Describing Data

Once data is collected it has to be summarised. Two numbers do most of the work: one for the centre and one for the spread. Choosing the right pair, and knowing what each hides, is the substance of this topic.

Measures of central tendency

For the data $4, 7, 9, 12, 18$:

$$\text{mean } \bar{x} = \frac{\sum x}{n} = \frac{50}{5} = 10$$

The median is the middle of the ordered data: $9$.

The mode is the most frequent value — here there is none, since every value occurs once.

For grouped data the mean is estimated using class midpoints:

$$\bar{x} = \frac{\sum fx}{\sum f}$$

Measures of spread

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

For $4, 7, 9, 12, 18$ the range is $18 - 4 = 14$. Simple, but determined entirely by the two most extreme values.

$$\text{interquartile range} = Q_3 - Q_1$$

For $3, 5, 7, 8, 10, 12, 15, 18$: the median is $9$, $Q_1 = \frac{5+7}{2} = 6$ and $Q_3 = \frac{12+15}{2} = 13.5$, so

$$\text{IQR} = 13.5 - 6 = 7.5$$

The IQR describes the middle half of the data and is unaffected by extremes — which is exactly why it pairs with the median.

Variance and standard deviation

The variance is the mean squared deviation from the mean:

$$\sigma^2 = \frac{\sum(x - \bar{x})^2}{n} = \frac{\sum x^2}{n} - \bar{x}^2$$

The second form is almost always faster by hand.

For $4, 7, 9, 12, 18$ with $\bar{x} = 10$:

$$\sum x^2 = 16 + 49 + 81 + 144 + 324 = 614$$ $$\sigma^2 = \frac{614}{5} - 10^2 = 122.8 - 100 = 22.8$$

The standard deviation is its square root:

$$\sigma = \sqrt{22.8} = 4.77 \text{ (3 s.f.)}$$

Check by the definition: $\frac{36+9+1+4+64}{5} = \frac{114}{5} = 22.8$ ✓

For a frequency table:

$$\sigma^2 = \frac{\sum fx^2}{\sum f} - \bar{x}^2$$

Stem-and-leaf diagrams and box plots

A stem-and-leaf diagram shows the shape of the distribution while keeping every original value, so the median and quartiles can be read straight off. It must have a key — "$2 \mid 4$ means 24" — or it means nothing.

A box plot shows five numbers: minimum, $Q_1$, median, $Q_3$, maximum. The box spans the IQR with the median marked inside it, and whiskers reach to the extremes (or to the outlier boundaries, with any outliers plotted separately).

Box plots are at their most useful when two or more are drawn on the same scale: differences in centre and spread become visible at a glance.

Outliers and skewness

An outlier is conventionally a value

$$\text{below } Q_1 - 1.5 \times \text{IQR} \qquad \text{or} \qquad \text{above } Q_3 + 1.5 \times \text{IQR}$$

For the data above, with $Q_1 = 6$, $Q_3 = 13.5$ and IQR $= 7.5$:

$$6 - 11.25 = -5.25 \qquad 13.5 + 11.25 = 24.75$$

Every value lies between those bounds, so there are no outliers.

Skewness describes asymmetry:

Income data is the standard example of positive skew: a few very high earners pull the mean above the median, which is why "median income" is the figure usually reported.

Coding

Substituting $y = \dfrac{x - a}{b}$ makes awkward numbers manageable. The statistics transform back as:

$$\bar{x} = a + b\bar{y} \qquad \sigma_x = b\,\sigma_y$$

If $x$ is coded as $y = x - 100$, with $\bar{y} = 3.2$ and $\sigma_y = 1.5$:

$$\bar{x} = 100 + 3.2 = 103.2 \qquad \sigma_x = 1.5$$

So if 5 is added to every value, the mean rises by 5 and the standard deviation does not move. If every value is doubled, the mean doubles and so does the standard deviation — while the variance is multiplied by $4$.