The lengths of extra-time, $t$ minutes, played in the first and second halves of $100$ football matches are summarised in the following table.

Extra-time ($t$ min)	$t⩽1$	$t⩽2$	$t⩽4$	$t⩽5$	$t⩽7$	$t⩽9$
First halves $\left(cf\right)$	$24$	$62$	$80$	$92$	$97$	$100$
Second halves $\left(cf\right)$	$6$	$17$	$35$	$82$	$93$	$100$

The first-half median is exactly $100$ seconds.

Explain why the mean must be greater than the median for the extra-time played in the first halves.

Eighty candidates took an examination in Astronomy, for which no candidate scored more than $80\%.$ The examiners suggest that five grades, $A, B, C, D$ and $E,$ should be awarded to these candidates, using upper grade boundaries $64, 50, 36$ and $26$ for grades $B, C, D$ and $E,$ respectively. In this case, grades $A, B, C, D$ and $E,$ will be awarded in the ratio $1: 3: 5: 4: 3.$

Using the examiners' suggestion, represent the scores in a cumulative frequency polygon and use it to estimate the median score.

Eighty candidates took an examination in Astronomy, for which no candidate scored more than $80\%.$ The examiners suggest that five grades, $A, B, C, D$ and $E,$ should be awarded to these candidates, using upper grade boundaries $64, 50, 36$ and $26$ for grades $B, C, D$ and $E,$ respectively. In this case, grades $A, B, C, D$ and $E,$ will be awarded in the ratio $1: 3: 5: 4: 3.$

All of the grade boundaries are later reduced by $10\%.$ Estimate how many candidates will be awarded a higher grade because of this.

The values of $x$ shown in the following table are to be represented in a bar chart.

Find the mode, mean and the median of $x.$

The values of $x$ shown in the following table are to be represented in a bar chart.

The two smallest values of $x$ (i.e. $5$ and $5$) are changed to $21$ and $31.$ Investigate the effect that this has on the mode, the mean, the median and on the shape of the curve.

The values of $x$ shown in the following table are to be represented in a bar chart.

If, instead, the two largest values of $x$ (i.e. $11$ and $11$) are changed to $-9$ and $b,$ so that the mean of $x$ decreases by $1,$ find the value of $b$ and investigate the effect that this has on the mode, the median and the shape of the curve.

Students' marks in a Biology examination are shown by percentage in the following table.

Without drawing an accurate histogram, describe the shape of the set of marks. What does the shape suggest about the values of the mean, the median and the mode?

Students' marks in a Biology examination are shown by percentage in the following table.

Information is provided about the marks in examinations in two other subjects:

Chemistry: mode $>$ median $>$ mean

Physics: mean $>$ median $>$ mode

Sketch a curve to show the shape of the distribution of marks in each of these exams.