A shop has in its stock$80$ rectangular celebrity posters. All of these posters have a width to height ratio of $1:\sqrt{2},$ and their mean perimeter is $231.8 \mathrm{cm}.$ Given that the sum of the squares of the widths is $200120{ \mathrm{cm}}^2,$ find the standard deviation of the widths of the posters.

Find the range and the interquartile range of the following sets of data.

$5, 8,13,17,22,25,30$

Find the range and the interquartile range of the following sets of data 

$7,13,21,2,37,28,17,11,2$

Find the range and the interquartile range of the following sets of data.

$42, 47, 39, 51, 73, 18, 83, 29, 41, 64$

Find the range and the interquartile range of the following sets of data.

$113, 97, 36, 81, 49, 41, 20, 66, 28, 32, 17, 107$

At a village fair, visitors were asked to guess how many sweets are in a glass jar. The best six guesses were $180, 211, 230, 199, 214$ and $166.$

Show that the mean of these guesses is $200,$ and use $\mathrm{SD}=\sqrt{\frac{\Sigma (x-\overline{x}{)}^2}{n}}$ to calculate the standard deviation.

At a village fair, visitors were asked to guess how many sweets are in a glass jar. The best six guesses were $180, 211, 230, 199, 214$ and $166.$

The jar actually contained$202$sweets. Without further calculation, write down the mean and the standard deviation of the errors made by these six visitors. Explain why no further calculations are required to do this.

The number of women in senior management positions at a number of companies was investigated. The number of women at each of the $25$ service companies and at each of the $16$ industrial companies are denoted by $w_{\mathrm{s}}$ and $w_1,$ respectively. The findings are summarised by the totals:
$\Sigma {\left(w_{\mathrm{s}}-5\right)}^2=28,\Sigma \left(w_{\mathrm{s}}-5\right)=15,\Sigma {\left(w_1-3\right)}^2=12$ and $\Sigma \left(w_1-3\right)=-4$

Show that there are, on average, more than twice as many women in senior management positions at the service companies than at the industrial companies.

The number of women in senior management positions at a number of companies was investigated. The number of women at each of the $25$ service companies and at each of the $16$ industrial companies are denoted by $w_s$ and $w_1,$ respectively. The findings are summarised by the totals:
$\Sigma {\left(w_{\mathrm{s}}-5\right)}^2=28,\Sigma \left(w_{\mathrm{s}}-5\right)=15,\Sigma {\left(w_1-3\right)}^2=12$ and $\Sigma \left(w_1-3\right)=-4$

Show that $\Sigma w_{\mathrm{s}}^2\ne {\left(\Sigma w_{\mathrm{s}}\right)}^2$ and that $\Sigma w_1^2\ne {\left(\Sigma w_{\mathrm{I}}\right)}^2.$