SO5005 Chap.4 Variables, Descriptions and Distributions
Variables, Descriptions and Distributions
Describe before testing
Variable scale determines which summaries and displays are meaningful. Counts and percentages describe categories; numerical measures require centre and spread. Mean with standard deviation suits a roughly symmetric distribution, while median with interquartile range resists a strong tail.
Every summary needs a denominator, unit and missing-data count.
Shape is substantive information
Extreme observations may be errors, unusual valid cases or the event the study needs to explain. Inspect the distribution and measurement process before excluding anything. Descriptive analysis is the first result, not decoration placed before an inferential test.
What this chapter covers
- 01
Categorical and numerical variables
- 02
Continuous and discrete measurement
- 03
Counts and denominators
- 04
Mean and standard deviation
- 05
Median and interquartile range
- 06
Missing and extreme values
Worked example · free
One long wait shifts the mean
- 2Order the values and calculate the mean.
- 2Find the median and quartiles.
- 2Explain the long right tail.
Key terms
- Mean
- The sum of numerical values divided by their count.
- Interquartile Range
- The distance from the first to the third quartile, covering the middle half of ordered values.
- Missingness
- The pattern and amount of unavailable observations for a variable or time point.
Variables, Descriptions and Distributions FAQ
When should the median lead?
Use the median when a numerical distribution is markedly skewed or contains influential extreme values that make the mean a poor description of a typical observation. Pair it with quartiles or the interquartile range and still explain the full shape.
Why must percentages show denominators?
The denominator identifies the population represented by the percentage and reveals missing responses. Twelve positive responses can mean very different things out of fifteen, thirty-five or one hundred eligible observations, so reporting only the percentage hides essential context.
Should an outlier always be deleted?
No. Check whether it is impossible, miscoded, duplicated or a valid unusual case. Document any correction or exclusion and consider how conclusions change with the value included. Automatic deletion can remove the operational event the research question should explain.
Assessment move
Take raw values and compute centre and spread by hand. Sketch the shape, identify any impossible or influential observation, then write one result sentence containing n, unit, missingness and the chosen summary pair. Begin every dataset by building a variable dictionary with name, meaning, unit, permissible values and missing-value code.
Many statistical errors start when a numeric code is mistaken for a measured quantity or when an impossible value is treated as genuine. For categorical variables, produce counts first and calculate percentages from a printed denominator. Recalculate one percentage manually to confirm the software is using valid responses rather than every row.
For numerical variables, order a small sample and find the median and quartiles before calculating the mean. Change one observation to an extreme value and watch which summaries move. This experiment makes resistance memorable and prevents an automatic mean-and-standard-deviation response.
Practise describing histograms without labels such as normal or non-normal: state the number of modes, direction of any tail, visible gaps and influential values. Then connect the description to a reporting choice. Keep an audit log for corrections, exclusions and derived variables, with the original value and reason preserved.
When a value looks strange, ask whether it is impossible, improbable or substantively important; those categories require different actions. Finish by drafting a compact table title and footnote that identify the analysed sample, units, summary format and missing observations. A reader should be able to reconstruct the denominator without searching the methods.
Before inferential analysis, explain aloud how the descriptive results support or challenge the planned procedure. If the answer mentions only a significance test, return to the distribution and measurement process. Build a results template that begins with participant flow and data completeness before any outcome.
For each numerical variable, write the plausible range and unit from the collection instrument, then compare it with the observed minimum and maximum. Recalculate one mean from a small subset and one percentage from raw counts to detect coding or denominator errors. Prepare two versions of a table, one suitable for symmetric data and one for skewed data, and explain why the chosen version fits.
Finally, ask what decision a reader could make from the descriptive section alone. If the answer is nothing, the table may be full of numbers without addressing the research question. Remove any summary that lacks a clear interpretive role, and add the comparison or denominator a decision-maker would need to understand the remaining pattern. Check that the text and table use identical denominators.
Resolve every discrepancy before inference. Then archive the corrected table with notes explaining every revision and exclusion.
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