GEA1000 Chap.3 Numerical Data: Summaries and Relationships
Numerical Data: Summaries and Relationships
Summaries keep the distribution attached
The mean and standard deviation describe centre and spread around the mean, while the median and interquartile range are more resistant to skew and extreme observations.
Histograms and boxplots reveal shape that no pair of summary statistics can preserve by itself.
The mean uses every value, so a long tail or extreme observation can pull it away from the middle of most cases. Standard deviation measures spread around that mean and shares the variable's unit. The median and quartiles depend on order, making them more resistant without making them universally superior.
Choose the pair that answers the question and suits the distribution.
Transformations create fast checks
Adding a constant shifts mean, median and quartiles while leaving standard deviation and interquartile range unchanged. Multiplying by a positive constant scales all of them. These properties make unit conversions and spreadsheet outputs auditable.
A fixed offset that changes spread, or a scale conversion that adds to standard deviation, signals an incorrect formula or cell reference.
Relationships need plots
Correlation measures linear direction and strength without units. Regression adds a prediction line with a slope that keeps the measurement units.
Curvature, separated groups, influential points and aggregation can all make a single coefficient misleading.
Describe a scatterplot through form, direction, strength and unusual features before quoting correlation. A high-leverage point lies far in the explanatory direction and can rotate a fitted line. A response outlier sits far above or below the line.
Neither should be deleted automatically; compare the analysis with and without the point and investigate what it represents.
Aggregation changes the analysis level
A country-level or group-level relationship concerns those aggregates. It need not describe individuals inside them. Separate subgroup patterns can also differ from the pooled line because group centres carry the overall association.
Colouring or slicing by group is analytical work that tests whether aggregation is hiding the relationship of interest.
Regression stays inside its range
The slope gives the average predicted response change for one explanatory-unit increase within the observed range. Extrapolation beyond that range can be unreliable even when the fitted calculation is correct.
End with the population, units, range and non-causal status of an observational relationship.
What this chapter covers
- 01
Mean and median
- 02
Variance and standard deviation
- 03
Quartiles and interquartile range
- 04
Histograms and boxplots
- 05
Scatterplots and correlation
- 06
Linear regression and aggregation
Worked example · free
Mean, median and sample spread
- 2Calculate the mean and median.
- 3Calculate the sample standard deviation from squared deviations.
- 1Connect the summaries to distribution shape.
Key terms
- Sample Mean
- The sum of observed numerical values divided by the number observed.
- Sample Standard Deviation
- A measure of observed spread around the sample mean using an n minus one denominator.
- Interquartile Range
- The distance from the first quartile to the third quartile.
- Outlier
- An observation unusually far from the rest of the response distribution.
- Correlation Coefficient
- A unit-free measure of the direction and strength of a linear relationship.
- Ecological Fallacy
- Assigning an aggregate relationship between groups to individuals within those groups.
Numerical Data: Summaries and Relationships FAQ
When should median and interquartile range be preferred?
They are often more informative when a distribution is strongly skewed or contains influential extremes, because they resist movement from a few unusual values. Mean and standard deviation remain useful when the inferential target or distributional shape calls for them.
Why must a scatterplot accompany correlation?
Correlation summarises linear direction and strength but cannot reveal curvature, separated clusters or the exact influence of unusual points. A near-zero value can coexist with a strong curved relationship, while one high-leverage point can create or erase an apparent linear pattern.
Why is aggregate correlation dangerous?
A relationship between group averages concerns groups, not automatically the individuals inside them. Group composition and other group-level variables can generate an aggregate pattern that differs from within-group relationships. Individual-level claims require individual-level data and an appropriate design.
Exam move
Create small numerical data sets that make one property visible at a time. Add a constant and verify that mean and median shift while standard deviation and interquartile range stay fixed. Multiply every value and verify that centre and spread all scale. Build two distributions with the same mean but different spread, then two with similar centre and spread but different shape.
For relationships, sketch the scatterplot before computing correlation, and describe form, direction, strength and unusual features in that order. Recalculate a fitted relationship with and without a high-leverage point and explain why deletion is a substantive decision rather than a cleaning reflex.
Use the chapter solutions to learn how intermediate calculations are shown, but replace their contexts and numbers in your practice. After every calculation, write one sentence containing the variable, unit, comparison and level of analysis. This turns a numerical result into the interpretation expected in longer responses.
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