function for$x^2$ is given below.

$\begin{array}{c}E\left(w\right)=25\\ {\sigma }_w=2\\ r_v=1.35\end{array}$