2Research methods
Describing data
Descriptive statistics summarise a sample without generalising beyond it. Which measure of central tendency and of spread to use depends on the level of measurement and the skew.
Practice
10 questions
Level
Foundation
Reading time
3 min
Levels of measurement
The level of measurement decides which statistics mean anything. Getting it wrong is a common way to choose the wrong test. The four levels go back to Stevens (1946).
Whether a mean suits Likert data is debated: Norman (2010) argues that parametric methods hold up well on it. The median uses the order of the categories without assuming the gaps between them are equal.
- Nominal: categories with no order, such as diagnosis, gender or treatment arm. Counts, percentages and the mode.
- Ordinal: ordered, but the intervals are not equal, such as Likert items or symptom severity ratings. Median and range.
- Interval: equal intervals and no true zero, such as temperature in Celsius and many standardised scores. Mean and standard deviation.
- Ratio: equal intervals and a true zero, such as reaction time or the number of sessions attended, so ratios make sense: 20 sessions is twice 10.
Central tendency
The mean uses every value, which makes it efficient and also sensitive to outliers. The median reports position and ignores magnitude, which makes it robust. For nominal data, the mode is the only one of the three that applies.
In a symmetrical distribution with one peak the mean, median and mode coincide. In a positively skewed distribution the mean is usually pulled above the median, towards the long tail; in a negatively skewed one, below it. The rule often fails, especially with discrete data such as counts, so look at the distribution itself (von Hippel, 2005).
Worked example:Sessions attended by five clients
- Five clients attend 2, 3, 3, 4 and 18 sessions. One long course of therapy gives the data a long tail to the right.
- The mean adds the values and divides by how many there are.(2 + 3 + 3 + 4 + 18) ÷ 5 = 6
- The median is the middle value once they are in order: the third of 2, 3, 3, 4 and 18.median = 3
- Four of the five attended fewer sessions than the mean of 6. The median of 3 describes the typical client, so report it with the interquartile range.
Source: Field (2024), checked .
Spread
The standard deviation is roughly the typical distance of scores from the mean, in the original units: the square root of the average squared distance, where a sample divides by n − 1 rather than n. Variance is its square: convenient mathematically, but not directly interpretable.
The interquartile range is the middle 50% of the data, and 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.
Source: Field (2024), checked .
Common mistakes
- Reporting a mean for a single Likert item without saying it assumes equal gaps between the categories.
- 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.
Sources
- Field (2024), Discovering Statistics Using IBM SPSS Statistics, sixth edition. SAGE. Checked .
- Norman (2010), Likert scales, levels of measurement and the ‘laws’ of statistics. Advances in Health Sciences Education, 15(5), 625–632. Checked .
- Stevens (1946), On the theory of scales of measurement. Science, 103(2684), 677–680. Checked .
- von Hippel (2005), Mean, median, and skew: correcting a textbook rule. Journal of Statistics Education, 13(2). Checked .