MATH2209 Chap.1 Data, Variables and Descriptive Statistics
Data, Variables and Descriptive Statistics
This chapter begins with the decision that controls every descriptive calculation: what kind of variable was measured. It separates nominal and ordinal categories from discrete and continuous quantities, then connects distribution shape with mean, median, variance, standard deviation and graphical displays.
The worked material shows why an extreme observation can move the mean without equally moving the median, and why an outlier should be investigated rather than silently removed.
What this chapter covers
- 01
Categorical and quantitative variables
- 02
Nominal and ordinal measurement
- 03
Discrete and continuous data
- 04
Mean, median and mode
- 05
Variance and standard deviation
- 06
Distribution shape and outliers
Worked example · free
Choose a centre for a right-skewed stay length
- 1The total is 70, so mean = 70 / 7 = 10 days.
- 1The fourth ordered value is the median, 6 days.
- 1The 35-day value creates a long right tail and pulls the mean upward.
- 1Report the median as the clearer typical stay and retain the unusual value for investigation.
Key terms
- Nominal Variable
- A categorical variable whose labels have no inherent ranking.
- Ordinal Variable
- A categorical variable with ordered levels but no guaranteed equal spacing.
- Standard Deviation
- A measure of spread expressed in the original measurement unit.
Data, Variables and Descriptive Statistics FAQ
When should I report the median?
Use the median when a quantitative distribution is strongly skewed or contains influential extreme values. It depends on order rather than every magnitude, so one unusual observation moves it less than the mean. Pair it with the interquartile range and still show the distribution.
Does an outlier have to be deleted?
No. An outlier may be a recording error, a valid rare case or the observation that reveals an important mechanism. Check how it was produced, compare analyses with and without it when appropriate, and document any decision instead of removing it automatically.
Why classify variables before graphing?
Variable type determines meaningful geometry. Bar charts compare categorical frequencies, while histograms display ordered numerical intervals. Treating categories as a continuous scale or connecting unrelated levels can imply distances and trends that the measurement does not support.
How should I audit a descriptive report?
Start with the sample size, missing observations and the measurement scale of every variable. Check that each display matches that scale, axes carry units and categories have not been given artificial distances. Recalculate centre and spread from the ordered values, then compare those numbers with the visible shape.
Mean with standard deviation should be justified by approximate symmetry and lack of influential extremes; median with interquartile range is usually clearer for strong skew. Investigate unusual observations rather than deleting them automatically. Finish by naming the sample or population, describing shape and spread, and explaining why the selected summary answers the research question.
If the graph, calculation and sentence disagree, inspect data entry, sorting and axis construction before accepting any one of them.
What belongs in a variable decision table?
Record the variable name, its measurement level, meaningful arithmetic, suitable displays and the preferred centre and spread. Add the measurement unit and note whether missing values or impossible codes need investigation. For an ordinal variable, preserve category order without assuming equal gaps. For a count, keep the nonnegative support visible.
For continuous measurements, inspect shape and unusual observations before selecting mean or median. This compact table turns variable classification into a reproducible choice rather than a label written after the analysis.
Assessment move
Build a four-column table for each new variable: type, meaningful operations, display and summary. Begin each session by taking six examples from health or biology and classifying them without calculating anything. Include at least one nominal category, one ordered response, one count and one continuous measurement. For every choice, explain which arithmetic and graph are meaningful and which would create a false impression.
This trains the decision that precedes every formula. Next, work with small ordered datasets. Calculate mean, median, range, interquartile range, variance and standard deviation with visible intermediate steps. Change one central value, then change one extreme value, and record which summaries move most.
Sketch the before-and-after distributions so that resistance to outliers becomes a property you can see rather than a sentence to memorise. When a dataset is skewed, practise pairing median with interquartile range; when it is roughly symmetric without influential points, justify mean with standard deviation. Add units to every numerical summary. Use graph-selection drills as a separate pass.
Match categorical frequencies to bar charts, numerical intervals to histograms and paired quantitative variables to scatterplots. Audit axes, category ordering and scale before interpreting the picture. A technically polished chart should still be rejected if its geometry conflicts with the variable type. Finish with short written conclusions.
Name the population or sample, describe shape, identify any unusual observation and justify the reported centre and spread. Avoid calling an observation an error without evidence. Instead, state whether it needs verification and show how it affects the conclusion.
Once a week, reproduce the sample mean and variance formulas from the raw formula sheet, calculate them on five values, and check the standard deviation against the original measurement scale. Your final self-test is to read a paragraph, identify every variable, choose one display and one summary for each, and explain the choice before touching software.
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