IQR Calculator: Q1, Q3, Outliers & Steps | StatsUnlock

An IQR calculator finds the interquartile range: the third quartile minus the first quartile, IQR = Q3 − Q1.

Sort the data, find Q1 (25th percentile) and Q3 (75th percentile), then subtract. For 46, 47, 48, 49, 50, 50, 51, 52, 53, 54, 55, 56, 57, 58, 61, 82, Excel’s QUARTILE.INC gives Q1 = 49.75 and Q3 = 56.25, so IQR = 6.5. Values outside Q1 − 1.5 × IQR and Q3 + 1.5 × IQR are outliers, so 82 is flagged.

Rule of thumb: report the median with the IQR when data are skewed or contain outliers, and always say which quartile method you used, because Excel, a TI-84 and a textbook can give different Q1 and Q3 for the same numbers.

The interquartile range (IQR) is the distance between the first quartile (Q1) and the third quartile (Q3). Quartiles cut sorted data into four parts with the same number of values. Q1 has 25% of the data below it, the median (Q2) has 50% below it, and Q3 has 75% below it. The IQR is the width of the middle half.

What it measures: spread. A small IQR means the typical values sit close together. A large IQR means they are spread out. Because the IQR ignores the lowest 25% and the highest 25%, it is a robust measure: a single wrong value or extreme case barely moves it. The range and the standard deviation can both be pulled a long way by one point.

A worked one-liner: for 3, 5, 7, 8, 12, 13, 14, 21 (n = 8), the lower half 3, 5, 7, 8 has median 6 and the upper half 12, 13, 14, 21 has median 13.5. By hand, IQR = 13.5 − 6 = 7.5. Excel’s QUARTILE.INC gives 13.25 − 6.5 = 6.75. Both are correct, they simply use different rules.

Who uses it: clinicians report median (IQR) for length of stay and lab values, economists use it for income, ecologists use it for body size and home-range data, teachers use it for exam scores, and data scientists use the 1.5 IQR rule to clean data before modelling.

4.1 What the IQR actually tells you

The IQR is the width of the band that holds the middle half of your observations. If the IQR of systolic blood pressure is 18 mmHg, then half of the patients sit inside an 18 mmHg window. That is a concrete, easy statement, and it stays true whatever happens in the tails.

It does not tell you where the band is. An IQR of 18 could run from 110 to 128 or from 150 to 168. That is why the IQR is always reported with the median, and ideally with Q1 and Q3 as well, so the reader can see both the location and the width.

4.2 Reading Q1, the median and Q3 together

The three quartiles split the sorted data into four groups of equal size. Look at the two gaps inside the box. If the median sits in the middle of the box, the middle half is symmetric. If the gap from the median up to Q3 is larger than the gap down to Q1, the upper half is more spread out, which is right skew. Bowley skewness puts a number on this: (Q3 + Q1 − 2 × median) ÷ IQR. It runs from −1 to +1, and values beyond ±0.2 are worth mentioning.

4.3 The 1.5 IQR rule for outliers, and what to do with a flag

John Tukey proposed fences at 1.5 IQR below Q1 and above Q3. For normal data only about 0.7% of values fall outside them, so a value beyond a fence is unusual. Values beyond 3 IQR (the outer fences) are extreme. A flag is a question, not a verdict. First check for typing errors, unit mix-ups (grams entered as kilograms), and missing-value codes like −999. If the value is real, keep it and use methods that resist it, such as the median, the IQR, or a non-parametric test. Deleting real values because they are inconvenient is a form of bias.

The 1.5 IQR rule flags too many values in large samples. With n = 1,000 normal values you should expect about 7 flags even when nothing is wrong. For big datasets, or for heavily skewed data, consider the 2.2 multiplier of Hoaglin and Iglewicz or a log transform before screening.

4.4 Why your IQR may differ from Excel, a TI-84 or a textbook

There is no single agreed definition of a sample quartile. Hyndman and Fan (1996) listed nine. Excel’s QUARTILE.INC, R’s default and NumPy all use type 7, which interpolates at position 1 + (n − 1)p. Excel’s QUARTILE.EXC, Minitab and SPSS use type 6, position (n + 1)p. A TI-84 and most school textbooks take the median of the lower and upper halves. The differences are largest with small samples and odd n. With the textbook set 6, 7, 15, 36, 39, 40, 41, 42, 43, 47, 49, Q1 is 15 by hand but 25.5 in Excel. Neither is wrong. Pick one, state it, and use it for every group.

4.5 IQR versus standard deviation: which one to report

If the data are roughly normal, the SD and IQR carry the same information, and IQR ≈ 1.349 × SD. If the data are skewed, have outliers or are on an ordinal scale, the SD is inflated by the tail and describes nobody well. Then the median with IQR is the honest summary. The calculator reports the IQR ÷ SD ratio: well below 1.349 means heavy tails or outliers; near 1.349 means the two measures agree.

4.6 Comparing IQRs across groups

Comparing IQRs is a fair way to compare spread when groups differ in size or contain outliers. A group with an IQR twice as large has a middle half twice as wide. When the groups have very different medians, you can also compare the quartile coefficient of dispersion, (Q3 − Q1) ÷ (Q3 + Q1), which is unit-free. Remember that two different IQRs do not prove a real difference in spread; a Levene, Brown-Forsythe or Fligner-Killeen test does that.

4.7 Ties, ordinal data and an IQR of zero

With Likert scores or counts, many values tie. If more than half the values share one score, Q1 and Q3 can be equal and the IQR becomes 0. Then every other value lies outside the fences and the outlier rule flags them all, which is meaningless. When that happens, report the percentage in each category instead, or report the median and the full range.

4.8 Decision guide

4.9 Common mistakes this calculator prevents

• Forgetting to sort the data before finding Q1 and Q3. The steps panel shows the sorted list first.
• Mixing quartile methods between groups or between software. The method table shows all four.
• Reporting the range as the spread when one value dominates it. The range and IQR sit side by side.
• Treating −999 or text entries as data. Non-numeric entries are skipped and named; missing-value codes trigger a warning.
• Deleting flagged values automatically. The output labels them mild or extreme and asks you to check them.
• Reporting IQR as a single number with no quartiles. The report templates give median (IQR) [Q1, Q3].

4.10 Practical judgement

Look at the box plot before you look at any number. Then read the IQR with the median, check whether the fences flag anything, and decide whether those flags are errors or real. Finally, pick your quartile method once and keep it for the whole study. In twelve years of field data on animal movement and body size, the most common problem has not been the formula but a stray decimal point or a missing-value code, and the 1.5 IQR screen catches exactly those.

When reporting IQR calculator results in APA format, give the median with the IQR, add Q1 and Q3 when space allows, name the quartile method, and say how outliers were defined and handled. The five templates below fill in automatically after you press Calculate.

▶ Run the analysis above to auto-fill all five examples with your results.

This flowchart shows how the IQR is calculated, step by step, from a raw column of numbers to Q1, Q3, the fences and the outlier decision. Every box uses the same worked example as the Excel, R and Python sections: 16 leaf lengths in millimetres from Site A, with one long leaf at 82 mm.

Check any box yourself: the Excel sheet in Section 9, the R script in Section 10 and the Python script in Section 11 all return Q1 = 49.75, Q3 = 56.25 and IQR = 6.5 for this dataset.

1. Enter your data. Type or paste numbers into the first box, separated by commas as the placeholder shows: 52, 48, 55, 61, 47, ... Spaces, tabs and new lines also work, so a column copied from Excel is fine.
2. Name each group. The name above each box is editable. Type something meaningful, such as Site A or Control, and it flows into the tables, charts and report sentences.
3. Add more groups if you need them. Press Add cluster to compare up to eight groups side by side. Each box has its own Clear and Remove buttons.
4. Or upload a CSV or Excel file. On the Upload tab, choose a file, then click every column you want. Each selected column is loaded as its own cluster, named after its header.
5. Or use the manual table. Type values into the grid, one column per group, and press Use this table.
6. Try a sample dataset. Ten built-in datasets cover clean data, skew, outliers, ties, small samples, three groups and a deliberately messy sensor log.
7. Choose the quartile method. Inclusive matches Excel QUARTILE.INC, R and Python. Exclusive matches SPSS. Median of halves matches a TI-84 and most textbooks.
8. Choose the fence multiplier. 1.5 is the standard Tukey rule. Use 2.2 for large samples or 3 to flag only extreme outliers.
9. Press Calculate IQR. Nothing is computed until you press the button. Changing any input clears old results so you never read stale numbers.
10. Read, check and export. Read the IQR and outlier panel, look at the four charts, copy a report sentence from Section 5, and download the text report or PDF.

Excel has no IQR function, but two quartile functions and one subtraction do the job. The part that confuses people is that Excel has two quartile functions that give different answers, and neither matches the hand method taught in most schools. This section uses the 16 leaf lengths from Section 7 and shows the spreadsheet at every step.

9.1 The functions you need

9.2 Step by step

9.3 The awkward cases: median and IQR together, groups, and odd n

Median and IQR in one cell, ready for a table: =TEXT(MEDIAN(A2:A17),"0.0")&" ("&TEXT(C5,"0.0")&")" returns 52.5 (6.5).

IQR for one group in a long table (values in A, group names in B): Excel 365 can filter inside the function.

In older Excel, use a PivotTable to split the groups into columns first, then apply QUARTILE.INC to each column.

Matching the TI-84 or a textbook. Excel has no median-of-halves function. With n = 16, the lower half is A2:A9 once sorted, so =MEDIAN(SMALL(A2:A17,{1,2,3,4,5,6,7,8})) gives Q1 = 49.5 and =MEDIAN(LARGE(A2:A17,{1,2,3,4,5,6,7,8})) gives Q3 = 56.5. The easier route is to set the method in this calculator to Median of halves.

9.4 Why your Excel answer might differ from R, Python, SPSS or a calculator

9.5 Making a box plot in Excel

1. Select the data column, including the header.
2. Insert → Insert Statistic Chart → Box and Whisker (Excel 2016 and later).
3. Right-click the box, choose Format Data Series, and set Quartile Calculation to Inclusive median to match QUARTILE.INC, or Exclusive median to match QUARTILE.EXC.
4. Tick Show outlier points. Excel uses the 1.5 IQR rule for the whiskers.
5. For groups, put each group in its own column and select them all before inserting.

Chart 1 in Section 3 draws the same box plot with the fences marked, which Excel cannot show.

9.6 Excel errors you will probably hit

R has a built-in IQR() function, so the number itself is one word. The script below goes further: it prints Q1, the median, Q3, the IQR, both sets of fences and every outlier with a plain-English verdict, then saves one publication-ready box plot at 300 dpi whose box matches the printed IQR. It uses base R only.

10.1 The complete script

R, base only, saves iqr_plot.png

# ============================================================
# IQR, quartiles, Tukey fences and outliers with one box plot
# StatsUnlock - https://statsunlock.com/iqr-calculator/
# Base R only. No packages to install.
# ============================================================

# ---- 1. Your data ------------------------------------------
# Replace with your own values, or read a CSV:
#   x <- read.csv("data.csv")$column_name
x <- c(52, 48, 55, 61, 47, 50, 58, 53,
       49, 56, 51, 54, 46, 57, 50, 82)
x <- x[!is.na(x)]                    # drop missing values

label <- "Site A leaf length"
units <- "mm"
k     <- 1.5                         # fence multiplier (3 = extreme only)

# ---- 2. Quartiles and IQR ----------------------------------
q     <- unname(quantile(x, c(0.25, 0.50, 0.75), type = 7))
q1    <- q[1]; med <- q[2]; q3 <- q[3]
iqr_x <- IQR(x, type = 7)            # same as q3 - q1

# ---- 3. Fences and outliers --------------------------------
lower    <- q1 - k * iqr_x
upper    <- q3 + k * iqr_x
lower2   <- q1 - 2 * k * iqr_x       # outer fences
upper2   <- q3 + 2 * k * iqr_x
outliers <- sort(x[x < lower | x > upper])
extreme  <- sort(x[x < lower2 | x > upper2])

# ---- 4. Plain-English summary ------------------------------
cat(sprintf("n = %d\n", length(x)))
cat(sprintf("Q1 = %.3f   Median = %.3f   Q3 = %.3f\n", q1, med, q3))
cat(sprintf("IQR = %.3f %s\n", iqr_x, units))
cat(sprintf("Fences: %.3f to %.3f (outer %.3f to %.3f)\n",
            lower, upper, lower2, upper2))
if (length(outliers) == 0) {
  cat("Verdict: no outliers by the 1.5 x IQR rule.\n")
} else {
  cat(sprintf("Verdict: %d outlier(s): %s\n", length(outliers),
              paste(outliers, collapse = ", ")))
  if (length(extreme) > 0)
    cat(sprintf("Extreme (beyond outer fence): %s\n",
                paste(extreme, collapse = ", ")))
}
cat(sprintf("Report: Mdn = %.2f %s (IQR = %.2f, Q1 = %.2f, Q3 = %.2f, n = %d)\n",
            med, units, iqr_x, q1, q3, length(x)))

# ---- 5. One publication-ready box plot ---------------------
# bxp() draws the box from OUR type 7 quartiles. boxplot() would
# use Tukey hinges (49.5 and 56.5) and not match IQR().
inside <- x[x >= lower & x <= upper]
stats  <- matrix(c(min(inside), q1, med, q3, max(inside)), ncol = 1)

png("iqr_plot.png", width = 2400, height = 1200, res = 300)
par(mar = c(4.5, 2, 2.5, 1))
bxp(list(stats = stats, n = length(x), out = outliers,
         group = rep(1, length(outliers)), names = ""),
    horizontal = TRUE, boxfill = adjustcolor("#16a34a", 0.30),
    border = "#166534", medlwd = 3, outpch = 19, outcol = "#c2410c",
    outcex = 1.6, ylim = range(c(x, lower, upper)), axes = FALSE)
axis(1)
box(bty = "l")
set.seed(42)                         # reproducible jitter
points(x, 1 + runif(length(x), -0.12, 0.12), pch = 16,
       col = adjustcolor("grey30", 0.45), cex = 0.9)
abline(v = c(lower, upper), lty = 2, lwd = 1.5, col = "#9333ea")
title(main = sprintf("%s: IQR = %.2f %s", label, iqr_x, units),
      xlab = sprintf("Value (%s)", units), font.main = 1)
dev.off()
cat("Saved iqr_plot.png\n")

Run it with Rscript iqr_plot.R, or paste it into the RStudio console. The figure appears as iqr_plot.png in the folder shown by getwd().

10.2 What each part does

10.3 What the figure shows

The figure is a horizontal box plot. The box runs from Q1 = 49.75 to Q3 = 56.25 with a thick line at the median, 52.5. The whiskers stop at 46 and 61, the last values inside the fences. The dashed purple lines are the fences at 40 and 66, and the orange point at 82 is the outlier. Grey jittered dots show every raw value so readers can see the sample size.

10.4 Common R problems and fixes

10.5 Useful one-liners

In Python the IQR is one line with NumPy, SciPy or pandas. The script below prints the same Q1, median, Q3, IQR, fences and outliers as R and Excel, then saves a 300 dpi box plot with the raw values and fences drawn on.

Install once with pip install numpy scipy matplotlib pandas.

11.1 The complete script

Python 3, saves iqr_plot.png

# ============================================================
# IQR, quartiles, Tukey fences and outliers with one box plot
# StatsUnlock - https://statsunlock.com/iqr-calculator/
# pip install numpy scipy matplotlib
# ============================================================
import numpy as np
import matplotlib
matplotlib.use("Agg")                     # save without opening a window
import matplotlib.pyplot as plt
from scipy import stats

# ---- 1. Your data ------------------------------------------
x = np.array([52, 48, 55, 61, 47, 50, 58, 53,
              49, 56, 51, 54, 46, 57, 50, 82], dtype=float)
# Or read a CSV:
#   import pandas as pd
#   x = pd.read_csv("data.csv")["column_name"].to_numpy(dtype=float)
x = x[~np.isnan(x)]                       # drop missing values

label = "Site A leaf length"
units = "mm"
k = 1.5                                   # fence multiplier (3 = extreme only)

# ---- 2. Quartiles and IQR ----------------------------------
q1, med, q3 = np.percentile(x, [25, 50, 75])   # linear = Excel QUARTILE.INC
iqr = q3 - q1
assert np.isclose(iqr, stats.iqr(x))           # SciPy agrees

# ---- 3. Fences and outliers --------------------------------
lower, upper = q1 - k * iqr, q3 + k * iqr
lower2, upper2 = q1 - 2 * k * iqr, q3 + 2 * k * iqr
outliers = np.sort(x[(x < lower) | (x > upper)])
extreme = np.sort(x[(x < lower2) | (x > upper2)])

# ---- 4. Plain-English summary ------------------------------
print(f"n = {x.size}")
print(f"Q1 = {q1:.3f}   Median = {med:.3f}   Q3 = {q3:.3f}")
print(f"IQR = {iqr:.3f} {units}")
print(f"Fences: {lower:.3f} to {upper:.3f} (outer {lower2:.3f} to {upper2:.3f})")
if outliers.size == 0:
    print("Verdict: no outliers by the 1.5 x IQR rule.")
else:
    listed = ", ".join(f"{v:g}" for v in outliers)
    print(f"Verdict: {outliers.size} outlier(s): {listed}")
    if extreme.size:
        print("Extreme (beyond outer fence): " + ", ".join(f"{v:g}" for v in extreme))
print(f"Report: Mdn = {med:.2f} {units} (IQR = {iqr:.2f}, "
      f"Q1 = {q1:.2f}, Q3 = {q3:.2f}, n = {x.size})")

# ---- 5. One publication-ready box plot ---------------------
fig, ax = plt.subplots(figsize=(8, 4))
box_kw = dict(whis=k, widths=0.5, patch_artist=True,
              boxprops=dict(facecolor="#bbf7d0", edgecolor="#166534"),
              medianprops=dict(color="#111827", linewidth=3),
              flierprops=dict(marker="o", markerfacecolor="#c2410c",
                              markeredgecolor="#c2410c", markersize=9))
try:
    ax.boxplot(x, orientation="horizontal", **box_kw)   # Matplotlib 3.10+
except TypeError:
    ax.boxplot(x, vert=False, **box_kw)                 # older Matplotlib
rng = np.random.default_rng(42)           # reproducible jitter
ax.scatter(x, 1 + rng.uniform(-0.12, 0.12, x.size),
           s=18, color="0.3", alpha=0.45, zorder=3)
for fence in (lower, upper):
    ax.axvline(fence, linestyle="--", color="#9333ea", linewidth=1.5)
ax.set_yticks([])
ax.set_xlabel(f"Value ({units})")
ax.set_title(f"{label}: IQR = {iqr:.2f} {units}")
fig.savefig("iqr_plot.png", dpi=300, bbox_inches="tight")
print("Saved iqr_plot.png")

11.2 What each part does

11.3 What the figure shows

It is the same figure as the R version: a horizontal box from 49.75 to 56.25, a median line at 52.5, whiskers to 46 and 61, dashed fences at 40 and 66, the outlier at 82 in orange, and grey dots for every raw value.

11.4 Common Python problems and fixes

11.5 Useful one-liners

12.1 Quartile positions for small samples (type 7, Excel QUARTILE.INC)

Conclusion: a position of 4.75 means "value 4 plus three quarters of the gap to value 5". Whole-number positions need no interpolation.

12.2 One dataset, four methods (textbook set, n = 11)

Data: 6, 7, 15, 36, 39, 40, 41, 42, 43, 47, 49.

Conclusion: with small odd samples the IQR can change by more than half depending on the method. Always state it.

12.3 Normal distribution facts for the IQR

Conclusion: the 1.5 IQR fences sit close to the familiar ±2.7 SD, so about 7 in 1,000 normal values are flagged by chance.

12.4 Expected false flags from the 1.5 IQR rule in normal data

Conclusion: in large samples a few flags are normal. Use k = 2.2, or look at how far beyond the fence a value lies, before calling it an error. Figures use the asymptotic 0.70% rate; small samples flag somewhat more often.

12.5 Which spread measure goes with which centre

Conclusion: pair the median with the IQR and the mean with the SD. Mixing them, such as mean (IQR), confuses readers.

12.6 IQR converters at a glance

Conclusion: these conversions assume roughly normal data. Use them for planning and meta-analysis, never in place of the raw data.

Eight worked IQR calculation examples covering the cases you will meet: the textbook odd-n set, a clear outlier, an even set by hand, skewed income, clean normal data, a tiny sample, ordinal ties, and two groups. All numbers use the inclusive (type 7) method unless stated.

Study design: the IQR assumes a set of independent measurements of one variable on at least an ordinal scale, collected in a way that does not bias the tails.

1. Define the variable and its unit before collecting. Mixed units, such as some lengths in cm and some in mm, are the most common source of false outliers.
2. Decide the unit of observation. One value per animal, plant, patient or plot. Repeat readings of the same unit should be averaged or modelled, not entered as separate rows.
3. Use a random or systematic sample. A convenience sample can shift both quartiles and give a precise IQR of the wrong group.
4. Fix the measurement resolution. Rounding to whole numbers creates ties, and many ties can make the IQR 0.
5. Record missing values as blank cells. Never use 0 or −999; the calculator will treat them as real values.
6. Write the quartile method into the protocol. Choose type 7 if you will analyse in Excel, R or Python, or type 6 if you will use SPSS.
7. Write the outlier rule in advance. State k (1.5 is standard) and what you will do with flagged values: check, keep, winsorise or remove.
8. Plan the sample size. Aim for at least 20 per group for stable quartiles; with fewer than 8, report the raw values too.
9. Keep a data log. Note who measured what and when, so a flagged value can be traced back.
10. Store data one column per group, exactly as below, ready to paste or upload here.

One column per group, one row per unit, numbers only below the header row. Blank cells are ignored, so unequal group sizes are fine.

Minimum sample: the calculator needs 3 values, but quartiles from fewer than 8 values are unstable. A single new value can move Q1 or Q3 by a whole data step.

Pre-registration: record the quartile method, the fence multiplier and the outlier policy before seeing the data. Choosing them afterwards, to keep or drop a particular value, is a common hidden bias.

Common collection mistakes:

• Typing missing data as 0 or −999, which creates fake low outliers.
• Mixing units or decimal commas and points in one column.
• Rounding heavily at the point of measurement, which creates ties.
• Pooling two different populations in one column, which widens the IQR.
• Deleting values in the field because they "looked wrong", so the tails are never seen.

This free IQR calculator is designed for students, teachers, researchers and analysts who need a robust measure of spread, a quick outlier screen, or a box plot with the numbers behind it.

Use the IQR when:

• ✓ The data are skewed, such as income, length of stay, reaction time or counts.
• ✓ There may be outliers and you want a spread measure they cannot distort.
• ✓ The variable is ordinal, such as Likert ratings or pain scores.
• ✓ You are reporting the median and need the matching spread.
• ✓ You want to screen data for errors before analysis.
• ✓ You are building a box plot or a five-number summary.

Use something else when:

• ✗ The data are normal and you will run a t-test or ANOVA: report the SD with the mean.
• ✗ You need the precision of an estimate: use a confidence interval.
• ✗ You need to test whether spreads differ: use Levene’s or Brown-Forsythe test.
• ✗ The data are nominal categories: report counts and percentages.
• ✗ More than half the values tie: the IQR may be 0, so use percentages per value.

Real-world examples:

1. Clinical research: hospital stay of median 4 days (IQR 2 to 7), because a few patients stay for weeks.
2. Ecology: home-range size of collared animals, where one wide-ranging individual would inflate the SD.
3. Education: comparing the spread of exam scores between two teaching methods with a side-by-side box plot.
4. Data cleaning: flagging sensor spikes before fitting a regression.

Design guidance: collect at least 20 values per group, keep units consistent, and decide the quartile method before you analyse.

Decision tree: Is the variable numeric or ordinal? If not, use percentages. Is it roughly normal with no outliers? Report mean (SD). Otherwise report median (IQR). Comparing groups? Use a Mann-Whitney U (two groups) or Kruskal-Wallis (three or more) for location, and Brown-Forsythe for spread.

Assumptions

1. Values are ordered on a meaningful scale. Check: the variable is ordinal, interval or ratio. If it fails: quartiles of codes for unordered categories are meaningless.
2. Observations are independent. Check: one value per unit. If it fails: repeated readings of a few units narrow the IQR and make the data look less variable than the population.
3. The sample represents the population. Check: random or systematic sampling. If it fails: Q1 and Q3 shift toward the over-sampled part, biasing both location and spread.
4. Enough data for stable quartiles. Check: n of 8 or more, ideally 20. If it fails: the IQR swings widely between samples and the methods disagree.
5. Few exact ties at the quartiles. Check: the data-quality table. If it fails: the IQR shrinks toward 0 and the outlier rule over-flags.
6. One population per column. Check: the histogram for two peaks. If it fails: the IQR widens to span both groups and describes neither.

Limitations

• The IQR ignores the outer 50% of the data, so it says nothing about how far the tails reach. Report the range or the outliers as well, or use a box plot.
• Sample quartiles have no single definition. Results depend on the method; always state it.
• The 1.5 IQR rule assumes roughly symmetric data. For skewed data it over-flags the long tail; consider a log transform or the adjusted box plot of Hubert and Vandervieren.
• The IQR is descriptive. It gives no p-value or confidence interval. For inference about spread, use Levene’s or Brown-Forsythe test, or a bootstrap confidence interval for the IQR.
• The IQR is less efficient than the SD for truly normal data, so it needs a larger sample for the same precision.
• The mean and SD conversions assume normality. For skewed data they can be badly wrong; use the raw data when you have it.

An IQR calculator answers a simple question: how wide is the middle half of my data? Sort the values, find Q1 and Q3, and subtract. That single number is hard to fool, which is why it sits beside the median in medical tables, inside every box plot, and at the heart of the most widely used outlier rule in statistics.

The calculation is easy; the details are where people go wrong. Quartiles have several valid definitions, so Excel, SPSS, R and a TI-84 can give different answers for the same numbers. The cure is not to find the "right" one but to choose one, name it, and use it everywhere. The method table on this page makes that choice visible, so you never have to guess why two numbers differ.

The 1.5 IQR rule is a screen, not a verdict. A flagged value deserves a second look at the raw record. Many flags turn out to be typing errors or missing-value codes; the rest are real, and real values belong in the analysis. Report how many values were flagged and what you did with them, and readers can judge for themselves.

Finally, pair the IQR with the median, give Q1 and Q3 when space allows, and show a box plot whenever you can. Those three habits make your spread statistics honest, reproducible and easy to read.

Paste your numbers above, read the box plot, check the fences, and come back to this IQR calculator whenever you need quartiles you can trust.

These peer-reviewed sources support the methods in this IQR calculator: quartile definitions, the 1.5 IQR outlier rule, box plots, robust spread, and converting between median (IQR) and mean (SD).

1. Aguinis, H., Gottfredson, R. K., & Joo, H. (2013). Best-practice recommendations for defining, identifying, and handling outliers. Organizational Research Methods, 16(2), 270–301. https://doi.org/10.1177/1094428112470848
2. Dawson, R. (2011). How significant is a boxplot outlier? Journal of Statistics Education, 19(2). https://doi.org/10.1080/10691898.2011.11889610
3. Freedman, D., & Diaconis, P. (1981). On the histogram as a density estimator: L2 theory. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 57(4), 453–476. https://doi.org/10.1007/BF01025868
4. Frigge, M., Hoaglin, D. C., & Iglewicz, B. (1989). Some implementations of the boxplot. The American Statistician, 43(1), 50–54. https://doi.org/10.1080/00031305.1989.10475612
5. Hoaglin, D. C., & Iglewicz, B. (1987). Fine-tuning some resistant rules for outlier labeling. Journal of the American Statistical Association, 82(400), 1147–1149. https://doi.org/10.1080/01621459.1987.10478551
6. Hoaglin, D. C., Iglewicz, B., & Tukey, J. W. (1986). Performance of some resistant rules for outlier labeling. Journal of the American Statistical Association, 81(396), 991–999. https://doi.org/10.1080/01621459.1986.10478363
7. Hozo, S. P., Djulbegovic, B., & Hozo, I. (2005). Estimating the mean and variance from the median, range, and the size of a sample. BMC Medical Research Methodology, 5, 13. https://doi.org/10.1186/1471-2288-5-13
8. Hubert, M., & Vandervieren, E. (2008). An adjusted boxplot for skewed distributions. Computational Statistics & Data Analysis, 52(12), 5186–5201. https://doi.org/10.1016/j.csda.2007.11.008
9. Hyndman, R. J., & Fan, Y. (1996). Sample quantiles in statistical packages. The American Statistician, 50(4), 361–365. https://doi.org/10.1080/00031305.1996.10473566
10. Krzywinski, M., & Altman, N. (2014). Visualizing samples with box plots. Nature Methods, 11(2), 119–120. https://doi.org/10.1038/nmeth.2813
11. Langford, E. (2006). Quartiles in elementary statistics. Journal of Statistics Education, 14(3). https://doi.org/10.1080/10691898.2006.11910589
12. Leys, C., Ley, C., Klein, O., Bernard, P., & Licata, L. (2013). Detecting outliers: Do not use standard deviation around the mean, use absolute deviation around the median. Journal of Experimental Social Psychology, 49(4), 764–766. https://doi.org/10.1016/j.jesp.2013.03.013
13. Luo, D., Wan, X., Liu, J., & Tong, T. (2018). Optimally estimating the sample mean from the sample size, median, mid-range, and/or mid-quartile range. Statistical Methods in Medical Research, 27(6), 1785–1805. https://doi.org/10.1177/0962280216669183
14. McGill, R., Tukey, J. W., & Larsen, W. A. (1978). Variations of box plots. The American Statistician, 32(1), 12–16. https://doi.org/10.1080/00031305.1978.10479236
15. Rousseeuw, P. J., & Croux, C. (1993). Alternatives to the median absolute deviation. Journal of the American Statistical Association, 88(424), 1273–1283. https://doi.org/10.1080/01621459.1993.10476408
16. Rousseeuw, P. J., & Hubert, M. (2011). Robust statistics for outlier detection. WIREs Data Mining and Knowledge Discovery, 1(1), 73–79. https://doi.org/10.1002/widm.2
17. Schwertman, N. C., Owens, M. A., & Adnan, R. (2004). A simple more general boxplot method for identifying outliers. Computational Statistics & Data Analysis, 47(1), 165–174. https://doi.org/10.1016/j.csda.2003.10.012
18. Shi, J., Luo, D., Weng, H., Zeng, X.-T., Lin, L., Chu, H., & Tong, T. (2020). Optimally estimating the sample standard deviation from the five-number summary. Research Synthesis Methods, 11(5), 641–654. https://doi.org/10.1002/jrsm.1429
19. Wan, X., Wang, W., Liu, J., & Tong, T. (2014). Estimating the sample mean and standard deviation from the sample size, median, range and/or interquartile range. BMC Medical Research Methodology, 14, 135. https://doi.org/10.1186/1471-2288-14-135
20. Weissgerber, T. L., Milic, N. M., Winham, S. J., & Garovic, V. D. (2015). Beyond bar and line graphs: Time for a new data presentation paradigm. PLOS Biology, 13(4), e1002128. https://doi.org/10.1371/journal.pbio.1002128