A number needs a statistical meaning
A calculated centre is not complete until it is named, interpreted for the population and checked against the table or class from which it came.
Learning goals
- Use and rearrange the empirical relationship among mean, median and mode.
- Interpret a calculated central measure in context.
- Compare measures and validate their plausibility.
Concept lesson
Three centres, three meanings
Mean is a frequency-weighted balance point, median divides ordered observations into two halves, and mode represents the frequency peak.
Empirical relationship
For a moderately skewed distribution, Mode approximately equals 3 Median minus 2 Mean, equivalently 3 Median=Mode+2 Mean. Write the complete relation before rearranging.
Contextual interpretation
A median of 149.03 cm means about half the observations lie at or below and about half at or above that value; it does not say every observation is near it.
Solve for one missing measure
Substitute the two known measures into 3 Median=Mode+2 Mean, isolate the unknown and check the arithmetic by substituting it into the original relation.
Plausibility triangle
Check mean within the overall data range, median within its median class and mode within its modal class. For inverse frequency, require a non-negative admissible count.
Relation rearrangement grid
Derive a separate form for each unknown from the full empirical relation, then solve mixed prompts without memorising three disconnected formulae.
Write a complete interpretation
For each reported mean, median or mode, name the population, variable, unit and meaning of the measure while avoiding claims the statistic does not support.
Error-location audit
Given inconsistent results, diagnose whether the error lies in class marks, cumulative frequency, modal/median class, formula parameters, empirical rearrangement or arithmetic.
Statistical measure comparison report
Use two calculated measures to estimate the third, compare their locations, interpret the most relevant measure for the context and state the limits of the empirical estimate.
Relation and interpretation protocol
Write the full empirical relation before substitution, label the resulting measure and unit, then add a range or class plausibility check.
Statistical interpretation mastery
Solve empirical-relation items for each unknown, interpret a contextual statistic and diagnose an implausible grouped-data result.