Quantile checks: fix the interpolation rule before comparing
Two programs can return different sample quartiles from identical numbers without either having read the data incorrectly. Quantiles require a definition of where probabilities sit between ordered observations. Fix that definition before interpreting a difference as a change in the underlying population.
The R quantile reference documents nine algorithms and identifies type 7 as its default in the referenced version. Other tools require their own documented method. The worksheet compares type 1 and type 7 explicitly rather than relying on a generic “percentile” label.
Preserve the ordered sample
Record observation identifiers, raw values, units, inclusion rules, missing-value handling and whether weights apply. Sort a copy while preserving the identifiers. Repeated values remain observations unless the research definition says otherwise; deleting ties changes the empirical population.
State the probability p, sample size n and algorithm. For an unweighted sample and 0 < p ≤ 1, type 1 uses the inverse empirical distribution: select the ordered observation at ceiling(np). Its zero-probability endpoint is handled separately as the minimum. This method returns an observed value.
For type 7, calculate h = 1 + (n − 1)p. Let j be floor(h) and g = h − j; interpolate as (1 − g)x[j] + g x[j+1], with endpoint handling at the minimum and maximum. Indices here start at one. An interpolated result need not be an observed sample value.
Hypothetical four-value comparison
Assume the fictional sorted sample is 2, 4, 10 and 20. All values have the same unit and equal observation weight.
| Probability | Type 1 | Type 7 |
|---|---|---|
| 25% | ceiling(4 × 0.25) = 1, so 2 | h = 1.75, so 3.5 |
| 75% | ceiling(4 × 0.75) = 3, so 10 | h = 3.25, so 12.5 |
For the lower quartile, type 7 gives 2 + 0.75 × (4 − 2) = 3.5. For the upper quartile it gives 10 + 0.25 × (20 − 10) = 12.5. Type 1 instead selects the first and third observations.
An interquartile range is therefore 8 under type 1 and 9 under type 7. A downstream rule using that range can classify observations differently even though the input file is unchanged. Do not call that a new outlier event without checking the algorithm.
Reproduce the implementation boundary
Test a small deliberate sample at probabilities zero, one and an internal probability. Include a tie, a missing value and a one-observation sample if those cases occur in practice. Some software rejects missing values by default while another silently omits them; a changed n is a data-treatment difference, not merely an interpolation difference.
If weights are present, specify what the weights represent and use a documented weighted-quantile definition. The unweighted type-7 formula does not become weighted simply by substituting an effective sample size. Replicating rows according to arbitrary weights can also change the target.
Distinguish an exact quantile of the stored sample from an estimate of a population quantile. A deterministic algorithm returning many decimal places does not eliminate sampling uncertainty. For very high percentiles in a small sample, few tail observations constrain the result; avoid treating interpolation as newly observed tail data.
Save tool version and the explicit method parameter with the result. Rerun both methods on identical raw values to isolate algorithm differences before investigating provider revisions or extraction failures. Recheck whenever a workflow changes software, defaults or weighted-data treatment.
Quantile labels and threshold membership need separate validation. A rule selecting values strictly above the 75th percentile can produce a different count from a rule using greater than or equal to, particularly with ties. Neither necessarily selects exactly one quarter of a finite sample. Preserve the comparison operator instead of changing the quantile until the desired count appears.
For a reproducible handoff, include the small sorted validation vector, chosen probability, one-based or zero-based index convention and expected result. Ask the implementation to reproduce that result before processing the real table. If an approximate streaming algorithm is used for a large dataset, record its approximation properties separately; exact type-7 interpolation and an approximate percentile sketch are not the same computation.
FAQ
Is an interpolated quartile a fabricated observation?
No, if labeled as a statistic under a stated algorithm. It becomes misleading when presented as an actual observed row.
Is there one universal sample-quantile definition?
No. Several legitimate conventions exist. Choose one appropriate to the task and use it consistently across comparisons.