Statistics I

Statistics is about making a pile of numbers say something useful. The skills here are mechanical, which makes them reliable marks — but only if you are precise about which average a question wants and what the data actually is.

Types of data

Qualitative data describes a quality: colour, favourite subject, blood group. It is not a number.

Quantitative data is numerical, and splits in two:

• Discrete data can only take separate, countable values. The number of children in a family is 2 or 3, never 2.4.
• Continuous data can take any value in a range and comes from measuring. Height, mass, time and temperature are all continuous — a height of 1.632 m is perfectly possible.

Frequency tables

A frequency table records how many times each value occurs. Twenty students were asked how many siblings they have:

The frequencies add to $3 + 5 + 4 + 8 = 20$, which must match the number of students. That check costs two seconds and catches a miscount.

Bar charts and pie charts

A bar chart compares frequencies. The bars are all the same width, they have gaps between them, and the vertical axis starts at zero and is labelled.

A pie chart shows how a whole splits into parts. Each category gets a slice whose angle is its share of $360°$:

$$\text{angle} = \frac{\text{frequency}}{\text{total frequency}} \times 360°$$

Sixty students chose a sport. Football 25, cricket 15, netball 12, athletics 8. Find the angle of each slice.

• Football: $\dfrac{25}{60} \times 360 = 150°$
• Cricket: $\dfrac{15}{60} \times 360 = 90°$
• Netball: $\dfrac{12}{60} \times 360 = 72°$
• Athletics: $\dfrac{8}{60} \times 360 = 48°$

Going the other way, a slice's frequency is

$$\text{frequency} = \frac{\text{angle}}{360°} \times \text{total}$$

So a $72°$ slice of a chart representing 300 people stands for $\frac{72}{360} \times 300 = 60$ people.

Mean, median and mode

Three different "averages", each answering a different question.

The mean is the total shared out equally:

$$\text{mean} = \frac{\text{sum of the values}}{\text{number of values}}$$

The median is the middle value once the data is in order.

The mode is the value that occurs most often. Data can have more than one mode, or none at all.

Find the mean, median and mode of $5, 3, 8, 3, 6$.

Mean: $\dfrac{5 + 3 + 8 + 3 + 6}{5} = \dfrac{25}{5} = 5$

Median: in order the data is $3, 3, 5, 6, 8$. The middle of five values is the third, so the median is $5$.

Mode: $3$ appears twice and everything else once, so the mode is $3$.

An even number of values

With an even count there is no single middle, so take the mean of the middle two. For $2, 5, 8, 11$:

$$\text{median} = \frac{5 + 8}{2} = 6.5$$

Notice the median need not be one of the data values.

Mean from a frequency table

Do not add the values once each — each one occurs as many times as its frequency says. Multiply, then divide by the total frequency.

$$\text{mean} = \frac{\sum fx}{\sum f}$$

For the siblings table above:

$$\sum fx = (0 \times 3) + (1 \times 5) + (2 \times 4) + (3 \times 8) = 0 + 5 + 8 + 24 = 37$$

$$\sum f = 20$$

$$\text{mean} = \frac{37}{20} = 1.85$$

The range

$$\text{range} = \text{largest value} - \text{smallest value}$$

For $12, 7, 19, 4, 15$ the range is $19 - 4 = 15$.

The range measures spread, not position. A small range means the data is tightly bunched; a large one means it is scattered.

Choosing the right average

Questions often ask which average is most appropriate, and there are marks in the reason.

Example. Six workers earn \$400, \$420, \$430, \$440, \$450 and \$3000 a week. The mean is \$856.67 — more than every worker earns but one. The median, \$435, describes a typical wage far better. The single large salary is an outlier, and outliers are exactly why the median exists.