Declare the quartile rule before flagging a value

Educational reading copy of the original activity first published by Figran Team at https://rentry.co/figran-quartile-rules-iqr-flags-20261015. This directory version retains all fictional source records, formulas, calculations, seven tasks and worked answers. Tables are represented as labelled semicolon-separated rows. The commercial website link is omitted; no Figran account or purchase is needed. Declare the quartile rule before flagging a value By Figran Team. We are associated with Figran, a commercial AI-assisted visual creation service. This teaching activity was prepared with AI assistance and checked by its author for internal arithmetic consistency. Every strip label and length below is invented. These are not physical measurements, research findings, classroom results, or evidence about Figran's performance. No software account is required. A rule that flags values beyond 1.5 times the interquartile range needs more than the phrase "use the quartiles." Different stated rules can place the first and third sample quartiles at different points between the same ordered records. This activity keeps the records and the flagging threshold rule fixed while changing the quartile calculation. A flag is a consequence of those choices, not proof that a record is wrong. 1. Fix the records before comparing rules The base version contains six fictional strip lengths, in millimetres. The source rows are intentionally not arranged by length. Strip IDs continue to identify the same records throughout the activity. Source row; Strip ID; Invented length in mm 1; A; 10 2; B; 3 3; C; 24 4; D; 7 5; E; 11 6; F; 6 The ascending ID-value list is B:3, F:6, D:7, A:10, E:11, C:24. Write its ordered values as x[1] through x[6]: 3, 6, 7, 10, 11, 24. Positions start at 1. The source remains six records; an interpolated quartile is not an additional strip. Both calculations below include all six records exactly once. Do not sort the lengths and then leave their IDs attached to the old row positions. Do not drop C merely because it is the largest value. The observed range of the supplied base values is 24 - 3 = 21 mm under either quartile rule. 2. State two explicit interpolation rules Let p be the target probability: 0.25 for Q1, 0.50 for Q2, and 0.75 for Q3. Let n be the number of included records. For each rule, calculate a position h in the ordered list. Rule used in this activity; Position h; Definition reference Rule 6; (n + 1) * p; R quantile type 6 Rule 7; 1 + (n - 1) * p; R quantile type 7 These formulas follow the continuous sample-quantile definitions identified as types 6 and 7 in the R documentation linked in section 8. The type numbers identify algorithms, not six or seven observations. This activity uses only the three probabilities above with n = 6 or n = 8; every position used here lies inside the ordered list. It is not a specification of endpoint handling for arbitrary sample sizes or probabilities. If h is an integer, select x[h]. Otherwise, let j be the whole-number part of h and let t = h - j. Interpolate by Q(p) = x[j] + t * (x[j + 1] - x[j]). For example, position 1.75 is three quarters of the distance from the first value to the second. It does not mean select source row 1.75 or round the position to a whole record. Use one rule consistently for Q1 and Q3 in a reported IQR. An unexplained "automatic quartile" label is not enough to reproduce the calculation. Naming the two rules here does not establish which definition every spreadsheet, charting tool, or boxplot function uses. 3. Calculate the base-version quartiles With n = 6, Rule 6 uses h = 7 * p and Rule 7 uses h = 1 + 5 * p. Keep the interpolation position separate from the resulting length. Rule; Target probability; Position h; Bracketing ordered lengths in mm; Quantile in mm Rule 6; Q1, p = 0.25; 1.75; 3 and 6; 5.25 Rule 6; Q2, p = 0.50; 3.5; 7 and 10; 8.5 Rule 6; Q3, p = 0.75; 5.25; 11 and 24; 14.25 Rule 7; Q1, p = 0.25; 2.25; 6 and 7; 6.25 Rule 7; Q2, p = 0.50; 3.5; 7 and 10; 8.5 Rule 7; Q3, p = 0.75; 4.75; 10 and 11; 10.75 For Rule 6, Q1 = 3 + 0.75 * (6 - 3) = 5.25 mm, and Q3 = 11 + 0.25 * (24 - 11) = 14.25 mm. For Rule 7, Q1 = 6 + 0.25 * (7 - 6) = 6.25 mm, and Q3 = 10 + 0.75 * (11 - 10) = 10.75 mm. Both rules give Q2 = 7 + 0.5 * (10 - 7) = 8.5 mm for this version. Agreement on Q2 does not make the Q1 and Q3 definitions interchangeable. None of the six strips needs to equal a calculated quartile. The source lengths and IDs stay unchanged while the interpolation rule changes. 4. Hold the flagging rule fixed Define IQR = Q3 - Q1. In this activity the lower fence is Q1 - 1.5 * IQR, and the upper fence is Q3 + 1.5 * IQR. Flag a supplied record only when its value is strictly below the lower fence or strictly above the upper fence. A value exactly on a fence would not be flagged by this stated rule. Base-version quartile rule; IQR in mm; Lower fence in mm; Upper fence in mm; Flagged supplied IDs Rule 6; 9; -8.25; 27.75; None Rule 7; 4.5; -0.5; 17.5; C, with length 24 mm For Rule 6, 1.5 * IQR = 13.5 mm; for Rule 7, it is 6.75 mm. The thresholds therefore differ even though the six input values and the multiplier 1.5 are identical. Under Rule 6, 24 is below 27.75. Under Rule 7, 24 is above 17.5. All other supplied values are inside both pairs of fences. A negative lower fence is an arithmetic threshold, not an assertion that a negative strip was recorded. Likewise, 17.5 mm is a threshold, not a seventh strip or an observed endpoint. This activity defines flagging thresholds only; it does not prescribe how a particular plotting application draws boxplot whiskers. Choose and record a method before interpreting the flags. Do not select whichever quartile rule produces a preferred classification and then describe that classification as method independent. A flag can prompt a check of the record and context; the calculation alone does not establish an error, a cause, or a reason to delete the observation. 5. Seven learner tasks 1. Reconstruct the base version's ascending ID-value list. State n and the observed range. Explain why neither interpolation nor a flag changes those source facts. 2. Calculate h for Q1, Q2, and Q3 under each rule with n = 6. Show the interpolation for each different Q1 and Q3. Explain why agreement on Q2 does not settle which quartile rule was used. 3. Calculate both IQRs and both pairs of 1.5-IQR fences. Identify every flagged supplied ID under each rule. State what happens to a value exactly equal to a fence under this activity's strict comparison. 4. A draft caption says: "Rule 6 gives Q1 = 5.25 mm and Q3 = 10.75 mm, so IQR = 5.5 mm." Diagnose the problem. Does correct subtraction make the pair a valid Rule 6 summary? Supply the consistent Rule 6 and Rule 7 alternatives. 5. Create a separate eight-record version by retaining A through F and adding G = 4 mm and H = 16 mm. Sort it and recalculate Q1, Q2, Q3, IQR, and both fences for each rule. Which supplied IDs are flagged? Explain why the six-record fences cannot simply be relabelled as an eight-record result. 6. A reviewer says: "C is flagged by Rule 7, so its source value must be wrong; delete it. Also draw the upper fence as if a strip of that length had been observed." Identify what the calculations establish and what they do not. Distinguish the observed range from the IQR, and a threshold from a source record. 7. Write a reproducible caption for the six-record base version using Rule 7. Include the fictional status, included records and units, quartile formula, quartiles and IQR, fence rule and values, the flagged ID, and the limit on interpretation. Avoid a claim about a measured experiment, classroom benefit, or Figran's performance. 6. Worked answers 1. The ascending base list is B:3, F:6, D:7, A:10, E:11, C:24. There are six records and their range is 21 mm. Interpolation summarizes the existing ordered values without adding a record. A flag labels a supplied value under a stated screening rule; it does not itself remove that record or revise its recorded length. 2. Rule 6 gives positions 1.75, 3.5, and 5.25. Its quartiles are 5.25, 8.5, and 14.25 mm. Rule 7 gives positions 2.25, 3.5, and 4.75, producing 6.25, 8.5, and 10.75 mm. The four Q1/Q3 interpolation steps are shown in section 3. The common Q2 = 8.5 follows from the same middle position in this example; it does not identify the Q1/Q3 algorithm. State the rule rather than infer it from a single matching summary. 3. Rule 6 has IQR = 14.25 - 5.25 = 9 mm and fences 5.25 - 13.5 = -8.25 mm and 14.25 + 13.5 = 27.75 mm. No supplied record is outside. Rule 7 has IQR = 10.75 - 6.25 = 4.5 mm and fences 6.25 - 6.75 = -0.5 mm and 10.75 + 6.75 = 17.5 mm. Only C:24 exceeds a fence. Exact equality is not a flag under the strict rule. These statements use unrounded arithmetic, not rounded display values to decide the comparison. 4. The caption mixes Rule 6's Q1 with Rule 7's Q3. Although 10.75 - 5.25 = 5.5 is correct subtractio