The median is the middle value of a dataset once the numbers are sorted from smallest to largest.

Sort your values, then take the one in the centre. If there is an even count of values there is no single middle, so the median is the average of the two central values. Because the median depends only on position and not on magnitude, one wildly extreme value cannot move it. That robustness is why income, house prices and hospital waiting times are always reported as medians rather than means.

Rule of thumb: if the mean and median differ by more than about 10%, your data are skewed and the median is the honest summary to report.

The median splits a dataset exactly in half. Sort the values, walk to the middle, and whatever value you land on is the median: half the observations lie at or below it and half lie at or above it. That definition sounds almost too simple to be useful, but it produces a property the mean can never have. Because the median is determined by where a value sits in the ordering rather than how big it is, changing the largest observation from 100 to 100 million does not shift the median by a single unit.

What this calculator reports:

• Median, the middle value, with its position shown explicitly.
• Q1 and Q3, the quartiles, which are the medians of the lower and upper halves.
• Interquartile range, Q3 minus Q1, the spread of the middle 50% and the median's natural partner.
• Median absolute deviation (MAD), the robust equivalent of the standard deviation.
• Confidence interval for the median, computed distribution-free from order statistics, which almost no other calculator provides.
• Any percentile you ask for, plus the full five number summary.
• Mean comparison, so you can see at a glance whether the data are skewed.
• Grouped median from a frequency table, for data that arrive already binned.

A worked one-liner: for the seven values 2, 4, 5, 7, 8, 9, 12 the position is (7 + 1) ÷ 2 = 4, so the median is the fourth value, 7. Add one more observation to make eight values and the position becomes 4.5, meaning you average the fourth and fifth.

Who uses it: national statistics offices report median household income because a handful of very high earners would distort the mean. Estate agents quote median house prices for the same reason. Clinicians report median survival time and median length of stay, both heavily right skewed. Ecologists use it for species counts. And anyone working with rankings or Likert scales uses it because the mean of ordinal data is not strictly meaningful.

4.1 What the median actually tells you

The median is a statement about position. It says that if you lined every observation up in order, this value would sit in the middle, with as many observations below it as above. That is all it claims, and the modesty of the claim is exactly what makes it reliable. It does not attempt to summarise the total, it does not weigh large values more heavily than small ones, and it does not assume the data have any particular shape.

The practical reading is "a typical value". When a newspaper says the median house price is 320,000, it means half the houses sold for less and half for more. If it quoted the mean instead, a single mansion changing hands could lift the figure by tens of thousands without a single ordinary house becoming more expensive. The median describes the middle of the market; the mean describes the total divided by the count, which is a different and often less useful thing.

4.2 The position formula and why an even count is different

The median sits at position (n + 1) ÷ 2 in the sorted list. With nine observations that is position 5, a whole number, so the median is an actual observed value. With ten observations it is position 5.5, which does not exist, so you take the average of the fifth and sixth values.

This has a consequence worth noticing: with an even count the median may be a number that appears nowhere in your data. The median of 2, 4, 5 and 7 is 4.5, and no one measured 4.5. That is legitimate and expected, but it occasionally confuses people who assume the median must be one of the observations. It also means that with an even number of integer values you can end up with a median ending in .5, which is fine to report as long as your measurement precision justifies it.

4.3 Why the median resists outliers and the mean does not

Statisticians describe this with the idea of a breakdown point, the proportion of the data you would have to corrupt before a statistic becomes arbitrarily wrong. The mean has a breakdown point of zero: one single value, taken far enough, drags it anywhere you like. The median has a breakdown point of 50%, the highest possible. You would have to corrupt half the dataset before you could force the median to move arbitrarily.

You can see this directly. Take the values 1, 2, 3, 4, 5. The mean and median are both 3. Now change the 5 to 500. The mean jumps to 102, while the median stays at exactly 3. The median did not merely change less, it did not change at all, because 3 is still the third of five sorted values. This is why the median is the default for income, wealth, waiting times, insurance claims and any measurement with a long tail.

4.4 The mean versus median gap, read properly

Comparing the two takes seconds and tells you the shape of your distribution. If they sit close together, the data are roughly symmetric. If the mean is above the median, there is a tail of large values pulling it up, which is right skew. If the mean is below, there is a tail of small values, which is left skew.

A gap under about 10% of the median usually means the data are symmetric enough that either summary is defensible. Between 10% and 20% you should report both. Above 20% the mean has been captured by the tail and the median is the honest choice. This calculator computes the gap and states a recommendation, but the reasoning is worth internalising because it applies to any dataset you meet.

One caution: the gap test detects skew, not every problem. Two clusters of data sitting either side of an empty middle can produce a mean and median that agree perfectly while both fall in the gap where no observation exists. The charts, not the summary statistics, are what catch that.

4.5 The interquartile range, the median's proper partner

A median on its own is as uninformative as a mean on its own. It needs a spread measure beside it, and the correct one is the interquartile range: Q3 minus Q1, the width of the middle 50% of the data. Just as you would never report a mean without a standard deviation, you should never report a median without an IQR.

The IQR shares the median's robustness. It is built from two positional values, so extreme observations cannot inflate it. This is why the standard reporting convention in medical and social research is median (IQR), sometimes written as median with the Q1 to Q3 pair in brackets. Both forms are acceptable as long as you say which you used, because "IQR = 8" and "IQR 34 to 42" convey different information at a glance.

4.6 Median absolute deviation, the robust standard deviation

The MAD is the median of the absolute distances between each value and the median. It answers "how far is a typical observation from the centre" using the same positional logic that makes the median robust in the first place.

Multiplying the MAD by the constant 1.4826 rescales it so that, for normally distributed data, it estimates the standard deviation. That constant is not arbitrary: it is the reciprocal of the 75th percentile of the standard normal distribution. The value of this is that you get an SD-like number that a single outlier cannot inflate, which makes MAD-based outlier detection considerably more reliable than the common approach of flagging anything more than three standard deviations from the mean, since a large outlier inflates the very SD being used to detect it.

4.7 The confidence interval for the median

Most calculators stop at the point estimate, but your median is computed from a sample and therefore carries uncertainty. This tool reports a confidence interval built from order statistics and the binomial distribution, which means it makes no assumption whatsoever about the shape of your data. It is exact rather than approximate, and it is valid for skewed distributions where a normal-theory interval would be wrong.

The method works by asking how many positions in from each end you must move before the probability of the true median falling outside is small enough. Because it counts whole observations, the achievable confidence levels are discrete. You cannot get exactly 95%; with sixteen observations the closest available interval is roughly 97.9%. The calculator reports the actual coverage achieved rather than pretending it hit your requested level exactly, which is the honest thing to do.

Below about six observations no 95% interval exists at all, because even the widest possible interval, from the minimum to the maximum, does not reach 95% coverage. In that case the calculator gives you the widest interval available and tells you what its true coverage is.

4.8 When the median is the wrong choice

The median is not universally superior. It discards information: it uses only the position of the middle observation and ignores everything about the magnitudes of the rest. When the data are genuinely symmetric with no outliers, the mean is more statistically efficient, meaning it varies less from sample to sample and gives you more precision for the same sample size.

The median also does not add up. Total revenue is the mean order value times the number of orders; there is no equivalent identity for the median. If your question concerns a total, or if you plan to run a t-test, ANOVA or regression, those methods are built on means and the median cannot substitute.

Finally, the median is insensitive in a way that can hide real change. If a policy improves outcomes for the worst-off quarter of a population without shifting anyone across the middle, the median will not move at all even though something important happened.

4.9 Ordinal data and why the median is often the only honest option

For ordinal data such as Likert scales, rankings and pain scores, the categories have a clear order but the gaps between them are not necessarily equal. The distance from "agree" to "strongly agree" is not guaranteed to match the distance from "neutral" to "agree". Averaging such codes assumes equal spacing that has not been established.

The median needs no such assumption. It only requires that the values can be ranked, which ordinal data satisfies by definition. This is why the median with an interquartile range is the standard summary for survey responses, and why reviewers often object to a mean Likert score.

4.10 Practical judgement

Compute both the median and the mean every time; it costs nothing. If they agree, report the mean with its standard deviation for the extra efficiency, and mention the median. If they disagree, lead with the median and its IQR, and say why. Always give n, because a median from seven observations deserves far less confidence than one from seven hundred. And plot the data before you commit to any single summary, because both the mean and the median can agree perfectly on a dataset neither of them describes.

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

1. Enter your data. The default tab takes comma-separated numbers, exactly as the placeholder shows: 52, 48, 55, 61, 47, .... They do not need to be sorted; the calculator sorts them for you. Newlines, tabs and semicolons all work, so a column pasted from Excel is fine.
2. Name each cluster. The group name field above every textarea is editable. Type something meaningful such as Site A, north slope and it flows into the results table, all four charts and the exported report.
3. Add or remove clusters. Press Add cluster to compare medians across groups. Every column has its own Clear and Remove button, and Remove is disabled on the last remaining column.
4. Or upload a file. On the Upload tab, choose a CSV or Excel file and click the column names you want. Every column you click becomes its own cluster with its own median, quartiles and confidence interval.
5. Use the Grouped Frequency tab if your data arrive already binned into classes. Enter each class's lower boundary and frequency plus the class width, and the calculator applies the interpolation formula and shows the working.
6. Try a sample dataset. Ten built-in datasets cover the cases that matter: symmetric data, a single dramatic outlier, heavy income skew, an even count where the median falls between two values, ordinal Likert responses, and a bimodal set where median and mean agree but both mislead.
7. Choose your quartile method. This only affects Q1 and Q3, never the median itself. Pick Tukey to match a TI-84 or a textbook, inclusive to match Excel's QUARTILE.INC or R's default.
8. Set the confidence level for the median interval. Because the method counts whole observations, the achievable levels are discrete, and the calculator reports the actual coverage rather than pretending it hit your target exactly.
9. Press Calculate Median. Nothing is computed until you do, and changing any input or setting clears the results so you never read stale numbers.
10. Read the position box and sorted strip first. They show exactly which observation the median came from, with the middle highlighted, so you can verify the answer by eye before trusting anything else on the page.

Excel gives you the median in one function, and unlike the mean there is no denominator convention to trip over, so MEDIAN agrees with R, Python, SPSS and this page on every dataset. What does cause trouble is everything around it: conditional medians, blanks versus zeros, and the quartile functions, which genuinely disagree with each other.

8.1 The functions you need

8.2 Step by step

8.3 The quartile trap

The median is identical in every piece of software ever written. The quartiles are not. Excel offers two functions that disagree, and neither matches the method most textbooks teach.

The median under all three is exactly 52.00. Only Q1 and Q3 move. If your homework expects the Tukey rule, use the quartile method selector in section 2 of this page, since Excel cannot produce it directly.

8.4 Conditional medians and the blanks-versus-zeros trap

This is the workaround for the missing MEDIANIF. For two conditions, nest the IF statements or multiply the conditions together:

8.5 Median in Google Sheets, SPSS and on a calculator

8.6 Excel errors you will probably hit

R gives you the median with median(x), and unlike the mean there is no denominator convention to worry about, so the answer matches Excel, Python and SPSS exactly. The script below is a complete, copy-and-run program that computes the median, quartiles, IQR, MAD and an exact distribution-free confidence interval, then saves one publication-ready figure at 300 dpi. Base R only, nothing to install.

9.1 The complete script

R, base only, saves median_plot.png

# ============================================================
# Median, quartiles and IQR with a single publication-ready plot
# StatsUnlock - https://statsunlock.com/median-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 <- x[!is.na(x)]
x <- c(52, 48, 55, 61, 47, 50, 58, 53,
       49, 56, 51, 54, 46, 57, 50, 52)

label <- "Site A - north slope"
units <- "mm"

# ---- 2. Median, quartiles, MAD -----------------------------
xs     <- sort(x)                  # sorting first is essential
n      <- length(xs)
med    <- median(xs)
qs     <- quantile(xs, c(0.25, 0.75), type = 7)   # type 7 = Excel INC
q1     <- as.numeric(qs[1]); q3 <- as.numeric(qs[2])
iqr    <- q3 - q1

# CAREFUL: R's mad() multiplies by 1.4826 by default.
mad_raw <- mad(xs, constant = 1)   # the plain median absolute deviation
mad_sd  <- mad(xs)                 # already scaled to estimate the SD

mean_x <- mean(xs)
gap    <- abs(mean_x - med) / med * 100

lo_fence <- q1 - 1.5 * iqr
hi_fence <- q3 + 1.5 * iqr
outliers <- xs[xs < lo_fence | xs > hi_fence]

# ---- 3. Distribution-free CI for the median ----------------
# Exact, from binomial order statistics. No normality assumed.
median_ci <- function(v, conf = 0.95) {
  v <- sort(v); m <- length(v); alpha <- 1 - conf
  cum <- 0; k <- 0
  for (i in 0:m) {
    p <- dbinom(i, m, 0.5)
    if (cum + p > alpha / 2) break
    cum <- cum + p; k <- i + 1
  }
  exact <- TRUE
  if (k < 1) { k <- 1; cum <- dbinom(0, m, 0.5); exact <- FALSE }
  list(lo = v[k], hi = v[m - k + 1], coverage = 1 - 2 * cum, exact = exact)
}
ci <- median_ci(xs, 0.95)

cat(sprintf("n            = %d\n", n))
cat(sprintf("position     = (n+1)/2 = %.1f\n", (n + 1) / 2))
cat(sprintf("median       = %.4f\n", med))
cat(sprintf("Q1, Q3       = %.4f, %.4f    IQR = %.4f\n", q1, q3, iqr))
cat(sprintf("MAD          = %.4f   (scaled -> %.4f)\n", mad_raw, mad_sd))
cat(sprintf("mean         = %.4f   gap = %.1f%%\n", mean_x, gap))
cat(sprintf("%s CI = [%.4f, %.4f]   actual coverage %.4f\n",
            if (ci$exact) "95%" else "widest", ci$lo, ci$hi, ci$coverage))
cat(sprintf("verdict      : %s\n",
    if (gap < 10) "symmetric, mean is fine too" else "skewed, report the median"))

# ---- 4. One figure, two stacked panels ---------------------
png("median_plot.png", width = 2700, height = 1800, res = 300)

layout(matrix(c(1, 2), nrow = 2), heights = c(1.5, 1))
par(mar = c(0.6, 5, 5, 2), family = "sans")

xr <- range(pretty(xs))

# upper panel: horizontal box plot
boxplot(xs, horizontal = TRUE, ylim = xr, axes = FALSE,
        col = "#e9d5ff", border = "#9333ea", lwd = 2,
        medlwd = 4, outpch = 21, outbg = "#fff7ed", outcol = "#c2410c",
        outcex = 1.3, boxwex = 0.55)

# CI band for the median, drawn behind by re-plotting the box on top
rect(ci$lo, 0.55, ci$hi, 1.45,
     col = adjustcolor("#9333ea", alpha.f = 0.10), border = NA)
boxplot(xs, horizontal = TRUE, add = TRUE, axes = FALSE,
        col = adjustcolor("#e9d5ff", alpha.f = 0.85), border = "#9333ea",
        lwd = 2, medlwd = 4, outpch = 21, outbg = "#fff7ed",
        outcol = "#c2410c", outcex = 1.3, boxwex = 0.55)
abline(v = mean_x, col = "#c2410c", lwd = 2.4, lty = 2)

title(main = sprintf("%s: median, quartiles and spread", label),
      adj = 0, cex.main = 1.15, font.main = 2, line = 3)
title(main = sprintf("n = %d   median = %.2f %s   IQR = %.2f   Q1 = %.2f   Q3 = %.2f",
                     n, med, units, iqr, q1, q3),
      adj = 0, cex.main = 0.95, font.main = 1, line = 1.6)

legend("topright", bty = "n", cex = 0.92,
       lwd = c(8, 2.4, 4), lty = c(1, 2, 1),
       col = c(adjustcolor("#9333ea", alpha.f = 0.15), "#c2410c", "#581c87"),
       legend = c(sprintf("%.1f%% CI for the median", ci$coverage * 100),
                  sprintf("Mean = %.2f", mean_x),
                  sprintf("Median = %.2f", med)))

# lower panel: every raw value, jittered so ties are visible
par(mar = c(4.6, 5, 0.6, 2))
set.seed(42)                       # seed = reproducible jitter
is_out <- xs %in% outliers
plot(xs, runif(n, -0.3, 0.3), xlim = xr, ylim = c(-1, 1),
     yaxt = "n", bty = "n", xlab = sprintf("Value (%s)", units), ylab = "",
     pch = 21, cex = 1.5, cex.lab = 1.05,
     bg = ifelse(is_out, "#fff7ed", adjustcolor("#9333ea", alpha.f = 0.7)),
     col = ifelse(is_out, "#c2410c", "white"), lwd = ifelse(is_out, 2, 1.1))
mtext("Raw\ndata", side = 2, line = 1.6, las = 1, cex = 0.8)
abline(v = med, col = "#581c87", lwd = 2.6)
abline(v = c(q1, q3), col = "#2563eb", lwd = 1.7, lty = 3)

dev.off()
cat("\nSaved median_plot.png\n")

Run it with Rscript median_plot.R, or paste it into the RStudio console. The figure appears as median_plot.png in your working directory, which getwd() will tell you.

9.2 What each part does

9.3 What the figure shows

The upper panel is a horizontal box plot: the box spans Q1 to Q3, the thick line inside is the median, the whiskers reach the furthest values within 1.5 times the IQR, and anything beyond appears as an individual point. Behind the box sits a shaded band showing the confidence interval for the median, and a dashed orange line marks the mean so you can see at a glance whether it has been pulled away by skew.

The lower panel plots every observation with a small vertical jitter, sharing the x-axis, with outliers picked out in orange and dotted blue lines at the quartiles. The box plot tells you the summary; the strip tells you the truth, including ties, gaps and clumping that binning would hide.

9.4 Common R problems and fixes

9.5 Useful one-liners

Python's np.median(x) matches every other tool exactly, which makes the median far less error-prone than the mean and its ddof trap. The script below is complete and runnable: it computes the median, quartiles, IQR, MAD and an exact distribution-free confidence interval, then saves one 300 dpi figure ready for a paper or a slide.

Install with pip install numpy matplotlib if you do not already have them. No SciPy required.

10.1 The complete script

Python 3, saves median_plot.png

# ============================================================
# Median, quartiles and IQR with a single publication-ready plot
# StatsUnlock - https://statsunlock.com/median-calculator/
# ============================================================

import numpy as np
import matplotlib.pyplot as plt
from math import comb

# ---- 1. Your data ------------------------------------------
# Replace with your own values, or read a CSV:
#   import pandas as pd
#   x = pd.read_csv("data.csv")["column_name"].dropna().to_numpy()
x = np.array([52, 48, 55, 61, 47, 50, 58, 53,
              49, 56, 51, 54, 46, 57, 50, 52], dtype=float)

label = "Site A - north slope"
units = "mm"

# ---- 2. Median, quartiles, MAD -----------------------------
xs      = np.sort(x)                 # sorting first is essential
n       = xs.size
median  = np.median(xs)
q1, q3  = np.percentile(xs, [25, 75])
iqr     = q3 - q1
mad     = np.median(np.abs(xs - median))
mad_sd  = 1.4826 * mad               # robust estimate of the SD
mean_x  = np.mean(xs)
gap     = abs(mean_x - median) / median * 100

lo_fence, hi_fence = q1 - 1.5 * iqr, q3 + 1.5 * iqr
outliers = xs[(xs < lo_fence) | (xs > hi_fence)]
inliers  = xs[(xs >= lo_fence) & (xs <= hi_fence)]

# ---- 3. Distribution-free CI for the median ----------------
# Exact, from binomial order statistics. No normality assumed.
def median_ci(sorted_x, conf=0.95):
    m = sorted_x.size
    alpha = 1 - conf
    cum, k = 0.0, 0
    for i in range(m + 1):
        p = comb(m, i) / 2 ** m
        if cum + p > alpha / 2:
            break
        cum += p
        k = i + 1
    exact = True
    if k < 1:                        # sample too small for this level
        k, cum, exact = 1, comb(m, 0) / 2 ** m, False
    return sorted_x[k - 1], sorted_x[m - k], 1 - 2 * cum, exact

ci_lo, ci_hi, coverage, exact = median_ci(xs, 0.95)

print(f"n            = {n}")
print(f"position     = (n+1)/2 = {(n + 1) / 2}")
print(f"median       = {median:.4f}")
print(f"Q1, Q3       = {q1:.4f}, {q3:.4f}    IQR = {iqr:.4f}")
print(f"MAD          = {mad:.4f}   (x1.4826 -> {mad_sd:.4f})")
print(f"mean         = {mean_x:.4f}   gap = {gap:.1f}%")
print(f"{'95%' if exact else 'widest'} CI = [{ci_lo:.4f}, {ci_hi:.4f}]"
      f"   actual coverage {coverage:.4f}")
print("verdict      :",
      "symmetric, mean is fine too" if gap < 10 else "skewed, report the median")

# ---- 4. One figure, two stacked panels ---------------------
fig, (ax_box, ax_dot) = plt.subplots(
    2, 1, figsize=(9, 6.0), sharex=True,
    gridspec_kw={"height_ratios": [1.5, 1], "hspace": 0.12})

bp = ax_box.boxplot(xs, vert=False, widths=0.55, patch_artist=True,
                    whis=1.5, showfliers=True,
                    boxprops=dict(facecolor="#e9d5ff", edgecolor="#9333ea", lw=2),
                    medianprops=dict(color="#581c87", lw=3),
                    whiskerprops=dict(color="#9333ea", lw=1.8),
                    capprops=dict(color="#9333ea", lw=1.8),
                    flierprops=dict(marker="o", markerfacecolor="#fff7ed",
                                    markeredgecolor="#c2410c", markersize=8,
                                    markeredgewidth=1.8))

ax_box.axvspan(ci_lo, ci_hi, color="#9333ea", alpha=0.10, zorder=0,
               label=f"{coverage*100:.1f}% CI for the median")
ax_box.axvline(mean_x, color="#c2410c", lw=2.2, ls="--", zorder=4,
               label=f"Mean = {mean_x:.2f}")
ax_box.plot([], [], color="#581c87", lw=3, label=f"Median = {median:.2f}")

ax_box.set_yticks([])
ax_box.set_title(f"{label}: median, quartiles and spread\n"
                 f"n = {n}   median = {median:.2f} {units}   "
                 f"IQR = {iqr:.2f}   Q1 = {q1:.2f}   Q3 = {q3:.2f}",
                 fontsize=12, fontweight="bold", loc="left")
ax_box.legend(frameon=False, fontsize=9.5, loc="upper right")
ax_box.grid(axis="x", alpha=0.25, zorder=0)
ax_box.spines[["top", "right", "left"]].set_visible(False)

rng = np.random.default_rng(42)                    # seed = reproducible jitter
ax_dot.scatter(inliers, rng.uniform(-0.3, 0.3, inliers.size),
               s=55, color="#9333ea", alpha=0.7,
               edgecolor="white", linewidth=0.9, zorder=3, label="Values")
if outliers.size:
    ax_dot.scatter(outliers, rng.uniform(-0.3, 0.3, outliers.size),
                   s=80, facecolor="#fff7ed", edgecolor="#c2410c",
                   linewidth=2, zorder=4, label="Outliers")
ax_dot.axvline(median, color="#581c87", lw=2.4, zorder=2)
ax_dot.axvline(q1, color="#2563eb", lw=1.6, ls=":", zorder=2)
ax_dot.axvline(q3, color="#2563eb", lw=1.6, ls=":", zorder=2)
ax_dot.set_ylim(-1, 1)
ax_dot.set_yticks([])
ax_dot.set_xlabel(f"Value ({units})", fontsize=11)
ax_dot.set_ylabel("Raw\ndata", fontsize=9, rotation=0, ha="right", va="center")
ax_dot.grid(axis="x", alpha=0.25, zorder=0)
ax_dot.spines[["top", "right", "left"]].set_visible(False)
if outliers.size:
    ax_dot.legend(frameon=False, fontsize=9, loc="upper right")

fig.savefig("median_plot.png", dpi=300, bbox_inches="tight")
print("\nSaved median_plot.png")

Run it with python median_plot.py. In Jupyter, replace the savefig line with plt.show(), or keep both to display and save at once.

10.2 What each part does

10.3 What the figure shows

The same figure the R script produces, drawn by a different engine. Upper panel: a horizontal box plot with the CI band for the median shaded behind it and the mean marked with a dashed line. Lower panel: every raw observation jittered vertically, outliers picked out, quartiles marked with dotted lines.

Showing both is deliberate. A box plot compresses your data into five numbers, which is exactly what makes it useful and exactly what makes it capable of hiding things. Two clusters with an empty middle, heavy ties, or a gap will all look unremarkable in the box and obvious in the strip beneath it.

10.4 Common Python problems and fixes

10.5 Useful one-liners

11.1 How one value destroys the mean and leaves the median untouched

Conclusion: the median did not merely change less, it did not change at all. Three is still the third of five sorted values however large the fifth becomes. This is the entire argument for the median in three rows.

11.2 Odd versus even counts

Conclusion: a fractional position means average the two values either side. With an even count the median may be a number that appears nowhere in your data, which is correct rather than an error.

11.3 Confidence interval coverage by sample size

Requesting 95%. Because the method counts whole observations, the achievable coverage is discrete and almost never exactly 95%.

Conclusion: coverage is always at least 95% when the interval exists, so the method is conservative rather than optimistic. Below n = 6 no 95% interval is possible even using the full range, and the calculator says so rather than reporting something false.

11.4 Why MAD beats the standard deviation for outlier detection

Base data 10 to 19. In the second row the value 19 is replaced by 500.

Conclusion: the standard deviation grew fifty-fold while the MAD did not move at all. This is why flagging outliers as "more than three standard deviations from the mean" is self-defeating: the outlier inflates the very yardstick you are using to detect it. Use the median and MAD instead.

11.5 Median from grouped frequency data

Worked example using the interpolation formula, classes of width 10.

With n = 40, we need position n/2 = 20. The cumulative frequency first reaches 20 in the class 20 to 30, so L = 20, CF = 13, f = 15 and w = 10:

Conclusion: this interpolates within the median class, assuming the values are spread evenly across it. It is an estimate, not an exact median, and it is the best you can do when the raw values have been discarded.

11.6 Which average to report

Conclusion: the median is not automatically better, it is better under specific conditions. When the data are clean and symmetric the mean is more efficient and should be preferred.

Note: every example below uses the Tukey quartile method, the calculator default. The median is identical under all three methods; only Q1, Q3 and the IQR change.

Study design: the median assumes a set of independent observations measured on a scale that can at least be ranked. Unlike the mean it does not require an interval or ratio scale, so ordinal data are acceptable.

1. Confirm the values can be ordered. That is the only scale requirement. Ranks, grades, Likert responses and pain scores all qualify, and for those the median is often the only strictly defensible average.
2. Define the population and the unit of observation. Be explicit about whether one observation is one subject, one repeat reading, or an average of several. Mixing these inflates n and makes the confidence interval look narrower than it is.
3. Randomise the selection. A median from a convenience sample is a robust summary of the wrong group. Robustness protects against outliers, not against bias.
4. Plan the sample size against the interval you need. Below n = 6 no 95% confidence interval for the median exists at all. For a reasonably tight interval aim for 30 or more, and see table 11.3 for what each n actually delivers.
5. Fix the measurement resolution. Aggressive rounding creates ties, which is not fatal for the median but can collapse the IQR to zero and make the outlier fences unusable.
6. Record missing data as blank, never as zero. A zero is a real observation that shifts the ordering; a blank is correctly ignored. This is one of the few ways to corrupt a median without noticing.
7. Decide the quartile convention in advance and write it in the protocol. The median is identical everywhere, but Q1 and Q3 are not, and an unstated convention makes your IQR irreproducible.
8. Record raw values, not just summaries. If you keep only a frequency table you are forced into the grouped interpolation formula, which gives an estimate rather than the true median.
9. Lay the data out in 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. Blank cells are ignored, so unequal group sizes are fine.

Pre-registration: state before collection whether you will report the median or the mean, and on what basis. Switching to the median after seeing that the mean was inconvenient is a real form of unintentional bias, and it is invisible in the final paper unless you declared the rule in advance.

Common collection mistakes:

• Recording missing values as zero, which shifts the ordering and drags the median down.
• Pooling two genuinely different populations, producing a median that describes neither.
• Treating repeat readings of one subject as independent observations.
• Discarding raw values and keeping only a frequency table, which forces an estimated median.
• Rounding so heavily that most values tie, collapsing the IQR to zero.

The median is the right choice more often than most people are taught, but it is not universally superior and knowing when it costs you something matters.

Use the median when:

• ✓ The data are skewed, as income, waiting times, survival and claim sizes always are.
• ✓ There are outliers you want to describe honestly rather than let dominate the summary.
• ✓ The data are ordinal, such as Likert scales, rankings or grades.
• ✓ You want a summary a single extreme value cannot move.
• ✓ Some values are censored, such as "over 100", where a mean cannot be computed at all but a median often can.
• ✓ You are reporting to a general audience, where "half were above and half below" is intuitive.

Prefer the mean when:

• ✗ The data are symmetric and clean, where the mean is more statistically efficient and gives more precision for the same n.
• ✗ You need a total, since the mean times n gives the sum and the median has no such identity.
• ✗ You plan a t-test, ANOVA or regression, all of which are built on means.
• ✗ Small shifts across the whole distribution matter, since the median can miss real change that does not cross the middle.
• ✗ The data are already known to be normally distributed, where the mean is the optimal estimator.

Real-world examples:

1. Economics, median household income and median house price, both reported precisely because the mean misleads.
2. Medicine, median survival time and median length of stay, always right skewed.
3. Ecology, median body mass or count per plot, where a few large individuals dominate.
4. Survey research, median Likert response, where a mean assumes equal spacing.
5. Software engineering, median response time, and the p50 that sits alongside p95 and p99 in every latency dashboard.
6. Education, median exam mark when a few very low scores would drag the mean below the typical student.

Decision rule: compute both. If they agree, report the mean with its standard deviation and mention the median. If they disagree by more than about 10%, lead with the median and its IQR and say why. If the data are ordinal, use the median regardless.

Assumptions

1. The values can be ranked. Check the measurement scale. This is the only requirement, and it is why ordinal data are acceptable here but not for the mean.
2. Observations are independent. Check the design. Repeated measurements of the same unit are not independent and make the confidence interval falsely narrow.
3. The sample represents the population. Check the sampling frame. Robustness protects the median from outliers, not from selection bias.
4. The sample is large enough for what you claim. Check n. Below six observations no 95% confidence interval exists, and below about ten the median shifts noticeably when one value changes.
5. The data come from one population. Check chart 1. A bimodal dataset produces a median in the empty middle that describes nothing.
6. For the confidence interval only, the distribution is continuous. Heavy ties make the interval conservative, so its true coverage exceeds the nominal level.

Limitations

• The median discards information. It uses only the middle position and ignores the magnitudes of everything else, which makes it less efficient than the mean on clean symmetric data.
• It does not add up. Total revenue is the mean order value times the count; there is no equivalent identity for the median, so it cannot be used to recover totals.
• It is insensitive to change in the tails. A policy that improves the worst-off quarter without moving anyone across the middle will not shift the median at all.
• It cannot detect bimodality. Two clusters either side of an empty middle produce a median that matches the mean perfectly while describing no observation.
• Most standard tests are built on means. If you need a t-test, ANOVA or regression, you need the mean, or a rank-based alternative such as Mann-Whitney.
• The confidence interval has discrete coverage. You cannot request exactly 95%; the achievable levels are set by the binomial distribution and the calculator reports what it actually reached.
• The grouped-data median is an estimate, since interpolating within the median class assumes values are spread evenly across it.
• This tool assumes an unweighted simple sample. Weighted, stratified and clustered survey designs need a design-consistent quantile estimator.

A median calculator does something deceptively modest: it sorts your numbers and reports the one in the middle. That modesty is the source of its power. Because the median depends on position rather than magnitude, no single observation, however extreme, can move it. You could replace your largest value with a billion and the median would not shift by a unit. No other common summary statistic offers that guarantee.

The habit worth building is to compute the median and the mean together every time, then compare them. When they sit close, your data are roughly symmetric and the mean is the more efficient summary. When they pull apart, the mean has been captured by a tail and the median is the honest choice. That comparison costs nothing, takes two seconds, and prevents the single most common misreporting in applied statistics, which is quoting a mean for skewed data and calling it typical.

Report the median with its interquartile range, exactly as you would report a mean with a standard deviation. A median alone tells the reader where the middle is but nothing about how tightly the data cluster around it, and a median with no n attached invites more confidence than it has earned. The confidence interval this calculator provides is worth including too, particularly because it is exact and assumption-free, which is a genuinely strong claim that most intervals cannot make.

Two cautions are worth carrying away. The first is that robustness protects you from outliers, not from bias: a median computed from the wrong sample is precisely the wrong answer. The second is that the mean-median comparison detects skew but not bimodality, and two clusters with an empty middle will produce a median and a mean that agree perfectly while both describe a value nobody recorded. Plot the data. The charts catch what the summary statistics cannot.

Paste your numbers into the calculator above, check the position box to see exactly which observation the median came from, read the recommendation before the answer, and copy whichever reporting sentence matches your journal. Then reach for this median calculator whenever a distribution looks lopsided, because that is precisely the situation the median was built for.

The following peer-reviewed references support the methods used in this median calculator, covering robust central tendency, distribution-free confidence intervals for the median, the median absolute deviation and best practice in reporting.

1. Thompson, W. R. (1936). On confidence ranges for the median and other expectation distributions for populations of unknown distribution form. The Annals of Mathematical Statistics, 7(3), 122–128. https://doi.org/10.1214/aoms/1177732502
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3. 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
4. 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
5. Wilcox, R. R., & Keselman, H. J. (2003). Modern robust data analysis methods: Measures of central tendency. Psychological Methods, 8(3), 254–274. https://doi.org/10.1037/1082-989X.8.3.254
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7. Manikandan, S. (2011). Measures of central tendency: Median and mode. Journal of Pharmacology and Pharmacotherapeutics, 2(3), 214–215. https://doi.org/10.4103/0976-500X.83300
8. Altman, D. G., & Bland, J. M. (1994). Statistics notes: Quartiles, quintiles, centiles, and other quantiles. BMJ, 309(6960), 996. https://doi.org/10.1136/bmj.309.6960.996
9. 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
10. 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
11. Tukey, J. W. (1977). Exploratory data analysis. Addison-Wesley. https://search.worldcat.org/title/3058187
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13. Jamieson, S. (2004). Likert scales: How to (ab)use them. Medical Education, 38(12), 1217–1218. https://doi.org/10.1111/j.1365-2929.2004.02012.x
14. Norman, G. (2010). Likert scales, levels of measurement and the laws of statistics. Advances in Health Sciences Education, 15, 625–632. https://doi.org/10.1007/s10459-010-9222-y
15. Huber, P. J., & Ronchetti, E. M. (2009). Robust statistics (2nd ed.). Wiley. https://doi.org/10.1002/9780470434697
16. 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
17. Weisstein, E. W. (n.d.). Statistical median. MathWorld, A Wolfram Web Resource. https://mathworld.wolfram.com/StatisticalMedian.html
18. American Psychological Association. (2020). Publication manual of the American Psychological Association (7th ed.). https://doi.org/10.1037/0000165-000
19. NIST/SEMATECH. (2013). e-Handbook of statistical methods. National Institute of Standards and Technology. https://www.itl.nist.gov/div898/handbook/
20. R Core Team. (2024). R: A language and environment for statistical computing. R Foundation for Statistical Computing. https://www.R-project.org/