Leptokurtic vs Platykurtic Kurtosis Calculator | StatsUnlock

Leptokurtic vs platykurtic describes the tails of a distribution: a leptokurtic distribution has heavy tails and positive excess kurtosis (above 0), a platykurtic distribution has light tails and negative excess kurtosis (below 0), and a normal, mesokurtic distribution has excess kurtosis of exactly 0.

Kurtosis is the average of the fourth power of the z-scores. Because each z-score is raised to the fourth power, a few extreme values decide almost the whole number. Kurtosis is a measure of tails and outliers, not a measure of how pointed the peak looks.

Rule of thumb: excess kurtosis above +0.5 is leptokurtic, below −0.5 is platykurtic, and between the two is close to mesokurtic. For a normality check, an absolute value up to 2 is usually acceptable, and a z value (kurtosis ÷ SE) beyond ±1.96 is significant at the 5% level.

Kurtosis is a number that describes the shape of a distribution's tails. It answers one question: compared with a normal curve that has the same mean and standard deviation, does this data set produce more extreme values or fewer? The three names come from Greek words. Lepto means slender, platy means broad and flat, and meso means middle.

• Leptokurtic (excess kurtosis > 0): heavy tails. Extreme values happen more often than the normal curve predicts. Stock returns, insurance claims and animal movement distances are classic examples. The Laplace distribution has excess kurtosis of +3, and Student's t with 5 degrees of freedom has +6.
• Mesokurtic (excess kurtosis ≈ 0): normal tails. The normal distribution is the reference, with excess kurtosis of exactly 0.
• Platykurtic (excess kurtosis < 0): light tails. Values are spread evenly with few extremes, often because a scale has hard limits. A uniform distribution has excess kurtosis of −1.2, and the lowest possible value is −2, reached by a coin toss coded 0 and 1.

What it actually measures: kurtosis is the average of the z-scores raised to the fourth power. A value 1 SD from the mean adds 1 to that average, but a value 3 SD away adds 81. So the number is driven almost entirely by the tails. The old textbook story that kurtosis measures "peakedness" is not correct, and Westfall (2014) shows that two distributions with the same peak can have very different kurtosis.

A worked one-liner: for Class A in the default data (n = 20, mean 51.25), the fourth central moment is 4395.09 and the variance (m₂) is 27.69, so Pearson kurtosis = 4395.09 ÷ 27.69² = 5.733 and excess kurtosis g₂ = 5.733 − 3 = 2.733. The bias-corrected value that Excel prints is G₂ = 3.936. Both say: leptokurtic.

Who uses it: psychologists and social scientists check kurtosis before running t-tests, ANOVA or structural equation models. Finance teams use it to judge "fat tail" risk. Ecologists see leptokurtic dispersal kernels, where most seeds or animals move a short distance and a few travel very far. Quality engineers watch for platykurtic output, which can mean two machines are mixed in one batch.

Kurtosis vs skewness: skewness (the third moment) tells you whether one tail is longer than the other. Kurtosis (the fourth moment) tells you how heavy both tails are together. A distribution can be perfectly symmetric (skewness 0) and still be strongly leptokurtic, as the Laplace and t distributions are.

4.1 What your kurtosis value actually tells you

Kurtosis compares the tails of your data with the tails of a normal curve that has the same mean and the same standard deviation. A positive excess kurtosis means your data produce extreme values more often than that normal curve would. A negative value means they produce fewer. That is the whole meaning of leptokurtic vs platykurtic, and nothing more.

Read the sign first, then the size. A value of +0.3 is a small departure that you can usually ignore. A value of +4 is a strong signal that a few values sit far from the rest. A value of −1.2 is what a perfectly flat, uniform spread of values gives, and values close to −2 almost always mean the data cluster at two ends, such as two separate groups mixed together.

4.2 Why kurtosis is about tails, not the peak

Many textbooks say a leptokurtic curve is "tall and thin" and a platykurtic curve is "flat-topped". This is the most common mistake about kurtosis. Because kurtosis averages z⁴, values within one standard deviation of the mean contribute at most 1 each to the total, while a single value 3 SD away contributes 81. Westfall (2014) proved that the central part of the data can only add a tiny, bounded amount to kurtosis.

The table Which values drive the kurtosis? in Section 3 shows this on your own data. In the default Class A data, the scores 66 and 38 together supply 88.9% of Σz⁴. Delete those two values and Class A becomes close to mesokurtic. So when you report a leptokurtic result, you are really reporting that your data contain a few extreme values.

4.3 Which kurtosis number did your software give you?

There are three formulas in common use, and they give different numbers on the same data. G₂ is the bias-corrected estimator printed by Excel's KURT, SPSS, SAS and pandas. g₂ is the plain moment estimator printed by SciPy's default and by R's moments package after you subtract 3. b₂ (Type 3) is printed by Minitab and by R's e1071 package by default. Joanes and Gill (1998) compared all three.

For large samples they agree. For small samples they do not: with n = 20 in Class A, G₂ = 3.94, g₂ = 2.73 and b₂ = 2.17. Always name the formula in your Methods section. And check whether a program prints excess kurtosis (normal = 0) or Pearson kurtosis (normal = 3), because a Pearson value of 2.8 is slightly platykurtic, not leptokurtic.

4.4 Is the result statistically significant?

A sample can show non-zero kurtosis just by chance, especially when it is small. Two tests answer whether the departure from 0 is real. The z ratio divides G₂ by its standard error and compares the result with ±1.96. It is simple and it is what SPSS users report, but it is too generous for small samples because the sampling distribution of kurtosis is strongly skewed. The Anscombe-Glynn test transforms the kurtosis first so the z value really follows a normal distribution, and it is what SciPy and R use.

In the default data, Class A has a z ratio of 3.97 but an Anscombe-Glynn z of only 2.52 (p = .012). Both say significant, but the gap shows how much the simple ratio can overstate the evidence. Class B is platykurtic by its value (−1.14) but not significant (p = .118), because 20 values are too few to tell a uniform spread from a normal one with confidence.

4.5 How large a value is a problem?

There is no single cut-off, and the right rule depends on sample size. For a quick normality screen, many methods texts accept an absolute excess kurtosis of up to 2. Kim (2013) gives a sample-size rule: for n under 50 use |z| < 1.96, for n from 50 to 300 use |z| < 3.29, and above 300 ignore z (it becomes significant for trivial departures) and look at the value itself, with an absolute kurtosis up to 7 still acceptable. The same limit of 7 is widely used before structural equation modelling. The checks table in Section 3 applies these rules for you.

4.6 What a leptokurtic result means for your analysis

Heavy tails matter more than light tails. Tests about means, such as the t-test and ANOVA, are fairly robust to kurtosis when groups are similar in size. But tests and intervals about variances are not. The F-test for equal variances, Bartlett's test and confidence intervals for a standard deviation can be badly wrong under leptokurtosis (Box, 1953). In structural equation modelling, positive kurtosis inflates the chi-square fit statistic and shrinks standard errors (Curran, West and Finch, 1996).

Good responses are: check the extreme values for errors, use robust methods such as trimmed means or Yuen's test, use Levene's or the Brown-Forsythe test instead of an F-test for variances, or bootstrap your confidence intervals.

4.7 What a platykurtic result means for your analysis

Light tails are usually harmless for tests on means; they make them slightly conservative. The bigger message is about where the data came from. Strong platykurtosis often means a bounded scale (a 1 to 7 rating scale used evenly), a floor or ceiling effect, or two subgroups mixed together. A value near −2 with a histogram that shows two humps is almost always a mixture. Darlington (1970) showed that negative kurtosis is closely linked to bimodality, so plot the data and look for a grouping variable you forgot to separate.

4.8 Kurtosis and skewness together

Always read kurtosis beside skewness. Skewness shows whether one tail is longer than the other; kurtosis shows how heavy the tails are overall. A long right tail (positive skew) usually also produces positive kurtosis, as in the sloth bear movement sample. The D'Agostino-Pearson K² test and the Jarque-Bera test combine both into one normality test. The calculator reports both, so you can see whether the shape problem is mainly asymmetry, mainly tail weight, or both.

4.9 Sample size changes everything

Kurtosis is the least stable of the common descriptive statistics. Its standard error is about √(24 ÷ n), so with n = 20 it is close to 1 and with n = 100 it is still about 0.48. Small samples often hide heavy tails completely, because the rare extreme values simply have not been drawn yet. Large samples flag tiny departures as significant. Report the value, the standard error and the confidence interval, and do not rely on the p-value alone.

4.10 Practical judgement

Use the shape label to describe your data, use the standard error and confidence interval to judge how sure you are, and use the driver table to find out which values are responsible. If one or two values supply most of Σz⁴, go back to the raw records before doing anything else. If the data are genuinely heavy-tailed, choose methods built for that. If they are platykurtic, check for a mixture or a bounded scale. Then write the result in plain words, as the templates in Section 5 show.

When reporting leptokurtic vs platykurtic results in APA format, give the kurtosis value, name the formula, add the standard error, and state the shape in words. Kurtosis is usually reported in the descriptive statistics part of the Results, next to the mean, SD and skewness.

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

5.1 APA 7th edition results sentence

5.2 Thesis or dissertation Methods and Results

5.3 Plain-language summary

5.4 Structured abstract or conference poster

5.5 Replication or pre-registration statement

5.6 Table and figure conventions

In a descriptive statistics table, add two columns after M and SD: Skewness (SE) and Kurtosis (SE), for example 3.94 (0.99). Add a note under the table: "Kurtosis is excess kurtosis (G₂); the normal distribution has a value of 0." For a figure, a normal Q-Q plot is the clearest picture of kurtosis. Plot 2 in Section 3 is ready to use.

5.7 The Discussion sentence

Link the shape to its consequence: "Because scores were leptokurtic, variance-based tests were replaced with Levene's test, and confidence intervals were bootstrapped." Reviewers want to know what you did about the shape, not only that you measured it.

5.8 Reporting checklist

• The kurtosis value, with the formula named (G₂, g₂ or b₂) and "excess" stated.
• The standard error, and n for each group.
• The test used (z ratio or Anscombe-Glynn), the z value and the p value.
• The shape word, leptokurtic, platykurtic or mesokurtic, and the rule you used for it.
• Skewness next to kurtosis, and what you did if the data failed your normality rule.

This flowchart shows how kurtosis is calculated, step by step, from a raw column of numbers to a leptokurtic, platykurtic or mesokurtic verdict. Every box uses the same worked example as the Excel, R and Python sections: the 20 exam scores of Class A.

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 G₂ = 3.9365, g₂ = 2.7332 and SEK = 0.9924 for Class A, and G₂ = −1.1397 for Class B.

1. How to use this leptokurtic vs platykurtic calculator: enter your data. The default tab takes comma-separated numbers, exactly as the placeholder shows: 52, 48, 55, 61, 47, .... New lines, tabs, semicolons and spaces also work.
2. Name each group. The name field above every box is editable. Type something meaningful, such as Class A, morning, and it appears in the tables, charts and downloads.
3. Add or remove groups. Press Add group to compare the shape of several samples side by side. Each group has its own Clear and Remove buttons.
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 group, named after its header.
5. Or type into the manual table and press Use this table to send it to the calculator.
6. Try a sample dataset. Ten built-in data sets cover clear leptokurtic data (stock returns, bear movements), clear platykurtic data (uniform waiting times, mixed heights), a near-normal set, and a data-entry error that shows how one wrong value explodes kurtosis.
7. Choose the kurtosis formula. Keep G₂ to match Excel and SPSS, or switch to g₂ or b₂ to match SciPy, R moments or Minitab.
8. Set the shape band, the test, alpha and the bootstrap. The default ±0.5 band, the Anscombe-Glynn test, alpha = 0.05 and 2,000 seeded resamples suit most work.
9. Press Run Kurtosis Analysis. Nothing is computed before you do, and changing any input clears the old results so you never read stale numbers.
10. Read the verdict, check the plots and export. Start with the shape label and the driver table, look at the Q-Q plot, then copy a sentence from Section 5 or download the report as a text file or PDF.

Excel has one direct function for kurtosis, KURT, and it returns the bias-corrected excess kurtosis G₂, the same number SPSS prints. That part is easy. What trips people up is everything around it: Excel gives no standard error, no p-value and no shape label, and it has no function at all for the moment version g₂ that SciPy and many papers use. This section builds all of them, with the spreadsheet shown at each stage, using the Class A and Class B scores.

9.1 The functions you need

9.2 Step by step

9.3 The awkward cases

The p-value. Excel only gives you the simple z ratio. The Anscombe-Glynn test that SciPy and R use needs a longer formula. For Class A, Excel's z ratio of 3.97 gives p = .00007, while the Anscombe-Glynn test gives z = 2.52 and p = .012. Both are "significant", but the Excel version overstates the evidence for small samples.

Pearson kurtosis (normal = 3). If a paper or Stata reports kurtosis around 3, compare it with g₂ + 3, not with KURT.

For Class A this returns 5.7332. That is 2.7332 + 3, not a different result.

Fewer than four values. KURT returns #DIV/0! when there are fewer than four numbers or when every value is the same, because the variance is zero.

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

9.5 Charting kurtosis in Excel

1. Select the data column and choose Insert → Charts → Histogram (Excel 2016 or later).
2. Right-click the horizontal axis, choose Format Axis, and set a sensible bin width so you can see the tails.
3. For a Q-Q plot, sort the data in a new column, add =NORM.S.INV((ROW()-1-0.375)/(COUNT(A:A)+0.25)) beside it, standardise the data with =STANDARDIZE(x, mean, sd), and draw an XY scatter of the two columns.
4. Add a straight line through the origin with slope 1. Points that curve away at both ends show leptokurtosis; points that bend inward show platykurtosis.

Plots 1 and 2 in Section 3 draw the histogram with a normal curve and the Q-Q plot automatically, which is much faster than building them in Excel.

9.6 Excel errors you will probably hit

R has no built-in kurtosis function in base R, and the two popular packages disagree with each other and with Excel. The script below avoids that confusion. It uses base R only, computes all three kurtosis conventions, the standard error, the z ratio and the Anscombe-Glynn test, prints a plain-English verdict for every group, and saves one publication-ready figure at 300 dpi.

10.1 The complete script

R, base only, saves kurtosis_plot.png

# ============================================================
# Leptokurtic vs platykurtic: kurtosis, SE, tests and one plot
# StatsUnlock - https://statsunlock.com/leptokurtic-vs-platykurtic-kurtosis-calculator/
# Base R only. No packages to install.
# ============================================================

# ---- 1. Your data ------------------------------------------
# Replace with your own values, or read a CSV:
#   d <- read.csv("data.csv")
#   groups <- list("Class A" = d$score[!is.na(d$score)])
groups <- list(
  "Class A" = c(52, 48, 55, 61, 47, 50, 51, 49, 53, 50,
                52, 51, 50, 49, 52, 50, 51, 38, 66, 50),
  "Class B" = c(41, 44, 46, 48, 50, 52, 54, 56, 58, 60,
                62, 43, 47, 51, 55, 59, 63, 45, 49, 57)
)

# ---- 2. Kurtosis in all three conventions ------------------
kurt_all <- function(x) {
  x  <- x[!is.na(x)]                      # drop missing values
  n  <- length(x)
  d  <- x - mean(x)
  m2 <- mean(d^2); m3 <- mean(d^3); m4 <- mean(d^4)
  g2 <- m4 / m2^2 - 3                                       # moments, SciPy default
  G2 <- ((n + 1) * g2 + 6) * (n - 1) / ((n - 2) * (n - 3))  # Excel KURT, SPSS
  b2 <- (g2 + 3) * (1 - 1 / n)^2 - 3                        # type 3, e1071 default
  G1 <- m3 / m2^1.5 * sqrt(n * (n - 1)) / (n - 2)           # skewness, Excel SKEW
  ses <- sqrt(6 * n * (n - 1) / ((n - 2) * (n + 1) * (n + 3)))
  sek <- 2 * ses * sqrt((n^2 - 1) / ((n - 3) * (n + 5)))
  # Anscombe-Glynn test, same as moments::anscombe.test and scipy kurtosistest
  b    <- m4 / m2^2
  E    <- 3 * (n - 1) / (n + 1)
  varb <- 24 * n * (n - 2) * (n - 3) / ((n + 1)^2 * (n + 3) * (n + 5))
  xs   <- (b - E) / sqrt(varb)
  sb1  <- 6 * (n^2 - 5 * n + 2) / ((n + 7) * (n + 9)) *
          sqrt(6 * (n + 3) * (n + 5) / (n * (n - 2) * (n - 3)))
  A    <- 6 + 8 / sb1 * (2 / sb1 + sqrt(1 + 4 / sb1^2))
  den  <- 1 + xs * sqrt(2 / (A - 4))
  t2   <- sign(den) * ((1 - 2 / A) / abs(den))^(1 / 3)
  z_ag <- (1 - 2 / (9 * A) - t2) / sqrt(2 / (9 * A))
  list(n = n, g2 = g2, G2 = G2, b2 = b2, G1 = G1, sek = sek,
       z = G2 / sek, z_ag = z_ag, p_ag = 2 * pnorm(-abs(z_ag)))
}

shape <- function(k, band = 0.5) {
  if (k > band) "leptokurtic (heavy tails)"
  else if (k < -band) "platykurtic (light tails)"
  else "mesokurtic (close to normal)"
}

# ---- 3. Print the results ----------------------------------
res <- lapply(groups, kurt_all)
for (g in names(res)) {
  r <- res[[g]]
  cat(sprintf("\n%s (n = %d)\n", g, r$n))
  cat(sprintf("  Excess kurtosis G2 = %.4f   g2 = %.4f   b2 = %.4f\n", r$G2, r$g2, r$b2))
  cat(sprintf("  SE of kurtosis     = %.4f   z = G2/SE = %.4f\n", r$sek, r$z))
  cat(sprintf("  Anscombe-Glynn z   = %.4f   p = %.4f\n", r$z_ag, r$p_ag))
  cat(sprintf("  Skewness G1        = %.4f\n", r$G1))
  cat(sprintf("  Verdict: %s, %s at alpha = 0.05\n", shape(r$G2),
      if (r$p_ag < 0.05) "significantly different from normal"
      else "not significantly different from normal"))
}

# ---- 4. Seeded bootstrap 95% CI for G2 ---------------------
set.seed(42)
for (g in names(groups)) {
  bs <- replicate(2000, kurt_all(sample(groups[[g]], replace = TRUE))$G2)
  ci <- quantile(bs, c(0.025, 0.975), na.rm = TRUE)
  cat(sprintf("Bootstrap 95%% CI for G2, %s: %.2f to %.2f\n", g, ci[1], ci[2]))
}

# ---- 5. One figure: histogram + normal curve per group -----
png("kurtosis_plot.png", width = 1200 * length(groups), height = 1100, res = 300)
par(mfrow = c(1, length(groups)), mar = c(4.2, 4.2, 3, 1))
for (g in names(groups)) {
  x <- groups[[g]]; r <- res[[g]]
  col <- if (r$G2 > 0.5) "#9333ea" else if (r$G2 < -0.5) "#0d9488" else "#16a34a"
  h <- hist(x, breaks = "Sturges", plot = FALSE)
  xs <- seq(min(x) - sd(x), max(x) + sd(x), length.out = 200)
  ys <- dnorm(xs, mean(x), sd(x))
  plot(h, freq = FALSE, col = adjustcolor(col, 0.55), border = "white",
       ylim = c(0, max(h$density, ys) * 1.1), xlim = range(xs),
       main = sprintf("%s: G2 = %.2f", g, r$G2), xlab = "Value", cex.main = 0.95)
  lines(xs, ys, col = "#2563eb", lwd = 2)
  mtext(shape(r$G2), side = 3, line = 0.2, cex = 0.7)
}
invisible(dev.off())
cat("\nSaved kurtosis_plot.png\n")

Run it with Rscript kurtosis.R from a terminal, or paste it into the RStudio console. Class A prints G2 = 3.9365, g2 = 2.7332, SE = 0.9924 and Anscombe-Glynn p = 0.0118. Class B prints G2 = −1.1397 and p = 0.1182. The bootstrap interval differs a little from this page because R uses a different random generator.

10.2 What each part does

10.3 What the figure shows

Two panels sit side by side. For Class A the purple bars pile up near 50 and two isolated bars sit far out at 38 and 66, beyond where the blue normal curve has almost no height: that is the leptokurtic signature. For Class B the teal bars are spread evenly and stop before the normal curve's tails do: that is the platykurtic signature.

10.4 Common R problems and fixes

10.5 Useful one-liners

e1071::kurtosis(x, type = 2)                 # same as Excel KURT
moments::kurtosis(x) - 3                     # g2, the SciPy default
moments::anscombe.test(x)                    # Anscombe-Glynn test
sapply(split(d$y, d$group), function(v) kurt_all(v)$G2)   # per group
qqnorm(x); qqline(x)                         # quick Q-Q plot
write.csv(do.call(rbind, lapply(res, unlist)), "kurtosis.csv")

Python's scipy.stats.kurtosis(x) is one line, but its default is the moment estimator g₂, not the bias-corrected G₂ that Excel and SPSS print. The script below sets it explicitly, adds the standard error, the z ratio, the Anscombe-Glynn test and a seeded bootstrap interval, prints a plain verdict for each group, and saves one 300 dpi figure.

Install with pip install numpy scipy matplotlib. Add pandas only if you read a CSV.

11.1 The complete script

Python 3, saves kurtosis_plot.png

# ============================================================
# Leptokurtic vs platykurtic: kurtosis, SE, tests and one plot
# StatsUnlock - https://statsunlock.com/leptokurtic-vs-platykurtic-kurtosis-calculator/
# ============================================================
import numpy as np
from scipy import stats
import matplotlib.pyplot as plt

# ---- 1. Your data ------------------------------------------
# Or read a CSV:
#   import pandas as pd
#   df = pd.read_csv("data.csv")
#   groups = {"Class A": df["score"].dropna().to_numpy()}
groups = {
    "Class A": [52, 48, 55, 61, 47, 50, 51, 49, 53, 50,
                52, 51, 50, 49, 52, 50, 51, 38, 66, 50],
    "Class B": [41, 44, 46, 48, 50, 52, 54, 56, 58, 60,
                62, 43, 47, 51, 55, 59, 63, 45, 49, 57],
}

def shape(k, band=0.5):
    if k > band:
        return "leptokurtic (heavy tails)"
    if k < -band:
        return "platykurtic (light tails)"
    return "mesokurtic (close to normal)"

rng = np.random.default_rng(42)          # seeded bootstrap
fig, axes = plt.subplots(1, len(groups), figsize=(4.6 * len(groups), 4.0))
axes = np.atleast_1d(axes)

for ax, (name, values) in zip(axes, groups.items()):
    x = np.asarray(values, dtype=float)
    x = x[~np.isnan(x)]                   # drop missing values
    n = len(x)

    # ---- 2. Kurtosis in all three conventions ---------------
    G2 = stats.kurtosis(x, fisher=True, bias=False)   # Excel KURT, SPSS, pandas
    g2 = stats.kurtosis(x, fisher=True, bias=True)    # SciPy default
    b2 = (g2 + 3) * (1 - 1 / n) ** 2 - 3              # type 3, Minitab
    G1 = stats.skew(x, bias=False)                     # Excel SKEW

    ses = np.sqrt(6 * n * (n - 1) / ((n - 2) * (n + 1) * (n + 3)))
    sek = 2 * ses * np.sqrt((n ** 2 - 1) / ((n - 3) * (n + 5)))
    z_ag, p_ag = stats.kurtosistest(x)                # Anscombe-Glynn

    boot = [stats.kurtosis(rng.choice(x, n, replace=True), bias=False)
            for _ in range(2000)]
    lo, hi = np.nanpercentile(boot, [2.5, 97.5])

    # ---- 3. Print the results -------------------------------
    print(f"\n{name} (n = {n})")
    print(f"  Excess kurtosis G2 = {G2:.4f}   g2 = {g2:.4f}   b2 = {b2:.4f}")
    print(f"  SE of kurtosis     = {sek:.4f}   z = G2/SE = {G2 / sek:.4f}")
    print(f"  Anscombe-Glynn z   = {z_ag:.4f}   p = {p_ag:.4f}")
    print(f"  Skewness G1        = {G1:.4f}")
    print(f"  Bootstrap 95% CI for G2: {lo:.2f} to {hi:.2f}")
    verdict = "significantly" if p_ag < 0.05 else "not significantly"
    print(f"  Verdict: {shape(G2)}, {verdict} different from normal at alpha = 0.05")

    # ---- 4. Histogram with a matching normal curve ----------
    col = "#9333ea" if G2 > 0.5 else ("#0d9488" if G2 < -0.5 else "#16a34a")
    ax.hist(x, bins="sturges", density=True, color=col, alpha=0.6, edgecolor="white")
    xs = np.linspace(x.min() - x.std(ddof=1), x.max() + x.std(ddof=1), 200)
    ax.plot(xs, stats.norm.pdf(xs, x.mean(), x.std(ddof=1)), color="#2563eb", lw=2)
    ax.set_title(f"{name}: G2 = {G2:.2f}\n{shape(G2)}", fontsize=10)
    ax.set_xlabel("Value")
    ax.set_ylabel("Density")

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

11.2 What each part does

11.3 What the figure shows

The same two panels as the R figure: purple bars for Class A with isolated values at 38 and 66 sitting in the far tails of the blue normal curve, and teal bars for Class B that fill the middle evenly and stop short of the normal tails.

11.4 Common Python problems and fixes

11.5 Useful one-liners

stats.kurtosis(x, bias=False)                     # same as Excel KURT
stats.kurtosis(x, fisher=False)                   # Pearson kurtosis, normal = 3
stats.kurtosistest(x)                             # Anscombe-Glynn z and p
stats.normaltest(x)                               # D'Agostino-Pearson K2
stats.jarque_bera(x)                              # Jarque-Bera test
df.groupby("group")["y"].apply(lambda s: s.kurt())  # per group, pandas
stats.probplot(x, dist="norm", plot=plt)          # quick Q-Q plot

Run it with python kurtosis.py. Class A prints G2 = 3.9365, g2 = 2.7332, SE = 0.9924, z = 2.5173 and p = 0.0118, the same as R, Excel and this page.

12.1 Kurtosis of common distributions

What to conclude: a symmetric distribution can be either leptokurtic or platykurtic, and the theoretical floor is −2. Values follow from the standard moment formulas; see Balanda and MacGillivray (1988) and DeCarlo (1997).

12.2 Standard error of kurtosis and the critical value by sample size

What to conclude: with 20 values you need |G₂| near 2 to call kurtosis significant, while with 1,000 values even 0.3 is significant. This is why Kim (2013) changes the rule with sample size.

12.3 How to describe the size of excess kurtosis

What to conclude: these are practical labels, not formal tests. The limits of 2 and 7 come from the normality rules used in Kim (2013) and the simulation design of Curran, West and Finch (1996).

12.4 Acceptable kurtosis range for normality, by rule

What to conclude: always state which rule you used. The checks table in Section 3 applies the z rule, Kim's rule and the ±2 rule at the same time.

12.5 Share of values in the centre and the tails

What to conclude: heavy-tailed distributions have more values beyond 3 SD (about five times as many for the Laplace) and also more near the centre, with fewer in the "shoulders" between. Plot 4 in Section 3 compares your data with the normal row.

12.6 Moors' robust kurtosis

What to conclude: Moors (1988) uses octiles instead of fourth powers, so one wild value cannot dominate it. If the moment kurtosis is large but Moors is near 1.233, a few outliers are doing all the work.

Eight worked examples show leptokurtic vs platykurtic results across the situations you will meet. Every value uses G₂ (Excel KURT) and the Anscombe-Glynn test unless stated. Load the same data from the sample dropdown in Section 2 to reproduce them.

Study design: kurtosis assumes a set of independent measurements of one numeric variable on an interval or ratio scale, collected from one well-defined population per group.

1. Decide before collecting why you need kurtosis. Is it a normality check before another test, a description of risk in the tails, or the main outcome? The answer sets your sample size and your cut-off rule.
2. Plan enough values for a leptokurtic vs platykurtic decision. Kurtosis needs at least 4 values to compute and about 20 for the Anscombe-Glynn test. For a confident shape verdict, aim for 50 or more per group, because rare tail values may not appear in small samples.
3. Define one group per column. Mixing two populations, such as two age groups or two sites, creates artificial platykurtosis. Record the grouping variable so you can split the data later.
4. Sample at random or systematically. Selecting "typical" cases removes the tails you are trying to measure and pushes kurtosis down.
5. Fix the instrument and its range. A scale with a ceiling (such as a 1 to 7 rating) or a detection limit cuts off the tails and produces platykurtosis that is not real.
6. Record values at full precision. Heavy rounding creates many tied values and can distort a kurtosis calculation in small samples.
7. Never use 0 or 999 for missing data. A code like 999 becomes an extreme value and can push kurtosis close to its maximum on its own. Leave the cell blank.
8. Double-enter or verify the extreme values. Because kurtosis is driven by the tails, check every value beyond 3 SD against the original record.
9. Write the outlier rule in advance. Decide how real extreme values will be handled before you see the kurtosis, so the decision cannot be tuned to the result.
10. Keep the time order if there is one. Kurtosis assumes independent values. If you measure the same subject repeatedly, note it, because correlated values make the standard error too small.

Minimum sample: 4 to compute, 20 for the Anscombe-Glynn test, 50 or more per group for a reliable verdict, and a few hundred to estimate the size of heavy-tailed kurtosis with any precision. Pre-registration line: "Excess kurtosis (G₂) will be computed per group and judged with the Anscombe-Glynn test at α = .05 and Kim's (2013) rule." Common mistakes: pooling groups, using codes for missing data, and deleting extreme values before calculating kurtosis.

This free leptokurtic vs platykurtic calculator is designed for anyone who needs to describe the tails of a numeric variable or check them before choosing a statistical method.

Use it when:

• You must report skewness and kurtosis in a descriptive statistics table.
• You are checking normality before a t-test, ANOVA, regression or structural equation model.
• You want to know whether extreme values are more common than a normal model predicts, as in finance, insurance, hydrology or animal movement.
• You need to compare the tail shape of several groups side by side.
• You suspect a data-entry error or a mixture of two populations.

Do not use it when:

• Your data are categories or ranks with only a few levels; kurtosis of a 3-point scale says little.
• You have fewer than 4 values (impossible) or fewer than about 20 (very imprecise).
• Your values are not independent, such as a time series with strong autocorrelation.
• You need a full test of normality on its own; add a Shapiro-Wilk or Kolmogorov-Smirnov test as well.

Real-world examples:

1. A psychology student checks that questionnaire scores are not too leptokurtic before running a confirmatory factor analysis.
2. A risk analyst shows that daily returns have fat tails, so a normal value-at-risk model would understate losses.
3. A quality engineer finds platykurtic output and discovers two machines feeding one production line.
4. An ecologist describes a leptokurtic dispersal kernel for seeds or radio-collared animals.

Decision tree: if you only need to know whether data are normal, pair this tool with a Kolmogorov-Smirnov or Shapiro-Wilk test. If one tail is longer than the other, use a skewness calculator too. If kurtosis is high because of outliers, use a robust measure such as Moors' kurtosis or a trimmed analysis. If data are heavy-tailed and you need to compare means, use a robust test such as Yuen's trimmed-mean test or a bootstrap. If kurtosis is strongly negative and the histogram has two humps, split the data by the hidden grouping variable.

Assumptions

1. Independent observations. Requires each value to come from a different unit. Check the design, or plot values in collection order. If it fails, the standard error is too small and tests become too significant.
2. One population per group. Requires that a group is not a hidden mixture. Check the histogram for two humps. If it fails, kurtosis is biased downward, toward platykurtic.
3. A numeric, interval-level variable. Requires that differences between values are meaningful. Check the measurement scale. If it fails, as with a few ordinal categories, kurtosis has no clear meaning.
4. Correct values in the tails. Requires the extreme values to be real. Check the driver table and original records. If it fails, kurtosis is biased strongly upward, toward leptokurtic.
5. For the tests: normality under the null hypothesis and n of at least 5 (Anscombe-Glynn) or about 20 (reliable). Check n. If n is small, the z ratio over-states significance and the Anscombe-Glynn p-value is approximate.

Limitations

• Kurtosis is unstable in small samples. Its standard error is about √(24 ÷ n). Use the bootstrap interval, or collect more data.
• It is dominated by a few values. Use Moors' octile kurtosis or L-kurtosis as a robust alternative.
• It describes tails, not the peak. To describe peakedness or bimodality, use a density plot or a bimodality coefficient.
• Sample kurtosis is bounded by n. With n values, g₂ can never exceed about n − 5 (about 15 for n = 20), so very heavy tails are under-estimated in small samples. Fit a heavy-tailed model instead.
• Significance depends on n. Large samples make trivial kurtosis significant. Judge the size against Section 12.3 rather than relying on p.
• Kurtosis alone is not a normality test. Combine it with skewness (D'Agostino-Pearson K² or Jarque-Bera) and a Q-Q plot, or use the Shapiro-Wilk or Kolmogorov-Smirnov test.

What it means in practice

The question behind leptokurtic vs platykurtic is always the same: are extreme values more or less common than a normal curve predicts? A leptokurtic result tells you to look at the tails, to check the extreme values, and to protect any analysis that relies on variances or normal-theory standard errors. A platykurtic result tells you to look at where the data came from, for a bounded scale or a hidden mixture of groups. A mesokurtic result means the tails give you no reason for concern.

Cautions

Kurtosis is the most fragile of the everyday descriptive statistics. It is unstable in small samples, it can be created almost entirely by one value, and its meaning is about tails, not the shape of the peak. Always report it with its standard error, name the formula, and look at a Q-Q plot before you draw a conclusion.

Next steps

Paste your own data into the calculator, read the shape verdict and the driver table, check plots 1 and 2, and copy the APA sentence from Section 5. Then come back to this leptokurtic vs platykurtic calculator whenever you need to check the tails of a new data set.

The following peer-reviewed references support the methods used in this leptokurtic vs platykurtic kurtosis calculator, covering the meaning of kurtosis, the three sample estimators, the significance tests, normality cut-offs, robust alternatives and real-world heavy tails.

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