1.1. Each member of a group of 27 people was timed when completing a puzzle.

The time taken, $x$ minutes, for each member of the group was recorded.

For these 27 people $\underset{}{\sum}x=607.5$ and $\underset{}{\sum}{x}^{2}=17623.25$

Calculate the mean time taken to complete the puzzle.

1.2. Calculate the standard deviation of the times taken to complete the puzzle.

1.3. The times are summarised in the following box and whisker plot.

Taruni defines an outlier as a value more than 3 standard deviations above the mean.

State how many outliers Taruni would say there are in these data, giving a reason for your answer.

1.4. Adam and Beth also completed the puzzle in $a$ minutes and $b$ minutes respectively, where $a>b$.

When their times are included with the data of the other 27 people

• the median time increases

• the mean time does not change

Suggest a possible value for $a$ and a possible value for $b$, explaining how your values satisfy the above conditions.

1.5. Without carrying out any further calculations, explain why the standard deviation of all 29 times will be lower than your answer to part (b).