Question 1

How do you calculate the IQR step by step?

Accepted Answer

1) Sort the data in ascending order. 2) Find Q2 (median) — the middle value. 3) Find Q1 — the median of values below Q2 (excluding Q2 for odd n). 4) Find Q3 — the median of values above Q2. 5) IQR = Q3 − Q1. Example: {2, 4, 5, 7, 8, 10, 11} — Q2=7, lower half {2,4,5} → Q1=4, upper half {8,10,11} → Q3=10. IQR = 10 − 4 = 6.

Question 2

How is the IQR used to detect outliers?

Accepted Answer

Tukey's fences: Lower = Q1 − 1.5×IQR, Upper = Q3 + 1.5×IQR. Any value outside these fences is an outlier. A 3×IQR multiplier identifies extreme outliers. Example: Q1=20, Q3=60, IQR=40 → fences: −40 and 120. Values below −40 or above 120 are outliers. This rule is used by default in box plot rendering in R, Python matplotlib, and most statistical software.

Question 3

When should I use IQR instead of standard deviation?

Accepted Answer

Use IQR when: (1) data is skewed (income, house prices, response times); (2) outliers should not distort the spread measure; (3) you are constructing a box plot. Use standard deviation when: data is approximately normal and you need it for parametric tests (t-test, z-score, regression). For symmetric distributions: SD ≈ IQR / 1.35 (the normal distribution relationship).

Question 4

What is a five-number summary?

Accepted Answer

The five-number summary: minimum, Q1, Q2 (median), Q3, maximum. These five values define the box plot. The box spans Q1 to Q3 (the IQR width) with a line at the median. Whiskers extend to the last non-outlier value within the fences. Points beyond the whiskers are plotted individually as outlier markers.

Question 5

Why do different tools give different quartile values?

Accepted Answer

Multiple quartile calculation methods exist, especially for small datasets. The main difference: whether to include the median in both halves (inclusive) or exclude it (exclusive). Excel and the inclusive method split differently from R's default (type 7, linear interpolation). For large datasets (n > 30) the differences are negligible. This calculator uses linear interpolation — consistent with R's default type 7 method.

Question 6

What does a large IQR indicate?

Accepted Answer

A large IQR relative to the median means the middle 50% of data has high spread. Example: median salary $60K with IQR $40K means the middle 50% earn $40K–$80K. Compare to IQR $10K: middle 50% earn $55K–$65K — much tighter. For relative comparison across datasets with different scales, use the coefficient of variation (CV = SD/mean × 100%) instead.

Criterion	IQR	Range	Std Deviation
Outlier resistance	High — ignores extreme values	None — determined by extremes	Moderate — pulled by outliers
Best data shape	Skewed or unknown	Symmetric, no outliers	Symmetric, approximately normal
Used in	Box plots, exploratory analysis	Quality control, simple ranges	Hypothesis tests, CIs, z-scores
Meaningful for	Ordinal and interval data	Any numeric data	Interval and ratio data
Changes with n?	Stable (middle 50%)	Increases with n	Converges with n
For normal data	IQR ≈ 1.35 × SD	≈ 4–6 × SD for n=50–200	The primary metric

Interquartile Range Calculator

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IQR vs Range vs Standard Deviation — When to Use Each

Case Study: Software Engineer Salary Analysis

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