Describe center and variability without hiding the distribution

Learning goal: Calculate and interpret mean, median, range and interquartile range, then choose summaries that fit the distribution and context. A measure of center describes a typical or balancing location; a measure of variation describes how spread out the values are. One cannot replace the other.

Use the complete invented duration data set. The numbers were made up for this lesson to model attempts timed from first touch of the cards to “finished” and rounded to whole seconds: 2, 3, 3, 4, 4, 4, 5, 5, 6, 6, 7, 11. There are twelve observations totaling sixty seconds. The mean is 60 ÷ 12 = 5 seconds per observation. It can be interpreted as an equal share of the total duration across twelve observations.

An equal-share model redistributes sixty one-second counters among twelve positions until each holds five. The total remains sixty even though the individual original values change in the model. The mean does not claim every actual attempt lasted five seconds. It describes what the equal share would be if the total were redistributed.

The median is the middle of the ordered data. With twelve values, the middle positions are sixth and seventh, containing four and five. Their midpoint is (4 + 5)/2 = 4.5 seconds. At least half of the observations are at or below the median and at least half are at or above it; here exactly six lie below 4.5 and six above. A median need not be one of the recorded values.

Range is maximum minus minimum: 11 − 2 = 9 seconds. It uses the two endpoints and indicates the full observed span. It does not reveal how the ten interior observations are distributed. Two data sets can have the same range but very different clustering.

For this lesson, find quartiles by taking the median of each half of the ordered data. The lower six values are 2, 3, 3, 4, 4, 4, so Q1 = (3 + 4)/2 = 3.5. The upper six are 5, 5, 6, 6, 7, 11, so Q3 = (6 + 6)/2 = 6. The interquartile range, or IQR, is Q3 − Q1 = 2.5 seconds. It describes the width of the middle half of the ordered observations.

Quartile conventions can differ, particularly with odd-sized data sets or in computer software. State the method used so another person can reproduce the calculation. All examples here use the median-of-halves method. For an odd-sized set, this lesson leaves the overall median out of both halves: for 1, 2, 4, 5, 7, 8, 9 the median is 5, the halves are 1, 2, 4 and 7, 8, 9, so Q1 = 2, Q3 = 8 and IQR = 6. Do not mix quartiles from different procedures within one explanation.

Compare two additional fictional sets, each with six observations. Set A is 3, 4, 4, 5, 5, 6. Set B is 1, 2, 4, 5, 7, 8. Both total twenty-seven and have mean 4.5; both have median 4.5. Yet A has range three and IQR one, while B has range seven and IQR five. Equal centers do not imply equal variability.

For A, the lower half 3, 4, 4 has median four and the upper half 5, 5, 6 has median five. For B, the corresponding half-medians are two and seven. The diagram shows both sets on the same scale, making their equal central location and different spread visible. Changing the scale for one set would make a visual comparison harder to interpret.

Choose summaries with the shape and question in mind. In the twelve-value set, the relatively high eleven-second observation pulls the mean above the median. Median and IQR help describe the central portion without giving that endpoint the same influence on center as the mean does. Mean is still useful when an equal-share or total-based interpretation is relevant. Report the unusual value and shape rather than using a preferred summary to hide them.

A summary needs units and context. “The mean is five” is incomplete without saying five seconds per recorded attempt in this invented example. Mean and median are centers; range and IQR are measures of variation. None alone explains why the attempts differed or establishes performance for an unobserved population.

Worked endpoint change: If eleven were replaced by twenty-three, the total would rise by twelve to seventy-two and the mean would become six. The middle pair would remain four and five, so the median would remain 4.5. This illustrates different sensitivity to an extreme value; it does not authorize changing an observed value without evidence.

Worked choice: For a description of the middle half of attempts, use Q1 = 3.5, Q3 = 6 and IQR = 2.5, while also reporting the full two-to-eleven span. For equal sharing of the total recorded time, use the five-second mean and explain its interpretation.

Guided calculations: Count and sum the twelve values, model their mean as an equal share and mark the median positions. Split the ordered list into halves, calculate quartiles and distinguish range from IQR.

Independent performance: Calculate centers and variation for actual data from the investigation lesson and for a contrasting supplied set. State observation count, attribute, units and quartile method. Describe shape, clusters, gaps and any unusual values. Choose summaries suited to the question, explain the choice and show why two equal means do not guarantee similar distributions.