Reconstruct keyed stems and define percentile rank

Try

A synthetic sample of ten waiting times, measured in whole minutes, is 8, 12, 12, 15, 19, 21, 24, 24, 27, 30. A histogram would group these into bins; a stem-and-leaf display can preserve each recorded value while arranging the observations by place value.

Use the tens digit as the stem and the ones digit as the leaf. The key is essential: 2 | 4 means 24 minutes. A different measurement scale could use the same printed digits for 2.4 or 240, so the reader cannot infer units from layout alone. Predict where 8 belongs and whether the repeated 12 should appear once or twice.

Predict

Before counting, predict whether “the percentage below 24” equals “the percentage at or below 24.” Two observations equal 24, so the inclusion rule should matter. Does the phrase “24 is at the 80th percentile” unambiguously tell a reader which rule was used?

Also predict whether the median must equal a recorded observation. Here the sample size is even, so the median averages the fifth and sixth ordered values. A stem display does not change that definition.

Experiment

Construct the worked display, then reconstruct its values to check that no observations disappeared.

Tens stem	Ordered ones leaves	Values recovered
0	8	8
1	2 2 5 9	12,12,15,19
2	1 4 4 7	21,24,24,27
3	0	30

Key: stem 1 and leaf 2 represent 12 minutes. There are ten leaves, one per observation. Both copies of 12 and both copies of 24 remain. The fifth and sixth observations are 19 and 21, so the median is 20 minutes even though nobody in this sample waited exactly 20 minutes.

Keep the leaves, compare the ranks

Predict both percentages before changing a value. Every repeated observation must retain its own leaf. Inputs are dimensionless integers from 0 to 99.

Observations (up to 60)12, 14, 14, 18, 21, 23, 25, 25, 25, 31 Comparison value

Reset observationsTry all tied values

Key: 2 | 5 means 25. Stem = tens digit; leaf = ones digit. Leaves are ordered within each stem. A dash means no observations in that decade.

All 10 observations, retained with their multiplicities

Stem	Leaves
0	—
1	2 4 4 8
2	1 3 5 5 5
3	1
4	—
5	—
6	—
7	—
8	—
9	—

Strictly below 25: 6/10 ≈ 60.0%. At or below 25: 9/10 ≈ 90.0%.

Percentages are rounded to one decimal place. The gap is the fraction equal to the comparison value. These are two explicitly named empirical rank conventions; neither calculation finds an inverse percentile cutoff. A small sample’s rank is not automatically a population percentile.

Temporary exploration: these inputs do not save practice answers or award completion.

Enter the worked values into the explorer and query 24. First predict both counts, then inspect the ordered stems and the reported fractions. Change the query to 23 without editing the dataset. Finally, add a third observation of 24 and predict what happens to the numerator and denominator under each convention. These controls are temporary exploration, not evidence of independently completed practice.

Understand

For a sample of n values and a query q, define the strictly-below rank as 100 times the number of observations less than q divided by n. Define the at-or-below rank as 100 times the number less than or equal to q divided by n. Both are valid empirical summaries when clearly named; they are different at a tie. They describe this sample rather than an unknown population distribution.

At q=24, six observations are strictly below: 8,12,12,15,19,21. The below rank is 60%. Including both 24s gives eight observations, so the at-or-below rank is 80%. Their difference, 20 percentage points, is exactly the fraction of the sample tied at the query. At q=23, no observation equals the query, and both ranks are 60%.

If one more 24 is added, the sample size becomes 11. The below count remains six, giving 600/11%, approximately 54.55%; the at-or-below count becomes nine, giving 900/11%, approximately 81.82%. Adding a tied observation can lower one rank while raising the other because both fractions have a changed denominator. Keep counts beside percentages so this behavior is inspectable.

A percentile cutpoint asks the reverse kind of question: given a proportion, which value should mark it? There are several quantile conventions. Under the explicitly chosen nearest-rank rule, the pth percentile for 0<p≤100 is ordered observation number ceil(np/100), where ceil means round up to an integer. In the worked sample the 80th-percentile cutpoint is observation 8, equal to 24. Its at-or-below rank is 80%, but its below rank is 60%. For the 70th percentile the selected observation is number 7, also 24. Ties therefore let the same cutpoint serve more than one requested percentile.

An interpolated quantile method can produce other cutpoints. That does not make a carefully labeled method wrong; it means the convention must accompany the result. Do not assume that a rank-of-value calculation and a value-at-percentile calculation are inverse functions, or that either must match an unstated software default. The nearest-rank rule above also deliberately excludes p=0; a separate endpoint convention would be needed there.

Stem displays are most useful for modest numerical datasets at a declared precision. A zero leaf matters: stem 3, leaf 0 is the observation 30. An empty stem between occupied ones makes a gap visible. Sorting leaves and retaining duplicates preserve the raw data; omitting duplicates or shifting the key changes the dataset rather than merely restyling it. The explorer uses integers from 0 to 99, so it does not silently round decimal or negative measurements into that range.

Apply

Construct the fresh display in practice and include its place-value key and units when supplied. Count its leaves, reconstruct its sorted values, and compare the result with the original list. For percentile queries, state the exact comparison rule before counting, retain repeated observations, and show the fraction before converting to a percentage.

For the mixed task, report a nearest-rank cutpoint and a median separately. Even if their numerical values happen to agree, explain their different definitions. A percentile summary does not establish normality, measurement accuracy or population representativeness.

Continue

Keep the raw observations, display key and percentile convention together in your evidence. Subsequent comparisons of groups can then distinguish a genuine distributional difference from a changed display scale, denominator or quantile convention. These are public guided tasks; accurate reconstruction and a justified interpretation require review beyond merely opening the explorer.