Interquartile Range Calculator: Q1, Q3, IQR, and Outliers

This tool exposes three definitions of Q1 and Q3 instead of hiding one. The exclusive median-of-halves method (the default) sorts the data, finds the median, splits the values into a lower and an upper half, and for an odd count leaves the median out of both halves; Q1 and Q3 are the medians of those halves. This is the method most US intro-statistics courses teach, and it matches the TI-83/84 1-Var Stats output and Moore & McCabe's textbook Introduction to the Practice of Statistics. Tukey's hinges use the same half-split but keep an odd-count median in both halves, following Tukey's Exploratory Data Analysis (1977); several textbooks and packages use it. The interpolated type 7 method skips the half-split entirely: Q1 and Q3 come from linear interpolation at the 25th and 75th percentiles using the rank h = 1 + (n - 1)p, which is R's default quantile and Excel's QUARTILE.INC / PERCENTILE.INC. Because popular calculators pick different defaults, two of them can report different IQRs for the same numbers and both be correct. The fix is not to argue about which is right - it is to choose the method your assignment, report, or software uses, then confirm the sorted data and quartiles this tool shows.

Once Q1 and Q3 are set, the tool builds two screening boundaries: a lower fence at Q1 - 1.5 x IQR and an upper fence at Q3 + 1.5 x IQR. The NIST/SEMATECH e-Handbook calls these the inner fences and flags any point beyond them as a mild outlier; the 1.5 multiplier traces back to Tukey's box-plot work. Take 10, 12, 13, 14, 15, 16, 17, 18, 100 on the default method: Q1 = 12.5 and Q3 = 17.5, so IQR = 5 and the fences sit at 12.5 - 7.5 = 5 and 17.5 + 7.5 = 25. The value 100 is past the upper fence, so the tool lists it as a possible outlier - and stops there. Following NIST, a flagged point is something to investigate (a typo, a rare event, or a real value from a skewed distribution), not something to auto-delete. This tool ships the 1.5 x IQR inner fence only, not the wider 3 x IQR outer fence some texts use to separate mild from extreme outliers.

IQR measures the spread of the middle 50% of the data, from Q1 to Q3, so a single extreme value barely moves it. The standard deviation squares every deviation from the mean, so one large outlier can inflate it sharply. That makes IQR the better spread summary when your data are skewed or contain outliers - incomes, house prices, response times, or scores with a few very high or very low values. It is also the engine behind the box plot: the box spans Q1 to Q3, the line inside is the median, and the whiskers usually stop at the 1.5 x IQR fences. Reach for the standard deviation instead when the data are roughly symmetric and you need a measure that feeds into means, z-scores, or normal-distribution work. A quick tell: when the mean and median are far apart the distribution is skewed, and IQR will describe the spread more honestly than the standard deviation.

A few habits keep the result honest. First, read the sorted data the tool echoes back - a mistyped or missing value shifts the quartiles silently. Second, confirm the method matches your source before comparing numbers; a homework key usually wants the exclusive default, while a spreadsheet wants type 7. Third, keep genuine duplicate values - they are real data and belong in the quartile calculation unless they are entry errors. Remember the tool needs at least four numbers, works on a raw list rather than grouped frequency tables or class intervals, and uses the 1.5 x IQR inner fence only. For anything you will report or grade, match the exact convention your teacher, publisher, or software requires rather than assuming every calculator agrees.