The lengths of $45$ ropes used at an outdoor recreational centre can be extended by $30\%$ when stretched. The sum of the squares of their stretched lengths is $0.0507{ \mathrm{km}}^2$ and their natural lengths, $x$ metres, are summarised by $\Sigma {\left(x-20\right)}^2=1200.$ Find the mean natural length of these $45$ ropes.

Three boys and seven girls are asked how much money they have in their pockets. The boys have $\$2.50$ each and the mean amount that the $10$ children have is $\$3.90.$

Show that the girls have a total of $\$31.50.$

Three boys and seven girls are asked how much money they have in their pockets. The boys have $\$2.50$ each and the mean amount that the $10$ children have is $\$3.90.$

Given that the seven girls have equal amounts of money, find the standard deviation of the amounts that the $10$ children have.

Jean Luc was asked to record the times of $20$athletes in a long distance race. He started his stopwatch when the race began and then went to sit in the shade, where he fell asleep. On waking, he found that $x$ athletes had already completed the race but he was able to record the times taken by all the others.

State the possible value(s) of $x$ if Jean Luc was able to use his data to calculate: the variance of the times taken by the $20$ athletes.

Jean Luc was asked to record the times of $20$athletes in a long distance race. He started his stopwatch when the race began and then went to sit in the shade, where he fell asleep. On waking, he found that $x$ athletes had already completed the race but he was able to record the times taken by all the others.

State the possible value $\left(s\right)$ of $x$ if Jean Luc was able to use his data to calculate:

the interquartile range of the times taken by the $20$ athletes.[Answer $0,1,2,3,4$ or $0,1,2,3$] 

Two days before a skiing competition, the depths of snow $x$, metres, at $32$ points on the course were measured and it was discovered that the numerical values of $\Sigma x$ and $\Sigma x^2$ were equal.

Given that the mean depth of snow was $0.885 \mathrm{m},$ find the standard deviation of $x.$

Two days before a skiing competition, the depths of snow $x$, metres, at $32$ points on the course were measured and it was discovered that the numerical values of $\Sigma x$ and $\Sigma x^2$ were equal.

Given that the mean depth of snow was $0.885 \mathrm{m}$ 

Snow fell the day before the competition, increasing the depth over the whole course by $1.5 \mathrm{cm}.$ Explain what effect this had on the mean and on the standard deviation of $x.$