Left Skewed vs Right Skewed Calculator | StatsUnlock

Left skewed vs right skewed describes which way the long tail of a distribution points: a left skewed (negative) distribution has its tail on the left and mean < median, while a right skewed (positive) distribution has its tail on the right and mean > median.

Skewness measures lopsidedness, not spread. Income, waiting times and hospital stays are usually right skewed. Easy exam scores, age at death and 5-star ratings are usually left skewed. Skewness does not tell you about outliers on its own, or about two peaks, so always look at a histogram or box plot as well.

Rule of thumb: G1 below −0.5 is left skewed, above +0.5 is right skewed, and between −0.5 and +0.5 is approximately symmetric. Beyond ±1 the skew is high. For example, hard exam scores give G1 = 1.72 (right skewed) and easy exam scores give G1 = −1.63 (left skewed).

Skewness measures how asymmetric a distribution is. A symmetric distribution looks the same on both sides of its centre, like a bell curve. A skewed distribution has one side that stretches out further than the other. That stretched side is called the tail, and the tail gives the skew its name.

• Right skewed (also called positively skewed or skewed to the right): most values sit on the low side and a few large values make a long tail to the right. Skewness is greater than 0.
• Left skewed (also called negatively skewed or skewed to the left): most values sit on the high side and a few small values make a long tail to the left. Skewness is less than 0.
• Symmetric: both tails are about the same length. Skewness is close to 0, and the mean, median and mode are close together.

What it actually measures. The moment skewness cubes each value’s distance from the mean. Cubing keeps the sign and makes far-away values count much more. So values far above the mean push the total up (right skew), and values far below the mean push it down (left skew). The result is divided by the standard deviation cubed, so skewness has no units. You can compare the skewness of exam scores with the skewness of incomes.

A worked one-liner: the hard exam scores 35, 36, 37, … 71, 88, 95 (n = 20) have mean 50.35 and median 44.50. The two scores of 88 and 95 drag the mean to the right. The adjusted skewness is G1 = 1.72, so these scores are highly right skewed.

A memory trick. Think of the tail as a pointing finger. If the long tail points to the right (toward larger numbers), the data are skewed right. If it points to the left (toward smaller numbers), the data are skewed left. Many students get this backwards because they look at where the hump is. The hump of a right skewed histogram sits on the left.

Who uses this: teachers check whether a test was too easy (left skew) or too hard (right skew). Economists report median income because income is right skewed. Clinicians report median length of stay. Ecologists see right skew in animal movement, body mass and species counts. Quality engineers watch for skew that signals a process drifting against a limit. Every analyst checks skewness before a t-test, ANOVA or regression, because strong skew can bias those tests.

Skewness is not the same as kurtosis. Skewness asks “which side has the longer tail?”; kurtosis asks “how heavy are the tails overall?”. This tool reports both, but the verdict is based on skewness.

4.1 Reading the verdict: which way is the skew?

Start with the sign of the skewness. A positive value means right skewed: the data have a long tail toward larger numbers. A negative value means left skewed: the long tail points toward smaller numbers. A value near zero means the two tails balance.

The name always follows the tail, never the peak. A right skewed histogram has its tallest bars on the left and thin bars trailing off to the right. That is why “skewed right vs skewed left” confuses so many students: the bulk of the data sits on the opposite side to the name. In the worked example, most hard-test scores are between 35 and 55, but two students scored 88 and 95. Those two scores form the right tail, so the hard test is right skewed.

4.2 How big is the skew? Reading the size of G1

The size (absolute value) of the skewness tells you how lopsided the data are. This tool uses the common rule of thumb associated with Bulmer:

• |G1| below 0.5: approximately symmetric. The mean is a fair summary.
• |G1| from 0.5 to 1: moderately skewed. Report the median as well as the mean.
• |G1| above 1: highly skewed. Lead with the median and IQR, and think about a transform before parametric tests.

Some textbooks are more lenient and treat values up to ±1, or even ±2, as acceptable for parametric tests (George and Mallery’s guideline). You can switch to the lenient bands in the settings. Whichever rule you use, name it in your methods so readers can judge.

4.3 Is the skew real? The standard error, z and the D’Agostino test

A sample can look skewed by chance, especially a small one. The standard error of skewness (SE) tells you how much skewness bounces around from sample to sample when the population is perfectly symmetric. For n = 20 it is 0.51, for n = 100 it is 0.24, and for n = 1000 it is 0.08.

Dividing G1 by its SE gives a z ratio. If |z| is above 1.96 the skew is significant at the 5% level. The D’Agostino skewness test refines this with a transformation that works well from n = 8 upward. It is the test behind scipy.stats.skewtest and R’s moments::agostino.test. A p-value below α means a sample this lopsided would be unusual if the population were symmetric.

Kim (2013) suggests a sample-size rule for the z ratio: use |z| > 1.96 below n = 50, |z| > 3.29 for n = 50 to 300, and judge the raw size of G1 (above 2 is substantial) for very large samples, where even trivial skew becomes significant.

4.4 Statistical significance vs practical importance

These two answer different questions. Significance asks “could this be chance?”; size asks “does it matter?”. With 600 values, a skew of 0.25 is significant (p = .012) but too small to change any decision. With 12 values, a skew of 1.1 can be non-significant even though the histogram is clearly lopsided. Always report both the size (G1) and the test (z and p), and let the size guide what you do.

4.5 The mean, median and mode: what they really tell you

The tail pulls the mean toward it, because the mean uses every value. The median barely moves. So in right skewed data the mean is usually above the median, and in left skewed data the mean is usually below it. That is why Pearson’s second coefficient, 3(mean − median) ÷ s, is a quick skewness check.

One warning: “mean > median means right skew” is a rule of thumb, not a law. Von Hippel (2005) showed it fails for many discrete distributions and for data with one long tail and one heavy short tail. When the mean-median order and G1 disagree, trust the histogram first, then G1, then the mean-median gap.

4.6 What the four charts show

Chart 1, the histogram. Bars left of the median are blue and bars right of the median are orange. The colour that stretches further marks the tail. The green curve is a normal distribution with the same mean and SD: where your bars rise above the curve on one side and fall below it on the other, you are seeing skew.

Chart 2, the box plot. In a right skewed box plot the right whisker is long and the median sits closer to Q1 (the left edge of the box). In a left skewed box plot the left whisker is long and the median sits closer to Q3. The orange diamond is the mean: it sits on the tail side of the median.

Chart 3, the normal probability (Q-Q) plot. Each point pairs a sorted data value with the value a normal distribution would predict. For right skewed data the points curve up and away from the line at the top right, a convex “smile”. For left skewed data they drop below the line at the bottom left, a concave “frown”.

Chart 4, skewness compared. Bars above zero are right skewed and below zero are left skewed. The green band is the symmetric zone. The black error bar on G1 is a rough 95% interval (G1 ± 1.96 SE): if it crosses zero, the skew may be chance.

4.7 Why three skewness coefficients disagree

G1, Pearson Sk2 and Bowley look at different parts of the data. G1 is driven by the cube of the distances, so it is dominated by the most extreme values. Pearson Sk2 compares the mean and median. Bowley uses only the quartiles, so it ignores the outer 25% on each side completely.

When G1 is large but Bowley is near zero, the middle half of your data is balanced and one or two extreme values create the skew. When all three agree in sign, the skew runs through the whole distribution. The pilot dataset in the samples shows the extreme case: G1 is +2.59 but Bowley is −0.50, because one value of 48 creates all of the positive skew.

4.8 Comparing two or more groups: left skewed vs right skewed side by side

When one group leans left and another leans right, as with the easy and hard tests, their means are misleading in opposite ways. The easy test mean (86.25) sits below its median (90.00), and the hard test mean (50.35) sits above its median (44.50). The gap in means is 35.90 points, but the gap in medians is 45.50 points. Comparing means understates how different the two classes really were.

For groups with opposite skew, compare medians, use a rank-based test such as Mann-Whitney or Kruskal-Wallis, or transform each group in a way that suits its own direction. Do not apply one log transform to both: it will make the left skewed group even more skewed.

4.9 Outliers, small samples and two peaks

Skewness is very sensitive to outliers. A single typing error, such as 950 instead of 95, can turn a symmetric sample into a highly skewed one. Check the outlier count in the checks table before trusting any verdict. With fewer than 20 values, removing or adding one point can flip the sign.

Two peaks are a different problem. A bimodal distribution can have skewness near zero and still look nothing like a bell curve. The tool reports the bimodality coefficient; values above 0.555 suggest two peaks (Pfister et al., 2013). Strong skew also pushes the coefficient up, so the tool only warns about two peaks when |G1| is below 0.75. If you see two humps in Chart 1, split the groups rather than describe one skewness.

4.10 Kurtosis, the companion number

Excess kurtosis (G2) describes how heavy both tails are compared with a normal curve. It is 0 for a normal distribution, positive when the tails are heavy, and negative when they are light. High skewness usually comes with positive kurtosis, because one heavy tail also raises tail weight. For a full kurtosis analysis use the leptokurtic vs platykurtic calculator listed in the related tools.

4.11 What skewness means for your next analysis

Mild skew (|G1| below 1) rarely harms t-tests or ANOVA once each group has 30 or more values, because the sampling distribution of the mean becomes close to normal. Strong skew with small groups is a problem: confidence intervals become lopsided and p-values become unreliable. In that case you can transform (log or square root for right skew; reflect then log for left skew), use a rank-based test, use a generalised linear model with a gamma or log-normal family, or bootstrap.

4.12 Common mistakes this calculator prevents

• Naming the skew after the peak instead of the tail.
• Mixing up G1 (Excel SKEW) and g1 (Excel SKEW.P, SciPy default) and reporting two different numbers for the same data.
• Calling a skew of 0.3 “skewed” because it is significant in a huge sample.
• Trusting the mean vs median order alone, which can mislead with discrete or bounded data.
• Forgetting that blanks, text or “n/a” codes in a column change n. The tool lists every ignored entry.
• Log-transforming left skewed data, which makes it worse.

Skewness usually appears in two places in a paper: the Methods, where you say how you checked distribution shape, and the Results, where you report the numbers. The five templates below fill in automatically after you run the analysis. Edit the wording to fit your study, but keep the numbers.

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

5.1 Methods wording

Say three things in your methods: which skewness formula you used, how you judged it, and what you planned to do if data were skewed. For example: “Skewness was quantified with the adjusted Fisher-Pearson coefficient (G1; Joanes & Gill, 1998) and tested with the D’Agostino skewness test (D’Agostino et al., 1990). Distributions with |G1| ≥ 0.5 were treated as skewed; for these, medians and interquartile ranges are reported and group comparisons used the Mann-Whitney U test.”

5.2 The results sentence

Lead with the direction in words, then the numbers. Good: “Hard-test scores were right skewed (G1 = 1.72, SE = 0.51, z = 3.00, p = .003).” Weak: “Skewness was 1.722.” The weak version forces the reader to know that positive means right, and it gives no idea whether 1.722 is large or could be chance.

For two groups with opposite skew, put them in one sentence so the contrast is clear: “Easy-test scores were left skewed (G1 = −1.63, p = .004) whereas hard-test scores were right skewed (G1 = 1.72, p = .003).”

5.3 Tables and figures

In a descriptive table, add a column for skewness and, if space allows, its SE. Report mean (SD) for symmetric variables and median [IQR] for skewed ones, and say which is which in a footnote. For figures, a histogram or a box plot with raw points shows skew far better than a bar chart of means. Weissgerber et al. (2015) showed that bar graphs of means hide exactly the shape problems that skewness describes.

5.4 The discussion sentence

Explain what the skew means in your field. For example: “The left skew in easy-test scores reflects a ceiling effect: many students reached the maximum, so the test could not separate the strongest students.” Or: “Right skew in length of stay is driven by a small number of complicated cases, which matters for bed planning because the mean overstates the typical stay.”

5.5 Reporting checklist

• ✓ Direction in words: left skewed, right skewed or approximately symmetric.
• ✓ The coefficient with its name (G1, g1 or b1) and two decimals.
• ✓ The SE, z or the D’Agostino p-value, and n.
• ✓ Mean and median, so the reader sees the gap.
• ✓ The threshold you used and its source.
• ✓ What you did about it: transformation, non-parametric test, or a reason to do nothing.
• ✓ A histogram, box plot or Q-Q plot in the paper or supplement.

These are the exact formulas this left skewed vs right skewed calculator uses. Every card ends with the worked numbers for the hard-test scores, so you can check each step by hand.

This flowchart shows how to decide left skewed vs right skewed, step by step, from a raw column of numbers to a tested verdict. Every box uses the same worked example as the Excel, R and Python sections: the 20 hard-test scores (35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 47, 48, 50, 51, 55, 62, 71, 88, 95). The easy-test totals are given in the final box.

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 G1 = 1.7220, g1 = 1.5901, SE = 0.5121 and Z = 3.0036 for the hard test, and G1 = −1.6308 for the easy test.

1. Enter your data. In the Paste / Type tab, type or paste comma-separated numbers into a group box, for example 52, 48, 55, 61, 47. New lines, tabs and spaces also work.
2. Name each group. Click the group name above each box and type a meaningful name such as Easy test. The name flows into the results table, all four charts and the report.
3. Add more groups to compare. Press Add group to compare left skewed vs right skewed groups side by side. Each box has its own Clear and Remove button.
4. Or upload a CSV or Excel file. On the Upload tab choose a file, then click the columns you want. Every selected column is loaded as its own group.
5. Or type into the manual table. On the Manual Entry tab, each column is a group. Rename the headers, add rows or columns, then press Use this table.
6. Try a sample dataset. Ten built-in datasets cover right skew (income, waiting time), left skew (easy tests, age at death, ratings), symmetric data and a messy pilot study.
7. Choose your settings. Pick the skewness formula (G1 matches Excel SKEW), the size bands, the significance level, the histogram bins, decimals and units.
8. Press Run Analysis. Nothing is calculated until you press the button, and changing any input clears old results so you never read stale numbers.
9. Read the verdict and checks. Each group gets a coloured verdict card, the coefficients, the D’Agostino test and a list of data checks such as outliers and small samples.
10. Use the charts and export. Read the histogram, box plot, normal probability plot and skewness chart, copy a reporting template from Section 5, and download the Doc or PDF report.

Excel can tell you whether data are left skewed or right skewed with one function, SKEW. What trips people up is everything around it: Excel has two skewness functions that give different numbers, it has no built-in p-value for skewness, and the Analysis ToolPak reports skewness without a standard error. This section shows each step on the hard-test scores with the spreadsheet on screen.

9.1 The functions you need

9.2 Step by step

9.3 The awkward cases

Excel gives skewness but no p-value. Neither SKEW nor the ToolPak tests it. Use the z ratio from step 3 for a quick answer, or the eight-cell D’Agostino chain from step 4 for the proper test. The quick z ratio is a little too generous in small samples: for the hard test it gives p = 0.0008, while the D’Agostino test gives 0.0027.

SKEW or SKEW.P? SKEW is the sample statistic most journals and SPSS users expect. SKEW.P treats your data as the whole population. For n = 20 they differ by about 8%; for n = 200 by under 1%.

Fewer than 3 values. SKEW returns #DIV/0! with fewer than three numbers or when every value is the same. That is correct: skewness is undefined there.

9.4 Why your Excel answer might differ from R or Python

9.5 Making a left vs right skewed histogram and box plot in Excel

1. Select one column of values, then Insert → Charts → Statistic → Histogram (Excel 2016 and later).
2. Right-click the horizontal axis, choose Format Axis, and set a bin width that gives 5 to 10 bars. Too many bars hide the tail.
3. For a box plot select both columns, then Insert → Statistic → Box and Whisker. Excel draws the mean as an ×; it sits on the tail side of the median line.
4. Add the skewness to the chart title, for example “Hard test, G1 = 1.72”, so readers do not have to guess.
5. Set the gap width to 0% on a histogram so the bars touch, which is what reviewers expect.

Charts 1 and 2 in Section 3 draw both plots for every group automatically, with the mean, median and fences marked.

9.6 Excel errors you will probably hit

R has no skewness function in base R, so people install a package and are then surprised that e1071, moments and psych give three different numbers. The script below needs no packages. It computes G1 (Excel SKEW), g1, the standard error, the D’Agostino test, Pearson and Bowley skewness for every group, prints a plain-English verdict, and saves one publication-ready figure at 300 dpi with histograms, box plots and a normal probability plot.

10.1 The complete script

R, base only, saves skewness_plot.png

# ============================================================
# Left skewed vs right skewed: skewness, D'Agostino test and
# one publication-ready figure (histograms, box plot, Q-Q plot)
# StatsUnlock - https://statsunlock.com/left-skewed-vs-right-skewed/
# Base R only. No packages to install.
# ============================================================

# ---- 1. Your data -------------------------------------------
# Replace with your own groups, or read a CSV:
#   df <- read.csv("data.csv")
#   groups <- list("Easy test" = na.omit(df$easy), "Hard test" = na.omit(df$hard))
groups <- list(
  "Easy test" = c(95, 92, 88, 97, 90, 85, 93, 78, 96, 89,
                  62, 91, 94, 87, 70, 98, 90, 83, 55, 92),
  "Hard test" = c(42, 48, 35, 51, 38, 62, 45, 40, 88, 37,
                  44, 55, 39, 47, 71, 36, 43, 50, 95, 41)
)
alpha <- 0.05
units <- "%"

# ---- 2. Skewness functions ----------------------------------
skew_g1 <- function(x) {                 # moment skewness = Excel SKEW.P
  d <- x - mean(x)
  mean(d^3) / mean(d^2)^1.5
}
skew_G1 <- function(x) {                 # adjusted = Excel SKEW, SPSS, SAS
  n <- length(x)
  skew_g1(x) * sqrt(n * (n - 1)) / (n - 2)
}
se_skew <- function(n) sqrt(6 * n * (n - 1) / ((n - 2) * (n + 1) * (n + 3)))
dagostino <- function(x) {               # same as moments::agostino.test
  n <- length(x)
  if (n < 8) return(c(Z = NA, p = NA))
  y  <- skew_g1(x) * sqrt((n + 1) * (n + 3) / (6 * (n - 2)))
  b2 <- 3 * (n^2 + 27 * n - 70) * (n + 1) * (n + 3) /
        ((n - 2) * (n + 5) * (n + 7) * (n + 9))
  w2 <- -1 + sqrt(2 * (b2 - 1))
  delta <- 1 / sqrt(log(sqrt(w2)))
  a  <- sqrt(2 / (w2 - 1))
  Z  <- delta * log(y / a + sqrt((y / a)^2 + 1))
  c(Z = Z, p = 2 * pnorm(-abs(Z)))
}

# ---- 3. Results for every group -----------------------------
res <- lapply(groups, function(x) {
  x  <- x[!is.na(x)]                     # drop missing values
  n  <- length(x)
  q  <- quantile(x, c(0.25, 0.5, 0.75))  # type 7, same as Excel
  G1 <- skew_G1(x)
  dz <- dagostino(x)
  list(n = n, mean = mean(x), median = median(x), sd = sd(x),
       g1 = skew_g1(x), G1 = G1, se = se_skew(n), z = G1 / se_skew(n),
       Z = unname(dz["Z"]), p = unname(dz["p"]),
       pearson2 = 3 * (mean(x) - median(x)) / sd(x),
       bowley = unname((q[3] + q[1] - 2 * q[2]) / (q[3] - q[1])))
})

for (nm in names(res)) {
  r <- res[[nm]]
  shape <- if (abs(r$G1) < 0.5) "approximately symmetric" else
    paste(if (abs(r$G1) > 1) "highly" else "moderately",
          if (r$G1 > 0) "right skewed (positive)" else "left skewed (negative)")
  sig <- if (!is.na(r$p) && r$p < alpha) "significant" else "not significant"
  cat(sprintf("\n%s  (n = %d)\n", nm, r$n))
  cat(sprintf("  mean = %.4f   median = %.4f   SD = %.4f\n", r$mean, r$median, r$sd))
  cat(sprintf("  G1 = %.4f   g1 = %.4f   SE = %.4f   z = %.4f\n", r$G1, r$g1, r$se, r$z))
  cat(sprintf("  D'Agostino Z = %.4f   p = %.4f\n", r$Z, r$p))
  cat(sprintf("  Pearson Sk2 = %.4f   Bowley = %.4f\n", r$pearson2, r$bowley))
  cat(sprintf("  Verdict: %s, %s at alpha = %.2f\n", shape, sig, alpha))
}

# ---- 4. One figure, four panels -----------------------------
cols <- c("#2563eb", "#ea580c", "#16a34a", "#9333ea")
png("skewness_plot.png", width = 3000, height = 2250, res = 300)
par(mfrow = c(2, 2), mar = c(4.5, 4.5, 3, 1), family = "sans")

for (k in 1:2) {                         # panels 1 and 2: histograms
  x <- groups[[k]]; r <- res[[k]]
  hist(x, breaks = "Sturges", col = adjustcolor(cols[k], 0.35), border = cols[k],
       main = sprintf("%s: G1 = %.2f, p = %.3f", names(groups)[k], r$G1, r$p),
       xlab = sprintf("Value (%s)", units), ylab = "Frequency", las = 1, cex.main = 1)
  abline(v = r$mean,   col = "#c2410c", lwd = 2.4, lty = 2)
  abline(v = r$median, col = "#9333ea", lwd = 2.4)
  legend(if (r$G1 < 0) "topleft" else "topright", bty = "n", cex = 0.85, lwd = 2.4, lty = c(2, 1),
         col = c("#c2410c", "#9333ea"),
         legend = c(sprintf("Mean %.2f", r$mean), sprintf("Median %.2f", r$median)))
}

boxplot(groups, horizontal = TRUE, outline = FALSE, las = 1, col = "white",
        main = "Box plot: long whisker = tail side", cex.main = 1,
        xlab = sprintf("Value (%s)", units),
        ylim = range(unlist(groups)))
set.seed(42)                             # seed = reproducible jitter
for (k in seq_along(groups)) {
  x <- groups[[k]]
  points(x, k + runif(length(x), -0.15, 0.15), pch = 19,
         col = adjustcolor(cols[k], 0.6))
  points(res[[k]]$mean, k, pch = 23, bg = "#c2410c", col = "#c2410c", cex = 1.6)
}

qq <- lapply(groups, qqnorm, plot.it = FALSE)
plot(NULL, xlim = range(sapply(qq, function(q) range(q$x))),
     ylim = range(unlist(groups)), las = 1, main = "Normal probability plot",
     cex.main = 1, xlab = "Theoretical normal quantile",
     ylab = sprintf("Sample value (%s)", units))
for (k in seq_along(groups)) {
  points(qq[[k]]$x, qq[[k]]$y, pch = 19, col = cols[k])
  qqline(groups[[k]], col = cols[k], lty = 2)
}
legend("topleft", bty = "n", pch = 19, col = cols[seq_along(groups)],
       legend = names(groups), cex = 0.85)

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

Run it with Rscript skewness.R. It prints G1 = −1.6308 (Z = −2.8827, p = 0.0039) for the easy test and G1 = 1.7220 (Z = 3.0036, p = 0.0027) for the hard test, matching Excel and Python.

10.2 What each part does

10.3 What the figure shows

The top two panels are histograms. The easy test piles up near 90 to 100 with a few low scores trailing left: the mean line (dashed) sits left of the median line, the picture of left skew. The hard test piles up near 40 with a tail running right to 95: the mean sits right of the median, the picture of right skew.

The bottom left panel puts the two box plots on one scale. The easy test has its long whisker and loose points on the left; the hard test has them on the right. The bottom right panel is the normal probability plot: the hard-test points bend up away from their line at the right end (convex), and the easy-test points bend down away from their line at the left end (concave).

10.4 Common R problems and fixes

10.5 Useful one-liners

In Python, scipy.stats.skew(x) is one line, but its default is the moment estimate g1, which disagrees with Excel, SPSS and pandas. The script below computes both, runs the D’Agostino test with scipy.stats.skewtest, prints a verdict for each group, and saves one 300 dpi figure with histograms, box plots and a normal probability plot.

Install with pip install numpy scipy matplotlib if you do not already have them. Add pandas only if you use the CSV line.

11.1 The complete script

Python 3, saves skewness_plot.png

# ============================================================
# Left skewed vs right skewed: skewness, D'Agostino test and
# one publication-ready figure (histograms, box plot, Q-Q plot)
# StatsUnlock - https://statsunlock.com/left-skewed-vs-right-skewed/
# ============================================================
import numpy as np
from scipy import stats
import matplotlib
matplotlib.use("Agg")                     # draw to file, no window needed
import matplotlib.pyplot as plt

# ---- 1. Your data -------------------------------------------
# Replace with your own groups, or read a CSV:
#   import pandas as pd
#   df = pd.read_csv("data.csv")
#   groups = {c: df[c].dropna().to_numpy(float) for c in ["Easy test", "Hard test"]}
groups = {
    "Easy test": np.array([95, 92, 88, 97, 90, 85, 93, 78, 96, 89,
                           62, 91, 94, 87, 70, 98, 90, 83, 55, 92], dtype=float),
    "Hard test": np.array([42, 48, 35, 51, 38, 62, 45, 40, 88, 37,
                           44, 55, 39, 47, 71, 36, 43, 50, 95, 41], dtype=float),
}
alpha = 0.05
units = "%"

# ---- 2. Skewness for every group ----------------------------
def skew_report(x):
    x = x[~np.isnan(x)]                   # drop missing values
    n = len(x)
    g1 = stats.skew(x)                    # moment skewness = Excel SKEW.P
    G1 = stats.skew(x, bias=False)        # adjusted = Excel SKEW, SPSS, SAS
    se = np.sqrt(6 * n * (n - 1) / ((n - 2) * (n + 1) * (n + 3)))
    q1, med, q3 = np.percentile(x, [25, 50, 75])   # type 7, same as Excel
    out = dict(n=n, mean=x.mean(), median=med, sd=x.std(ddof=1),
               g1=g1, G1=G1, se=se, z=G1 / se,
               pearson2=3 * (x.mean() - med) / x.std(ddof=1),
               bowley=(q3 + q1 - 2 * med) / (q3 - q1),
               kurt=stats.kurtosis(x, bias=False))  # = Excel KURT
    if n >= 8:
        out["Z"], out["p"] = stats.skewtest(x)       # D'Agostino test
    else:
        out["Z"], out["p"] = np.nan, np.nan
    return out

results = {name: skew_report(x) for name, x in groups.items()}

for name, r in results.items():
    if abs(r["G1"]) < 0.5:
        shape = "approximately symmetric"
    else:
        size = "highly" if abs(r["G1"]) > 1 else "moderately"
        side = "right skewed (positive)" if r["G1"] > 0 else "left skewed (negative)"
        shape = f"{size} {side}"
    sig = "significant" if r["p"] < alpha else "not significant"
    print(f"\n{name}  (n = {r['n']})")
    print(f"  mean = {r['mean']:.4f}   median = {r['median']:.4f}   SD = {r['sd']:.4f}")
    print(f"  G1 = {r['G1']:.4f}   g1 = {r['g1']:.4f}   SE = {r['se']:.4f}   z = {r['z']:.4f}")
    print(f"  D'Agostino Z = {r['Z']:.4f}   p = {r['p']:.4f}")
    print(f"  Pearson Sk2 = {r['pearson2']:.4f}   Bowley = {r['bowley']:.4f}   kurtosis = {r['kurt']:.4f}")
    print(f"  Verdict: {shape}, {sig} at alpha = {alpha}")

# ---- 3. One figure, four panels -----------------------------
colors = ["#2563eb", "#ea580c", "#16a34a", "#9333ea"]
names = list(groups)
fig, ax = plt.subplots(2, 2, figsize=(10, 7.5))
rng = np.random.default_rng(42)           # seed = reproducible jitter

for k, name in enumerate(names[:2]):      # panels 1 and 2: histograms
    x, r, a = groups[name], results[name], ax[0, k]
    a.hist(x, bins="sturges", color=colors[k], alpha=0.35, edgecolor=colors[k])
    a.axvline(r["mean"], color="#c2410c", ls="--", lw=2, label=f"Mean {r['mean']:.2f}")
    a.axvline(r["median"], color="#9333ea", lw=2, label=f"Median {r['median']:.2f}")
    a.set_title(f"{name}: G1 = {r['G1']:.2f}, p = {r['p']:.3f}", fontsize=11, loc="left")
    a.set_xlabel(f"Value ({units})")
    a.set_ylabel("Frequency")
    a.legend(frameon=False, fontsize=9)

a = ax[1, 0]                              # panel 3: box plots with points
box_kw = dict(widths=0.5, showfliers=False, medianprops=dict(color="black", lw=2))
try:
    a.boxplot([groups[n] for n in names], orientation="horizontal", **box_kw)
except TypeError:                         # Matplotlib older than 3.10
    a.boxplot([groups[n] for n in names], vert=False, **box_kw)
for k, name in enumerate(names):
    x = groups[name]
    a.scatter(x, k + 1 + rng.uniform(-0.15, 0.15, len(x)),
              color=colors[k % 4], alpha=0.6, s=18, zorder=3)
    a.scatter(results[name]["mean"], k + 1, marker="D", color="#c2410c", s=50, zorder=4)
a.set_yticks(range(1, len(names) + 1))
a.set_yticklabels(names)
a.set_title("Box plot: long whisker = tail side", fontsize=11, loc="left")
a.set_xlabel(f"Value ({units})")

a = ax[1, 1]                              # panel 4: normal Q-Q plots
for k, name in enumerate(names):
    (osm, osr), (slope, icpt, _) = stats.probplot(groups[name], dist="norm")
    a.scatter(osm, osr, color=colors[k % 4], s=18, label=name)
    a.plot(osm, slope * osm + icpt, color=colors[k % 4], lw=1.2, ls="--")
a.set_title("Normal probability plot", fontsize=11, loc="left")
a.set_xlabel("Theoretical normal quantile")
a.set_ylabel(f"Sample value ({units})")
a.legend(frameon=False, fontsize=9)

fig.suptitle("Left skewed vs right skewed", fontsize=14, fontweight="bold")
fig.tight_layout()
fig.savefig("skewness_plot.png", dpi=300, bbox_inches="tight")
print("\nSaved skewness_plot.png")

Run it with python skewness.py. In Jupyter, delete the matplotlib.use("Agg") line and add plt.show() at the end to see the figure inline.

11.2 What each part does

11.3 What the figure shows

The four panels match the R figure. The histograms show the easy test leaning left and the hard test leaning right, with the mean line pulled toward each tail. The box plots put both groups on one axis so you can see the long whiskers on opposite sides. On the normal probability plot, the hard-test points curve upward at the right end and the easy-test points curve downward at the left end. Matplotlib picks slightly different histogram edges from R, so bar heights can differ, but every number printed is identical.

11.4 Common Python problems and fixes

11.5 Useful one-liners

12.1 How to read the size of skewness

What to conclude: the sign gives the direction (positive = right, negative = left) and these bands give the size. Name the rule you used. Kim (2013) and Blanca et al. (2013) discuss why real data rarely have skewness exactly 0.

12.2 Left skewed vs right skewed vs symmetric at a glance

What to conclude: if three or more clues point the same way, the direction is clear. If they disagree, look at the histogram and trust G1 over the mean-median order (von Hippel, 2005).

12.3 How skew looks in each graph

What to conclude: in every graph the tail, the sparse side, names the skew. A stem-and-leaf plot is a histogram turned on its side, so read it the same way.

12.4 Standard error of skewness and the smallest significant |G1|

What to conclude: with 20 values a skewness under about 1.00 can be chance; with 500 values even 0.21 is significant. That is why size matters more than p in large samples. These are z-ratio cut-offs; the D’Agostino test is slightly stricter in small samples.

12.5 Which software gives which skewness

What to conclude: the three always share the same sign, so they never disagree about left vs right. They only differ in size, by up to about 15% at n = 20 and under 1% at n = 1000 (Joanes & Gill, 1998).

12.6 Fixing skewed data before a parametric test

What to conclude: never log-transform left skewed data directly; it stretches the left tail further. After reflecting, the direction of effects flips, so say so when you report results (Box & Cox, 1964; Feng et al., 2014).

12.7 Real-world variables and their usual skew

What to conclude: a hard floor (zero, or a minimum) with no ceiling gives right skew; a hard ceiling (100%, 5 stars, a maximum lifespan) gives left skew. Micceri (1989) found most real psychological data are skewed to some degree.

Eight worked examples, from a clean textbook case to the situations that fool people. In every histogram, blue bars are below the median and orange bars are above it, the dashed line is the mean and the solid line is the median.

Study design: a left skewed vs right skewed analysis needs independent measurements of one numeric variable on an interval or ratio scale, collected in one or more clearly defined groups. Skewness describes the shape of each group, so each group must be a real, single population.

1. Decide which variable’s skewness you need, and why. Write down whether you are checking a test assumption, describing a population, or comparing the shape of groups. This decides your sample size and threshold.
2. Define one unit of observation. One person, one plot, one day. Repeated readings of the same unit are not independent and make a small sample look large.
3. Sample at random from each group. A convenience sample can create or hide skew; for example, surveying only daytime callers trims the long right tail of waiting times.
4. Plan enough values to measure skewness reliably. Aim for at least 20 per group, and 50 or more if you want to compare left vs right skewed groups. The D’Agostino test needs at least 8.
5. Record the full range. Do not cap, round or group values at collection time. Capping incomes at “100 or more” deletes the right tail you are trying to measure.
6. Use a scale with enough resolution. A 1 to 5 rating gives only five distinct values, so skewness mostly reflects the ceiling or floor. Record a finer measure when you can.
7. Keep missing values blank, never zero. A fake zero creates a false left tail and can flip a right skewed variable to symmetric.
8. Write the outlier rule before collecting. Decide in advance how you will check extreme values (for example 1.5 × IQR fences) and when a value is an error rather than a real tail.
9. Note known floors and ceilings. Zero, 100%, a maximum score. These predict the direction of skew and help readers judge it.
10. Store one column per group with the group name in the first row, exactly as the layout below shows, ready to paste or upload for the skewness calculation.

One column per group, one row per unit, numbers only below the header. Blank cells are ignored, so groups can have different sizes.

Minimum sample: 8 values per group for the D’Agostino test, 20 for a stable skewness estimate (SE = 0.51), and 100 or more if a moderate skew of 0.5 must be detected reliably (SE = 0.24).

Pre-registration: record the skewness formula (G1), the threshold (|G1| ≥ 0.5), the test (D’Agostino, α = 0.05), and what you will do if a group is skewed, before you look at the data.

Common collection mistakes:

• Typing 0 for missing data, which manufactures a left tail.
• Top-coding or capping values, which removes the right tail.
• Mixing two populations in one column, which gives two peaks rather than a skew.
• Counting repeat measurements as separate units.
• Collecting too few values and then reading too much into the sign of skewness.

Use it whenever you need to know which way your data lean, how strongly, and whether that matters for the analysis you plan next.

Use it when:

• ✓ You want to know if a variable is left skewed, right skewed or symmetric before choosing mean or median.
• ✓ You are checking the normality assumption for a t-test, ANOVA or regression residuals.
• ✓ You are comparing the shape of two or more groups, such as an easy and a hard version of a test.
• ✓ You need a skewness value, its SE and a p-value to report in a paper or thesis.
• ✓ You want to check whether a log or square-root transform actually removed the skew.

Do not use it when:

• ✗ Your data are categories with no order. Skewness needs numbers.
• ✗ You have fewer than about 8 values. Look at the raw numbers instead.
• ✗ The histogram shows two peaks. Describe the two groups, not a skewness.
• ✗ You need a full normality test. Use Shapiro-Wilk or the Kolmogorov-Smirnov test, which check skewness and kurtosis together.
• ✗ You want to compare the centres of groups. Use a t-test, or Mann-Whitney if the groups are skewed.

Real-world examples:

1. Education: a teacher finds the final exam left skewed (ceiling effect) and the mid-term right skewed (floor effect), and rescales the next test.
2. Health: length of stay is right skewed, so a hospital reports median days rather than the mean in its annual report.
3. Ecology: daily travel distance of a collared animal is right skewed, so movement models use a gamma or log-normal distribution.
4. Business: product ratings are left skewed, so the team tracks the share of 1 and 2-star reviews rather than the mean rating.

Study design guidance: measure skewness on each group separately, never on pooled groups, because pooling two symmetric groups with different means can create apparent skew or two peaks. For regression, check the skewness of the residuals, not of the raw outcome.

Decision tree:

• |G1| below 0.5 → treat as symmetric → t-test, ANOVA or regression as planned.
• |G1| 0.5 to 1 and n ≥ 30 per group → parametric tests are usually fine → report median as well.
• |G1| above 1 and right skewed → try log or square root → if still skewed, Mann-Whitney (2 groups) or Kruskal-Wallis (3 or more).
• |G1| above 1 and left skewed → reflect, then log → or go straight to rank-based tests.
• Groups skewed in opposite directions → compare medians with rank-based tests; do not transform both the same way.
• Two peaks → split the data or use a mixture model.

Assumptions

1. Values are independent. Check the design: one value per unit. If values cluster (repeat readings, siblings, plots in one field), the SE is too small and the test finds skew too often (bias toward false positives).
2. The variable is numeric on an interval or ratio scale. Check what the numbers mean. On ordinal scales skewness describes the coding, and the bias depends on the spacing you chose.
3. Each group is one population. Check Chart 1 and the bimodality coefficient. Mixing groups biases skewness toward zero or toward the smaller group’s side.
4. Enough values. Check n: at least 8 for the test, 20 for stable estimates. Small samples bias G1 toward zero on average and make its sign unreliable.
5. No data errors in the tail. Check the outliers listed. A typing error in the tail inflates |G1|, biasing the verdict toward “highly skewed”.
6. For the test, the sampling distribution is close to normal after the D’Agostino transform. This holds well from n = 8 upward; below that p-values are too small (bias toward false positives).

Limitations

• Skewness is sensitive to outliers; one value can dominate it. Alternative: Bowley or the medcouple, which are robust (Brys et al., 2004).
• It cannot detect two peaks or gaps. Alternative: look at the histogram and the bimodality coefficient, or fit a mixture model.
• It is one number for a whole shape. Alternative: Q-Q plots, quantile plots, or the full five-number summary.
• The D’Agostino test only checks skewness, not kurtosis or normality as a whole. Alternative: the D’Agostino-Pearson omnibus test, Shapiro-Wilk or Kolmogorov-Smirnov.
• The size bands are conventions, not laws. Alternative: decide your threshold from the method you plan to use next and state it.
• The tool assumes a simple random sample. Survey weights or clustered designs need weighted moments.
• The tool does not compare skewness between groups formally. Alternative: bootstrap the difference in G1 (see the R one-liner in 10.5).

Telling left skewed vs right skewed data apart is one of the first and most useful checks in any analysis. The rule is simple once you remember it: the skew is named after the long tail, not after the peak. A right skewed distribution has a tail toward large values, a positive skewness, and a mean pulled above the median. A left skewed distribution has a tail toward small values, a negative skewness, and a mean dragged below the median.

The size of the skew matters as much as its direction. Below 0.5 in absolute value the data are close enough to symmetric for the mean and standard deviation, and for most parametric tests. Between 0.5 and 1 the skew is moderate, so report the median as well. Above 1 the skew is high: lead with the median and IQR, show the shape in a figure, and consider a transform or a rank-based test.

Significance answers a separate question. The D’Agostino test tells you whether a sample this lopsided could come from a symmetric population by chance. In small samples a large skew can be non-significant, and in very large samples a tiny skew can be significant. Report both the coefficient with its standard error and the test, and let the size guide your decisions.

Finally, never trust a single number. Skewness can be created by one typing error, hidden by two peaks, or forced by the ceiling of a rating scale. Read it next to the histogram, the box plot and the normal probability plot, and check the outlier and bimodality warnings before you write it up. Use this left skewed vs right skewed calculator whenever you meet a new dataset, because knowing which way your data lean is the quickest way to choose the right summary and the right test.

These peer-reviewed articles support the methods in this left skewed vs right skewed calculator: the skewness formulas, the D’Agostino test, robust and quartile skewness, the mean-median rule, transformations and graphical checks.

1. Altman, D. G., & Bland, J. M. (1996). Detecting skewness from summary information. BMJ, 313(7066), 1200. https://doi.org/10.1136/bmj.313.7066.1200
2. Blanca, M. J., Arnau, J., López-Montiel, D., Bono, R., & Bendayan, R. (2013). Skewness and kurtosis in real data samples. Methodology, 9(2), 78–84. https://doi.org/10.1027/1614-2241/a000057
3. Box, G. E. P., & Cox, D. R. (1964). An analysis of transformations. Journal of the Royal Statistical Society: Series B, 26(2), 211–243. https://doi.org/10.1111/j.2517-6161.1964.tb00553.x
4. Brys, G., Hubert, M., & Struyf, A. (2004). A robust measure of skewness. Journal of Computational and Graphical Statistics, 13(4), 996–1017. https://doi.org/10.1198/106186004X12632
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6. D’Agostino, R. B., Belanger, A., & D’Agostino, R. B., Jr. (1990). A suggestion for using powerful and informative tests of normality. The American Statistician, 44(4), 316–321. https://doi.org/10.1080/00031305.1990.10475751
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9. Ghasemi, A., & Zahediasl, S. (2012). Normality tests for statistical analysis: A guide for non-statisticians. International Journal of Endocrinology and Metabolism, 10(2), 486–489. https://doi.org/10.5812/ijem.3505
10. Groeneveld, R. A., & Meeden, G. (1984). Measuring skewness and kurtosis. Journal of the Royal Statistical Society: Series D (The Statistician), 33(4), 391–399. https://doi.org/10.2307/2987742
11. 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
12. Joanes, D. N., & Gill, C. A. (1998). Comparing measures of sample skewness and kurtosis. Journal of the Royal Statistical Society: Series D (The Statistician), 47(1), 183–189. https://doi.org/10.1111/1467-9884.00122
13. Kim, H.-Y. (2013). Statistical notes for clinical researchers: Assessing normal distribution (2) using skewness and kurtosis. Restorative Dentistry & Endodontics, 38(1), 52–54. https://doi.org/10.5395/rde.2013.38.1.52
14. Krzywinski, M., & Altman, N. (2014). Visualizing samples with box plots. Nature Methods, 11(2), 119–120. https://doi.org/10.1038/nmeth.2813
15. Micceri, T. (1989). The unicorn, the normal curve, and other improbable creatures. Psychological Bulletin, 105(1), 156–166. https://doi.org/10.1037/0033-2909.105.1.156
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17. Pfister, R., Schwarz, K. A., Janczyk, M., Dale, R., & Freeman, J. B. (2013). Good things peak in pairs: A note on the bimodality coefficient. Frontiers in Psychology, 4, 700. https://doi.org/10.3389/fpsyg.2013.00700
18. von Hippel, P. T. (2005). Mean, median, and skew: Correcting a textbook rule. Journal of Statistics Education, 13(2). https://doi.org/10.1080/10691898.2005.11910556
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20. Wilk, M. B., & Gnanadesikan, R. (1968). Probability plotting methods for the analysis of data. Biometrika, 55(1), 1–17. https://doi.org/10.1093/biomet/55.1.1