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, whether the data clusters symmetrically or leans to one side — is the distribution of the data. Reading the shape of a distribution is a core interpretation skill: it tells you whether the data is symmetric, skewed, or has unusual values at the extremes.

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. A classic symmetrical shape is the bell curve, which you will encounter later in this lesson when looking at the Normal distribution.

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 to the right. Because those extreme high values pull the mean upwards, the mean is greater than the median in a positively skewed distribution.

A real-world example: household income in the UK is positively skewed. Most households earn in a moderate range, but a small number of very high earners stretch the tail. The mean income is pulled upwards by these high earners, making it higher than the median. This is why the median is often a more representative average for income data.

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 to the left. The mean is pulled downwards by those extreme low values, making it less than the median.

A real-world example: scores on an easy test might show negative skew. Most students score highly (the cluster is on the right), but a small number of students who struggled pull the tail to the left. The mean score would be lower than the median because of those very low scores.

Key Fact

Positive skew: tail on the right, mean > median. Negative skew: tail on the left, mean < median. A useful memory trick: the skew is named after the direction of the TAIL, not the main cluster.

Key Fact

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

Commenting on Outliers

An outlier is a value that stands apart from the rest of the data — either much larger or much smaller than typical. Outliers can arise from genuine extreme values (a student who scored 100% on a test where everyone else scored between 40% and 80%), measurement errors (a data entry typo turning 160 cm into 16 cm), or a genuinely different subgroup in the data.

When commenting on outliers in exam questions, you should: (1) identify the outlier by referring to its approximate value, (2) suggest whether it might be a genuine extreme value or a possible error, and (3) comment on its effect on any averages you have calculated. An outlier can pull the mean towards it. The median is usually less affected.

Students often say “positive skew has most values 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 TAIL goes right (in the positive direction) and the CLUSTER sits at the lower end.

A quick check: if someone earned the most money in a room, are they pulling the mean income up? Yes — so income data is likely positively skewed (long right tail, even though most people cluster at lower incomes). That is positive skew.

Binomial Distribution (Higher)

A binomial distribution describes the number of successes in a fixed number of independent trials, where each trial has the same probability of success. The word “binomial” will appear in your exam. The key characteristics are:

A fixed number of trials (e.g. flipping a coin 5 times).
Each trial has only two possible outcomes (success or failure).
The probability of success is constant across all trials.
The trials are independent of each other.

In your exam, the number of trials $n$ will be no greater than 5. The distribution is symmetric when the probability of success is 0.5, and skewed when the probability is not 0.5. You do not need to use formal notation for the binomial distribution — describing its characteristics in words is sufficient.

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

The Normal distribution is a symmetric, bell-shaped distribution where values cluster around the mean and become progressively rarer further away. It describes many real-world characteristics: the heights of adults, birth weights, exam scores, and measurement errors all tend to follow approximately Normal distributions.

The specification requires you to know two key results about the Normal distribution:

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.

Action and Warning Lines in Quality Assurance (Higher)

In manufacturing and quality control, products are sampled regularly to check they meet a standard. The Normal distribution is used to set warning lines (at two standard deviations from the target) and action lines (at three standard deviations from the target). If a sample mean falls outside the action lines, the production process is stopped and investigated, because such a result would be very unusual if the process were running correctly.

Standardising Data (Higher)

To compare values from two different distributions (e.g. a student's score in Maths and their score in English, which have different means and standard deviations), you can standardise by converting each value to a z-score: $$z = \frac{x - \bar{x}}{s}$$ where $x$ is the value, $\bar{x}$ is the mean, and $s$ is the standard deviation. The formula will be given in the question. A z-score tells you how many standard deviations above or below the mean a value lies, allowing fair comparison across different scales.

The heights of a group of adults are 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. Heights above 194 cm or below 146 cm (three standard deviations away) are very unusual.