Describing data

Levels of measurement

The level of measurement determines which statistics are even meaningful. Getting this wrong is the single most common way to choose the wrong test.

• Nominal — categories with no order. Diagnosis, gender, treatment arm. Mode only.
• Ordinal — ordered, but intervals are not equal. Likert items, symptom severity ratings. Median and range.
• Interval — equal intervals, no true zero. Temperature in Celsius, many standardised scores. Mean and standard deviation.
• Ratio — equal intervals with a true zero. Reaction time, number of sessions attended. All statistics available.

Central tendency

The mean uses every value, which makes it efficient and also makes it sensitive to outliers. The median ignores magnitude and reports position, which makes it robust. The mode is the only option for nominal data.

In a symmetrical distribution the mean, median and mode coincide. In a positively skewed distribution the mean is pulled above the median; in a negatively skewed one it is pulled below. That relationship is a reliable way to infer skew direction from summary statistics alone.

Spread

The standard deviation is the average distance of scores from the mean, in the original units. Variance is its square, which is mathematically convenient but not directly interpretable.

The interquartile range is the middle 50% of the data and is the right companion to the median when the distribution is skewed.

• SD is in the original units; variance is in squared units.
• Report mean with SD, or median with IQR. Mixing them is a common error.
• About 68% of a normal distribution lies within 1 SD of the mean, 95% within 2, 99.7% within 3.

Where marks get lost

• Reporting a mean for ordinal data such as a single Likert item.
• Using the mean with a strongly skewed distribution, where the median describes the typical case better.
• Treating a large standard deviation as an error. It is a finding about variability.