Reading Skew, Outliers and Distributions

What Is a Distribution?

When you collect a set of data, the values spread across a range. The way those values are spread — which values are common, which are rare, and whether the data clusters symmetrically or leans to one side — is called the distribution of the data. Reading the shape of a distribution lets you interpret results in context, which is exactly what AO2 marks reward.

Symmetrical Distributions

A symmetrical distribution is balanced: the left and right sides are roughly mirror images of each other. In a perfectly symmetrical distribution the mean, median and mode are all equal and sit at the centre. The bell-shaped Normal distribution, covered later in this lesson, is the classic example.

Positive Skew

A positively skewed distribution has a longer tail on the right. Most values cluster at the lower end of the scale, with a few unusually high values stretching the tail out to the right. Because those extreme high values pull the mean upwards, the mean is greater than the median.

A real example: UK household income is positively skewed. Most households earn a moderate amount, but a small number of very high earners stretch the tail. The mean is dragged above the median, which is why the median is usually reported as the ‘typical’ income figure.

Negative Skew

A negatively skewed distribution has a longer tail on the left. Most values cluster at the higher end of the scale, with a few unusually low values stretching the tail out to the left. The mean is pulled downwards, so it is less than the median.

A real example: scores on an easy test are often negatively skewed. Most students score highly, clustering to the right, but a small number of students who struggled pull the tail left. The mean score ends up lower than the median because of those few very low marks.

Key Fact

Positive skew: tail on the right, mean > median. Negative skew: tail on the left, mean < median. Symmetrical: mean = median. The skew is named after the direction of the TAIL, not the main cluster.

Identifying Outliers

An outlier is a value that sits well apart from the rest of the data — either much larger or much smaller than the typical values. By inspection, you can often spot an outlier just by looking at the data set or a diagram: it will be isolated from the rest of the values.

When commenting on an outlier, aim to make three points: name its approximate value, suggest whether it looks like a genuine extreme result or a possible recording error, and comment on its effect on any averages you have calculated. An outlier pulls the mean towards it; the median, based on position rather than value, is usually much less affected.

Students often say “positive skew means most values are on the positive side of the graph.” This is wrong. Positive skew is named after the direction of the TAIL, not where the main cluster sits. In a positively skewed distribution, the cluster sits at the lower end and the tail stretches out to the right.

Quick check: if one person in a room earns far more than everyone else, are they pulling the mean up? Yes — so income data is positively skewed, even though most people cluster at the lower end of the scale.

Key Fact

Comment on an outlier in three steps: (1) state its approximate value, (2) suggest whether it is genuine or an error, (3) explain its effect on the mean compared with the median.

Calculating Skew (Higher)

At Higher tier, skew can also be found by calculation rather than by inspection alone. The formula, given on the formulae sheet, is:

$$\text{Skew} = \frac{3(\text{mean} - \text{median})}{\text{standard deviation}}$$

A positive result confirms positive skew (tail to the right); a negative result confirms negative skew (tail to the left); a result close to zero suggests the distribution is close to symmetrical. You do not need to memorise this formula — it will be printed on the formulae sheet — but you must know how to substitute values into it correctly.

Identifying Outliers by Calculation (Higher)

Higher tier students are also expected to identify outliers using a calculation rather than by inspection alone. A value counts as an outlier if it falls outside these boundaries:

$\text{small outlier} < LQ - 1.5 \times IQR$, and $\text{large outlier} > UQ + 1.5 \times IQR$.

Alternatively, a value can be classed as an outlier if it lies outside three standard deviations of the mean, written as $\bar{x} \pm 3\sigma$. Whichever method a question specifies, always state the boundary value you calculated before deciding whether the value in question is genuinely an outlier.

Binomial Distribution (Higher)

A binomial distribution, written using the notation $B(n, p)$, describes the number of successes in a fixed number of trials. Four characteristics must all be true:

A fixed number of trials, $n$.
Each trial has only two possible outcomes: success or failure.
The probability of success, $p$, stays constant across all trials.
The trials are independent of each other.

In this exam, questions will not use a value of $n$ greater than 10. The mean of a binomial distribution is $np$. The distribution is symmetric when $p = 0.5$, and skewed when $p$ is not 0.5.

The Normal Distribution and the 68/95 Rule (Higher)

The Normal distribution, written using the notation $N(\mu, \sigma^2)$, is symmetric and bell-shaped: values cluster around the mean and become progressively rarer further away. Heights, birth weights and many measurement errors approximately follow this shape. You are not expected to use Normal distribution tables — only to apply the following results:

Approximately 68% of the data lies within one standard deviation of the mean.
Approximately 95% of the data lies within two standard deviations of the mean.
Values more than three standard deviations from the mean are very unusual.

The heights of a group of adults are approximately Normally distributed with a mean of 170 cm and a standard deviation of 8 cm.

Within one standard deviation (162 cm to 178 cm): approximately 68% of adults. Within two standard deviations (154 cm to 186 cm): approximately 95% of adults. A height above 194 cm or below 146 cm (three standard deviations away) would be very unusual.