IQR Calculator

IQR is evaluated from Data Values. The calculation reports Q1, Q2 - Median and Q3.

Results

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About the IQR Calculator

IQR is treated here as a quantitative relation between Data Values and Q1, Q2 - Median, Q3 and IQR = Q3 - Q1.

The calculator uses a multi formula configuration. Each reported value is read as a direct evaluation of the stored rules with the declared field formats and units.

Formula basis:
IQR = Q3 - Q1
Lower fence = Q1 - 1.5 x IQR
Upper fence = Q3 + 1.5 x IQR
Outliers: x < lower fence OR x > upper fence

Interpret the outputs in the order shown by the result fields. Optional inputs affect only the outputs that depend on those variables.

Formula & How It Works

The calculation applies the following relations exactly as recorded in the metadata:

IQR = Q3 - Q1
Lower fence = Q1 - 1.5 x IQR
Upper fence = Q3 + 1.5 x IQR
Outliers: x < lower fence OR x > upper fence

Each output field is produced by substituting the supplied inputs into the relevant relation and then applying the declared rounding or text format.

Worked Examples

Example 1: Basic Dataset — Textbook Example

Inputs

data: 7, 15, 36, 39, 40, 41, 42, 43, 47, 49
Q1: 36.75. Q2 - Median: 40.5. Q3: 42.75. IQR = Q3 - Q1: 6. Lower Fence: 27.75. Upper Fence: 51.75. Outliers: 7, 15

With Data Values = 7, 15, 36, 39, 40, 41, 42, 43, 47, 49 as the stated inputs, the result is Q1 = 36.75, Q2 - Median = 40.5 and Q3 = 42.75. Each value corresponds to the declared output fields.

Example 2: Household Incomes — Skewed Distribution

Inputs

data: 28000, 35000, 42000, 47000, 51000, 54000, 62000, 68000, 75000, 320000
Q1: 43,250. Q2 - Median: 52,500. Q3: 66,500. IQR = Q3 - Q1: 23,250. Lower Fence: 8,375. Upper Fence: 101,375. Outliers: 320000

With Data Values = 28000, 35000, 42000, 47000, 51000, 54000, 62000, 68000, 75000, 320000 as the stated inputs, the result is Q1 = 43,250, Q2 - Median = 52,500 and Q3 = 66,500. Each value corresponds to the declared output fields.

Example 3: Test Scores — Finding Spread

Inputs

data: 55, 62, 65, 70, 72, 74, 75, 78, 80, 83, 85, 88, 90, 95
Q1: 70.5. Q2 - Median: 76.5. Q3: 84.5. IQR = Q3 - Q1: 14. Lower Fence: 49.5. Upper Fence: 105.5. Outliers: None

With Data Values = 55, 62, 65, 70, 72, 74, 75, 78, 80, 83, 85, 88, 90, 95 as the stated inputs, the result is Q1 = 70.5, Q2 - Median = 76.5 and Q3 = 84.5. Each value corresponds to the declared output fields.

Example 4: COVID Test Turnaround Times (hours)

Inputs

data: 4, 6, 8, 9, 10, 11, 12, 12, 13, 14, 15, 16, 18, 42
Q1: 9.25. Q2 - Median: 12. Q3: 14.75. IQR = Q3 - Q1: 5.5. Lower Fence: 1. Upper Fence: 23. Outliers: 42

With Data Values = 4, 6, 8, 9, 10, 11, 12, 12, 13, 14, 15, 16, 18, 42 as the stated inputs, the result is Q1 = 9.25, Q2 - Median = 12 and Q3 = 14.75. Each value corresponds to the declared output fields.

Common Use Cases

  • Calculate IQR to describe data spread
  • Identify outliers using the 1.5×IQR rule
  • Find Q1, Q2, Q3 for box-and-whisker plot
  • Measure spread robust to extreme values