Control chart -- Volatile compounds in human blood serum were measured by purge and trap gas chromatography/mass spectrometry. For quality control, serum was periodically spiked with a constant amount of $\mathbf{1}\mathbf{,}\mathbf{2}$-dichlorobenzene and the concentration role="math" localid="1663334314380" $(\mathrm{ng}/g=\mathrm{ppb})$was measured. Find the mean and standard deviation for the following spike data and prepare a control chart. State whether or not the observations (Obs.) meet each criterion for stability in a control chart.

Mean:

The Mean or the arithmetic mean $\left(\overline{x}\right)$is given as the summation of measured values divided by the number of measurements $\left(n\right)$Mean is also called as average.

$\overline{x}=\frac{{\displaystyle \sum _i}xi}{n}$

Standard deviation:

The standard deviation $\left(s\right)$is given by the formula:

$s=\sqrt{\frac{{{\displaystyle \underset{i}{\sum \left(x_i-\overline{x}\right)}}}^2}{n-1}}$

Given table:

Using mean formula,

$\overline{\mathrm{x}}=\frac{{\displaystyle \sum _{\mathrm{i}}\mathrm{xi}}}{\mathrm{n}}=08411$

Mean of the samples calculated was 0.8411.

Using standard deviation formula,

$\mathrm{s}=\sqrt{\frac{{\displaystyle {\underset{\mathrm{i}}{\sum \left({\mathrm{x}}_{\mathrm{i}}-\overline{\mathrm{x}}\right)}}^2}}{\mathrm{n}-1}}=0.1888$

Standard The standard deviation of the samples calculated was 0.1888.

The stability criteria for the control chart are.

- There shouldn't be observations outside the action lines-on day 101 one observation lies above the line.

- There are no two out of 3 consecutive measurements between warning and action lines-OK.

- There are no seven7 consecutive measurements all above or below the centre line-OK.

- There are no six consecutive measurements all steadily increasing or all steadily decreasing, wherever they are located.

- There are no fourteen consecutive points alternating up and down, nevertheless of where they are located-OK.

- There is no obvious non-random pattern-OK.