Descriptive Statistics

Look at the distribution before you summarise it

Every choice on this page follows from the shape of the data, so the shape has to be looked at first. Plot a histogram. Plot a boxplot. Do this before calculating anything you intend to report.

What you are looking for:

• Skew — a long tail in one direction, which pulls the mean away from where most of the data sits.
• Multiple peaks — usually a sign that two different groups are mixed in one variable, in which case any single summary of the whole is misleading.
• Extreme values — genuine outliers, or data entry errors, and you need to know which before doing anything else.
• Floor and ceiling effects — scores bunched at the minimum or maximum, which limits what any analysis can detect.
• Missing data — how much, and whether it is concentrated anywhere in particular.

Doing this catches most of the problems that would otherwise reach your examiner, and it takes minutes.

Which measure of the middle

Three measures, and they answer slightly different questions.

The practical rule: if the mean and median differ noticeably, the distribution is skewed and the median is usually the more honest summary. Income data is the standard illustration — a small number of very high values pull the mean above what most people earn, so the median describes the typical case better.

Reporting both is often the best answer, and costs one extra column.

Which measure of spread

A summary of the middle without a summary of spread is close to useless: two datasets with identical means can look completely different. The choice of spread measure should follow the choice of central tendency.

• Standard deviation pairs with the mean. It describes typical distance from the mean, and like the mean it is sensitive to extreme values. Quoting it alongside a median is a mismatch that signals the choice was not thought about.
• Interquartile range pairs with the median. It covers the middle half of the data and is unaffected by extremes, which is precisely why it suits skewed distributions.
• Range is the distance between minimum and maximum. It is easy to report and highly unstable, since it depends entirely on two values. Useful as context, not as your main measure.

Report the actual minimum and maximum too where they matter — particularly if a variable has a possible range that your data did not cover, which is itself worth knowing.

What to report, by data type

The data type constrains what is meaningful, and this is where quiet errors appear.

Categorical data — report counts and percentages. Always give the count alongside the percentage, because “60%” means something different when it is 6 of 10 rather than 600 of 1,000. Where a category is small, percentages can badly overstate precision.

Ordinal data — the median and interquartile range, or a frequency table showing the whole distribution. A frequency table is often more informative than any summary and takes little space. Whether means may be reported for summed Likert-type scales is a genuine debate in the literature; a single item is ordinal, and treating one item’s mean as a meaningful quantity is harder to defend than doing so for a multi-item scale.

Continuous data — mean and standard deviation if roughly symmetric; median and interquartile range if not. Report the sample size for each variable, since missing data means it often differs between them.

Sample size — report it everywhere, and report the number the analysis actually ran on rather than the number recruited. Those diverge after missing data, and a reader cannot interpret anything without knowing which they are looking at.

Tables and figures

Choose by what you want the reader to do. Tables are for looking values up and comparing precisely. Figures are for seeing shape, pattern and relationship. Presenting the same numbers in both is padding, and examiners read it as such.

Points that repeatedly cost marks:

• Decimal places. More decimals do not mean more precision, they mean less readability. Two is usually plenty; matching the precision of your measurement is the principle.
• Axes starting somewhere other than zero. On a bar chart this exaggerates differences and is a recognised way of misleading. If there is a genuine reason, say so on the figure.
• Bar charts of means with no indication of spread. Two variables with identical means can have entirely different distributions. Error bars, or a plot showing the distribution, tell the reader far more — and if you use error bars, state what they represent, since standard deviations, standard errors and confidence intervals look identical and mean different things.
• Captions that describe rather than inform. “Figure 3: Results” wastes the line. Say what the figure shows.
• Tables that are not self-contained. A reader should understand a table without hunting through the text for what N, the units, or the abbreviations mean.

Follow your discipline’s reporting style exactly for table and figure formatting. It is the cheapest credibility available, and deviation is noticed.

Missing data and outliers

Both need a stated policy, and the policy should be decided before you see what each option does to your results. Deciding afterwards is how analysis drifts into selecting the outcome you preferred.

Missing data. Report how much there is and where it falls. The question that matters is whether it is concentrated — if the people who skipped an item differ systematically from those who answered, deleting them changes who your sample represents. Say what you did and why: excluded, retained, or handled by an imputation method with a citation.

Outliers. First establish whether the value is an error or a real observation. Errors get corrected or removed, and you say so. Genuine extreme values are a different matter: removing them because they are inconvenient is not defensible, and they are sometimes the most interesting cases in the dataset. Where an outlier materially changes a result, the honest approach is to report the analysis both with and without it and let the reader see the difference.