11.5 Measures of central tendency and variation, including standard deviation — Practise with help

Work each question closed book first. Say at the top whether your standard deviation divides by $n$ or by $n - 1$ .

Warm-up: remember

Work out $7^2$ and $(-4)^2$ .
Work out $\sqrt{49}$ and $\sqrt{\dfrac{100}{4}}$ .
Find the mean of $12, 15, 18$ .

Answers

$7^2 = 49$ and $(-4)^2 = 16$ .
$\sqrt{49} = 7$ and $\sqrt{25} = 5$ .
$\dfrac{12 + 15 + 18}{3} = \dfrac{45}{3} = 15$ .

If question 1 was hard, revise squaring negative numbers before you start.

Practise with help

1. Find the mean, median and mode of $2, 3, 3, 5, 7, 10$ .

Hint

Put the values in order first. The mode is the value that occurs most often.

Next step

The values are already in order. For the median, there are six values, so take the mean of the 3rd and 4th.

Full working

\text{mean} = \frac{2 + 3 + 3 + 5 + 7 + 10}{6} = \frac{30}{6} = 5.

The median is $\dfrac{3 + 5}{2} = 4$ . The mode is 3, because 3 occurs twice and no other value does.

2. Find $Q_1$ , $Q_3$ and the interquartile range of $2, 4, 5, 7, 9, 10, 12$ using the $(n + 1)$ positions.

Hint

There are 7 values. Use position $\dfrac{n + 1}{4}$ for $Q_1$ and $\dfrac{3(n + 1)}{4}$ for $Q_3$ .

Next step

The positions are $\dfrac{8}{4} = 2$ and $\dfrac{24}{4} = 6$ , so both are whole numbers. Read the 2nd and 6th values.

Full working

The 2nd value is 4, so $Q_1 = 4$ . The 6th value is 10, so $Q_3 = 10$ . The interquartile range is

\text{IQR} = 10 - 4 = 6.

The middle half of the data spans 6 units.

3. Find the variance and the standard deviation of $1, 3, 5, 7, 9$ , dividing by $n$ .

Hint

Find the mean first. Then square each deviation from the mean.

Next step

The mean is 5. The squared deviations are $16, 4, 0, 4, 16$ .

Full working

The sum of the squared deviations is 40, so

\text{variance} = \frac{40}{5} = 8, \qquad \sigma = \sqrt{8} = 2.83 \ (3 \text{ s.f.}).

The standard deviation is about 2.83.

4. A data set has $n = 20$ , $\sum x = 500$ and $\sum x^2 = 13000$ . Find the mean and the standard deviation, dividing by $n$ .

Hint

Use $\bar{x} = \dfrac{\sum x}{n}$ and $\sigma = \sqrt{\dfrac{\sum x^2}{n} - \bar{x}^2}$ .

Next step

The mean is $\dfrac{500}{20} = 25$ . Then work out $\sqrt{\dfrac{13000}{20} - 25^2}$ .

Full working

\bar{x} = \frac{500}{20} = 25, \qquad \sigma = \sqrt{\frac{13000}{20} - 25^2} = \sqrt{650 - 625} = \sqrt{25} = 5.

The mean is 25 and the standard deviation is 5.

5. Two runners have the same mean lap time of 60 seconds. Runner A has a standard deviation of 2 seconds and runner B has a standard deviation of 8 seconds. Compare the runners.

Hint

The means agree, so compare the spreads.

Next step

Decide which runner has lap times closer to the mean.

Full working

Both runners have the same mean lap time, so their typical laps agree. Runner A has the smaller standard deviation, so A's lap times are more consistent and cluster closer to 60 seconds. Runner B's laps vary more, with more very fast and very slow laps.

6. Every value in a data set is multiplied by 3. Explain what happens to the mean and the standard deviation.

Hint

Test it on a small set, such as $1, 2, 3$ .

Next step

The set $1, 2, 3$ becomes $3, 6, 9$ . Compare the means and the spreads of the two sets.

Full working

The original set $1, 2, 3$ has mean 2 and standard deviation $\sqrt{\dfrac{2}{3}} \approx 0.82$ . The new set $3, 6, 9$ has mean 6 and standard deviation $\sqrt{6} \approx 2.45$ . Multiplying every value by 3 multiplies the mean by 3 and the standard deviation by 3. The spread scales with the data.