Variance Calculator: Sample, Population & SD | StatsUnlock

A variance calculator measures spread as the average squared distance from the mean. Subtract the mean from each value, square the differences, add them, then divide by n − 1 for a sample or n for a population. For 16 leaf lengths (52, 48, 55, 61, 47, ...), the sample variance is 17.60 mm² and the SD is 4.19 mm.

Rule of thumb: use the sample variance (n − 1) when your data are a sample from a larger group, which is almost always. Use the population variance (n) only when you measured every member. Take the square root of the variance to get the standard deviation in the original units.

Variance is a number that tells you how spread out your data are. It is the average of the squared distances between each value and the mean. If all values are the same, the variance is zero. The more the values scatter away from the mean, the larger the variance becomes.

Why square the distances? Some values sit above the mean and some below, so the plain distances always add up to zero. Squaring makes every distance positive, and it gives big distances more weight than small ones. The price is that the answer is in squared units: if you measure in millimetres, the variance is in mm². That is why we usually also report the standard deviation, which is the square root of the variance and is back in the original units.

What this calculator reports for each group:

• Sample variance (s²), dividing by n − 1. Use it for samples.
• Population variance (σ²), dividing by n. Use it only when you have every member of the group.
• Sample and population standard deviation, the square roots of the two variances.
• Sum of squares (SS), the top line of both variance formulas.
• Coefficient of variation (CV), the SD as a percentage of the mean.
• Standard error of the mean, range, IQR and MAD for context.
• Confidence interval for the variance and SD, from the chi-square distribution.
• Equal-variance tests across groups: Levene's (Brown-Forsythe), Bartlett's and, for two groups, the F-test.

A worked one-liner: for the values 2, 4, 4, 4, 5, 5, 7 and 9, the mean is 5. The squared deviations are 9, 1, 1, 1, 0, 0, 4 and 16, which add up to 32. Divide by n = 8 and the population variance is 4 (SD = 2). Divide by n − 1 = 7 and the sample variance is 4.5714 (SD = 2.1381).

Who uses variance: scientists use it to describe how consistent their measurements are. Quality engineers use it to check whether a machine fills bottles precisely. Investors use the variance of returns as a measure of risk. Ecologists compare variance between sites to see which habitat is more uniform. It is also the building block of the t-test, ANOVA (analysis of variance), regression and almost every other statistical test.

Quick variance converters

Small tools for the most common side questions. Each one runs only when you press its button.

4.1 What the variance number actually tells you

The variance is the average squared distance from the mean. On its own the number is hard to picture, because it is in squared units. A variance of 17.6 mm² does not mean values sit 17.6 mm from the mean. Take the square root first: an SD of 4.19 mm means a typical value sits about 4 mm away from the mean. Always read the variance and the SD together. Use the variance for calculations and the SD for describing your data in words.

A variance of zero means every value is identical. There is no upper limit. A larger variance always means more spread, but only when you compare data measured on the same scale and in the same units.

4.2 Sample variance or population variance: which number do you report?

The two numbers differ only in the last step. The sample variance divides the sum of squares by n − 1. The population variance divides by n. Dividing by n − 1 is called Bessel's correction. It makes the sample variance an unbiased estimate of the true population variance. Without it, a sample would underestimate the spread on average, because the sample mean is always as close as possible to its own data.

Report the sample variance whenever your data are a sample and you want to say something about a bigger group, which is nearly every research study. Report the population variance only when you have measured every single member of the group you care about, such as all 30 students in one class when you only want to describe that class. The gap between the two shrinks as n grows: at n = 16 it is about 6%, at n = 100 it is 1%.

4.3 Turning the variance into a standard deviation and a CV

The standard deviation is the square root of the variance, so it is back in your original units. For roughly bell-shaped data, about 68% of values fall within one SD of the mean and about 95% within two SDs. That rule helps you picture the spread quickly.

The coefficient of variation (CV) is the SD divided by the mean, times 100. It has no units, so you can compare the spread of groups with very different means. As a rough guide for lab and field data, a CV under 10% is low, 10% to 30% is moderate, and over 30% is high. The CV only makes sense for data on a ratio scale with a true zero and a positive mean. Do not use it for temperatures in Celsius or for data with negative values (Reed et al., 2002).

4.4 How precise is your variance? Reading the confidence interval

A variance from a sample is only an estimate. The confidence interval shows the range of true population variances that fit your data. It comes from the chi-square distribution and is not symmetric: the upper limit is always further from the estimate than the lower limit. With small samples the interval is very wide. For n = 16, the 95% interval for the leaf data runs from 9.60 to 42.15, more than four times wider at the top than at the bottom.

This interval assumes the data come from a normal distribution, and it is sensitive to that assumption. If your data are skewed or have heavy tails, treat the interval as approximate, or use a bootstrap interval in R or Python.

4.5 Comparing variances between groups

When you enter two or more groups, the calculator tests whether their variances differ. Three tests are shown, and they answer the same question in different ways:

• Levene's test, median-centred (Brown-Forsythe). The robust choice. It works well even when the data are not normal. This is the default in R's car::leveneTest and SciPy's levene (Brown & Forsythe, 1974).
• Bartlett's test. More powerful when the data really are normal, but it gives false alarms when they are not (Box, 1953). Trust it only after checking normality.
• F-test (two groups only). The ratio of the two sample variances. It is the classic textbook test and what Excel's F.TEST returns, but it is very sensitive to non-normal data.

A p-value below your significance level means the variances are unlikely to be equal. A p-value above it means you have no strong evidence that they differ. It does not prove they are equal, especially with small groups.

4.6 What a large or small variance means in practice

Whether a variance is "large" depends entirely on the subject. In manufacturing, a fill-volume variance of 1.7 ml² may fail a quality standard. In exam scores, a variance of 68 points² may be completely normal. Judge your number against three things: the measurement precision of your instrument, published values for similar data, and the practical tolerance in your field. The CV helps here because it puts spread on a common percentage scale.

A small variance means your values are consistent. That is good news for a machine or a lab assay. It can be bad news for a study that needs variety, such as a sample meant to cover a wide range of ages.

4.7 Why one outlier can dominate the variance

Because distances are squared, a value that sits 10 units from the mean adds 100 to the sum of squares, while a value 1 unit away adds only 1. A single outlier can therefore make up most of the variance. In the skewed income sample on this page, one high earner contributes 91% of the total sum of squares. The calculator's checks table shows the largest single share, so you can see when this happens.

When one value dominates, check it first. If it is a data entry error, fix it. If it is real, report the variance with and without it, or report robust measures such as the IQR or the median absolute deviation (MAD) alongside (Leys et al., 2013).

4.8 Unequal variances and what to do next

Many tests, including the Student t-test and classic one-way ANOVA, assume that groups have equal variances. If Levene's test is significant, or if the largest variance is more than about four times the smallest, switch to a method that does not need equal variances:

• For two group means, use Welch's t-test instead of Student's t-test.
• For three or more group means, use Welch's ANOVA with Games-Howell post-hoc tests.
• In regression, use robust (heteroscedasticity-consistent) standard errors or model the variance directly.

Sometimes the difference in variance is the finding itself. A new teaching method that makes scores more consistent, or a treatment that stabilises blood pressure, is a real result worth reporting.

4.9 The ratio of largest to smallest variance

The calculator reports the variance ratio, also called Hartley's F-max. It is a quick rule of thumb. A ratio under 2 suggests the variances are similar. Between 2 and 4 is a grey zone. Above 4, most methods that assume equal variances become unreliable, especially when the group sizes are also unequal. Use it alongside the formal tests, not instead of them (Hartley, 1950).

4.10 Variance, standard error and sample size are different things

The variance describes how spread out individual values are. It does not shrink as you collect more data; it settles towards the true population value. The standard error of the mean, s ÷ √n, describes how precisely you know the mean, and it does shrink as n grows. Mixing these up is one of the most common errors in published papers (Altman & Bland, 2005). Use the SD or variance to describe your data, and the SE or a confidence interval to describe the precision of a mean.

4.11 Common mistakes this calculator prevents

• Using the wrong denominator. Both the n − 1 and n versions are shown, labelled with their Excel functions.
• Reporting variance in the wrong units. Units are shown squared for the variance and plain for the SD.
• Forgetting that NumPy defaults to the population variance. Sections 9.4 and 11 show the fix.
• Treating blanks as zeros. Blank cells are skipped, and non-numeric entries are listed so you can fix them.
• Missing a dominant outlier. The checks table flags any value that contributes over half of the sum of squares.
• Trusting Bartlett's test on skewed data. The calculator states when normality looks doubtful and points you to Levene's test.

4.12 A short decision guide

When reporting variance calculator results in APA format, give the variance or SD together with the mean and the sample size, state whether you used the sample (n − 1) or population (n) formula, and name the test if you compared groups. The five templates below fill in automatically after you run the calculator.

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

These are the formulas this variance calculator uses, in the order they are computed.

This flowchart shows how the variance is calculated, step by step, from a raw list of numbers to a checked, reportable result. Every box uses the same worked example as the Excel, R and Python sections: 16 leaf lengths in mm from a botany practical.

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 a sample variance of 17.5958 and a population variance of 16.4961 for this dataset.

1. Enter your data. Type or paste numbers separated by commas, exactly like the placeholder: 52, 48, 55, 61, 47, ... Spaces, tabs, semicolons and new lines also work, so a column copied from Excel is fine.
2. Name each group. The group name box above each data box is editable. A name such as Site A flows into the results table, the charts and the report text.
3. Add more groups to compare. Press Add group to compare the variance of two or more groups. Each group has its own Clear and Remove buttons.
4. Or upload a file. On the Upload tab, choose a CSV or Excel file, then click the columns you want. Every column you click becomes its own group, named after its header.
5. Or use the manual table. Type values into the grid, rename the column headers, and press Use this table.
6. Try a sample dataset. Ten built-in datasets cover lab data, quality control, finance, ecology and a skewed set with an outlier.
7. Choose your settings. Pick sample (n − 1) or population (n) as the main variance, the confidence level for the interval, the significance level for the equal-variance tests, and your units.
8. Press Calculate Variance. Nothing is computed until you do. Changing any data or setting clears old results, so you never read stale numbers.
9. Read the results. Check the variance and SD cards, the formula block with your numbers, the step-by-step table, and the equal-variance tests if you have two or more groups.
10. Check the charts and export. Use the four charts to see the spread, copy one of the five reporting templates in Section 5, and download a text report or a PDF.

Need only a quick conversion? The converters under the Calculate button turn an SD into a variance, work out a percent variance between a budget and an actual value, and find the variance of a probability distribution.

Excel has built-in functions for both kinds of variance, so the calculation itself takes one formula. What trips people up is choosing between the four variance functions, getting a confidence interval, comparing two groups, and the business meaning of "variance" in budgets. This section covers all of it, with the spreadsheet shown at each stage, using the same 16 leaf lengths as the rest of the page.

9.1 The functions you need

9.2 Step by step

9.3 The awkward cases

Variance from a standard deviation. If a paper gives only the SD, square it: an SD of 4.1947 gives a variance of 17.5958. If it gives a population SD and you need the sample variance, use =B2^2*n/(n-1).

Variance of grouped or frequency data. With values in A and their counts in B, the sample variance is =SUMPRODUCT(B2:B10,(A2:A10-SUMPRODUCT(A2:A10,B2:B10)/SUM(B2:B10))^2)/(SUM(B2:B10)-1).

Variance of a probability distribution. With values in A and probabilities in B, the mean is =SUMPRODUCT(A2:A5,B2:B5) and the variance is =SUMPRODUCT(B2:B5,(A2:A5-D2)^2), where D2 holds the mean. No n − 1 correction is used here.

Variance by group with a condition. Excel has no VARIF. Use =VAR.S(IF(B2:B50="Site A",A2:A50)), which works directly in Excel 365 and as an array formula (Ctrl + Shift + Enter) in older versions.

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

9.5 Charting variance in Excel

1. Put group names in one column and each group's VAR.S in the next.
2. Select both columns, then Insert → Charts → Clustered Column.
3. For the more common SD error bars on a mean chart, chart the means instead, click the chart, press +, tick Error Bars, then More Options → Custom → Specify Value and select the STDEV.S cells.
4. Say in the caption whether the bars show the SD, the SE or a confidence interval.
5. To show raw spread, make a scatter chart with the group number on the x-axis and the values on the y-axis. Chart 1 on this page does this automatically.

9.6 Excel errors you will probably hit

R gives you the sample variance in one word, var(x). The script below is complete and copy-and-run. It shows every step of the calculation, gives both the sample and population variance, builds the confidence interval, prints a plain-English check, and saves one 300 dpi figure. It uses base R only, so there is nothing to install. The commented block at the end compares two groups with the F-test, Bartlett's test and Levene's test.

10.1 The complete script

R, base only, saves variance_plot.png

# ============================================================
# Variance calculator: sample and population variance, SD,
# 95% CI for the variance, and one publication-ready figure
# StatsUnlock - https://statsunlock.com/variance-calculator/
# Base R only. No packages to install.
# ============================================================

# ---- 1. Your data ------------------------------------------
# Replace with your own values, or read a CSV:
#   x <- read.csv("data.csv")$column_name
x <- c(52, 48, 55, 61, 47, 50, 58, 53,
       49, 56, 51, 54, 46, 57, 50, 52)
x <- x[!is.na(x)]                  # drop missing values
label <- "Leaf length"
units <- "mm"
conf  <- 0.95

# ---- 2. Variance, step by step -----------------------------
n      <- length(x)
mean_x <- mean(x)
dev    <- x - mean_x               # deviations, they sum to 0
sq     <- dev^2                    # squared deviations
ss     <- sum(sq)                  # sum of squares

var_s <- var(x)                    # sample variance, n - 1 (Excel VAR.S)
var_p <- var_s * (n - 1) / n       # population variance, n (Excel VAR.P)
sd_s  <- sqrt(var_s)
sd_p  <- sqrt(var_p)
cv    <- sd_s / mean_x * 100

# ---- 3. Confidence interval for the variance ---------------
a      <- 1 - conf
chi_lo <- qchisq(a / 2, df = n - 1)
chi_hi <- qchisq(1 - a / 2, df = n - 1)
ci     <- c(ss / chi_hi, ss / chi_lo)

cat(sprintf("n                   = %d\n", n))
cat(sprintf("mean                = %.4f\n", mean_x))
cat(sprintf("sum of squares      = %.4f\n", ss))
cat(sprintf("sample variance     = %.4f   (n - 1)\n", var_s))
cat(sprintf("population variance = %.4f   (n)\n", var_p))
cat(sprintf("sample SD           = %.4f   population SD = %.4f\n", sd_s, sd_p))
cat(sprintf("CV                  = %.2f%%\n", cv))
cat(sprintf("%.0f%% CI for variance = %.4f to %.4f\n", conf * 100, ci[1], ci[2]))
cat(sprintf("%.0f%% CI for SD       = %.4f to %.4f\n", conf * 100, sqrt(ci[1]), sqrt(ci[2])))
share <- max(sq) / ss * 100
cat(sprintf("largest share of SS = %.1f%% -> %s\n", share,
    if (share > 50) "one value dominates, check it" else "no single value dominates the variance"))

# ---- 4. One figure, two panels -----------------------------
png("variance_plot.png", width = 3000, height = 1350, res = 300)
par(mfrow = c(1, 2), mar = c(5.5, 4.6, 3.2, 1), family = "sans")
obs <- seq_len(n)

# Left: each value, the mean, and its deviation
plot(obs, x, type = "n", las = 1, xlab = "Observation",
     ylab = sprintf("%s (%s)", label, units))
rect(par("usr")[1], mean_x - sd_s, par("usr")[2], mean_x + sd_s,
     col = adjustcolor("#16a34a", alpha.f = 0.10), border = NA)
segments(obs, mean_x, obs, x, col = "#9333ea", lwd = 1.6)
points(obs, x, pch = 19, col = "#2563eb", cex = 1.1)
abline(h = mean_x, col = "#c2410c", lwd = 2.2, lty = 2)
title(main = "Deviations from the mean", adj = 0, cex.main = 1)

# Right: squared deviations with the variance line
barplot(sq, names.arg = obs, col = "#0d9488", border = "#115e59", las = 1,
        xlab = "Observation", ylab = sprintf("Squared deviation (%s^2)", units))
abline(h = var_s, col = "#c2410c", lwd = 2.2, lty = 2)
title(main = sprintf("Squared deviations, s^2 = %.2f", var_s), adj = 0, cex.main = 1)

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

# ---- 5. Comparing groups (optional) ------------------------
# placebo   <- c(142, 138, 150, 145, 136, 155, 141, 148, 139, 152, 144, 147)
# treatment <- c(128, 131, 126, 135, 129, 133, 127, 130, 132, 128, 134, 129)
# var.test(placebo, treatment)                  # F-test, two groups
# y <- c(placebo, treatment)
# g <- factor(rep(c("Placebo", "Treatment"), each = 12))
# bartlett.test(y ~ g)                          # Bartlett's test
# z <- abs(y - ave(y, g, FUN = median))         # Brown-Forsythe Levene by hand
# anova(lm(z ~ g))                              # same as car::leveneTest(y ~ g)

Expected output for the leaf data: sample variance 17.5958, population variance 16.4961, SD 4.1947, 95% CI for the variance 9.6018 to 42.1481. These match Sections 7, 9 and 11.

10.2 What each part does

10.3 What the figure shows

The left panel plots each value in order. The dashed orange line is the mean, the shaded band is the mean plus or minus one SD, and each purple line is a deviation. The right panel squares those deviations into bars. The dashed line is the sample variance. You can see at once which values contribute most: the 61 mm leaf makes the tallest bar, about 28% of the total.

If one bar towers over all the others, a single value is driving your variance. Check it before you report anything.

10.4 Common R problems and fixes

10.5 Useful one-liners

In Python the sample variance is np.var(x, ddof=1). The ddof=1 part matters: without it NumPy returns the population variance. The script below is complete and runnable. It works through each step, prints both variances, the confidence interval and a plain-English check, and saves one 300 dpi figure.

Install with pip install numpy scipy matplotlib if you do not already have them.

11.1 The complete script

Python 3, saves variance_plot.png

# ============================================================
# Variance calculator: sample and population variance, SD,
# 95% CI for the variance, and one publication-ready figure
# StatsUnlock - https://statsunlock.com/variance-calculator/
# ============================================================
import numpy as np
import matplotlib
matplotlib.use("Agg")                 # save to file, no window needed
import matplotlib.pyplot as plt
from scipy import stats

# ---- 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)
x = x[~np.isnan(x)]                   # drop missing values
label = "Leaf length"
units = "mm"
conf = 0.95

# ---- 2. Variance, step by step -----------------------------
n = len(x)
mean_x = x.mean()
dev = x - mean_x                      # deviations, they sum to 0
sq = dev ** 2                         # squared deviations
ss = sq.sum()                         # sum of squares

var_s = np.var(x, ddof=1)             # sample variance, n - 1 (Excel VAR.S)
var_p = np.var(x, ddof=0)             # population variance, n (Excel VAR.P)
sd_s = np.sqrt(var_s)
sd_p = np.sqrt(var_p)
cv = sd_s / mean_x * 100

# ---- 3. Confidence interval for the variance ---------------
a = 1 - conf
chi_lo = stats.chi2.ppf(a / 2, n - 1)
chi_hi = stats.chi2.ppf(1 - a / 2, n - 1)
ci_lo, ci_hi = ss / chi_hi, ss / chi_lo

print(f"n                   = {n}")
print(f"mean                = {mean_x:.4f}")
print(f"sum of squares      = {ss:.4f}")
print(f"sample variance     = {var_s:.4f}   (n - 1)")
print(f"population variance = {var_p:.4f}   (n)")
print(f"sample SD           = {sd_s:.4f}   population SD = {sd_p:.4f}")
print(f"CV                  = {cv:.2f}%")
print(f"{conf:.0%} CI for variance = {ci_lo:.4f} to {ci_hi:.4f}")
print(f"{conf:.0%} CI for SD       = {np.sqrt(ci_lo):.4f} to {np.sqrt(ci_hi):.4f}")
share = sq.max() / ss * 100
verdict = ("one value dominates, check it" if share > 50
           else "no single value dominates the variance")
print(f"largest share of SS = {share:.1f}% -> {verdict}")

# ---- 4. One figure, two panels -----------------------------
order = np.arange(1, n + 1)
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(10, 4.2))

# Left: each value, the mean, and its deviation
ax1.axhspan(mean_x - sd_s, mean_x + sd_s, color="#16a34a", alpha=0.10,
            label="mean +/- 1 SD")
ax1.vlines(order, mean_x, x, color="#9333ea", lw=1.4, zorder=2)
ax1.scatter(order, x, color="#2563eb", s=36, zorder=3, label="value")
ax1.axhline(mean_x, color="#c2410c", lw=2, ls="--", label=f"mean = {mean_x:.2f}")
ax1.set_xlabel("Observation")
ax1.set_ylabel(f"{label} ({units})")
ax1.set_title("Deviations from the mean", loc="left", fontweight="bold")
ax1.legend(frameon=False, fontsize=8, ncol=3, loc="upper center",
           bbox_to_anchor=(0.5, -0.17))

# Right: squared deviations with the variance line
ax2.bar(order, sq, color="#0d9488", edgecolor="#115e59")
ax2.axhline(var_s, color="#c2410c", lw=2, ls="--",
            label=f"sample variance = {var_s:.2f}")
ax2.set_xlabel("Observation")
ax2.set_ylabel(f"Squared deviation ({units}\u00b2)")
ax2.set_title("Squared deviations", loc="left", fontweight="bold")
ax2.legend(frameon=False, fontsize=8, loc="upper center",
           bbox_to_anchor=(0.5, -0.17))

for ax in (ax1, ax2):
    ax.spines[["top", "right"]].set_visible(False)

fig.suptitle(f"{label}: n = {n}, s\u00b2 = {var_s:.2f} {units}\u00b2, SD = {sd_s:.2f} {units}",
             x=0.01, ha="left", fontsize=11)
fig.tight_layout()
fig.savefig("variance_plot.png", dpi=300, bbox_inches="tight")
print("\nSaved variance_plot.png")

Printed output for the leaf data: sample variance 17.5958, population variance 16.4961, SD 4.1947, CV 8.00%, 95% CI for the variance 9.6018 to 42.1481, and largest share of SS 27.8%.

11.2 What each part does

11.3 What the figure shows

The figure is the same two-panel design as the R version: deviations on the left and squared deviations on the right. Looking at the squares is the fastest way to see why variance reacts so strongly to extreme values. A value twice as far from the mean makes a bar four times as tall.

11.4 Common Python problems and fixes

11.5 Useful one-liners

12.1 Sample or population variance?

Conclusion: in research, the sample variance is almost always correct. The population version is for complete counts, such as every machine in one factory.

12.2 Chi-square critical values for the variance confidence interval

Conclusion: for a 95% interval, divide SS by the 0.975 value for the lower limit and by the 0.025 value for the upper limit. For the leaf data, df = 15: 263.9375 ÷ 27.4884 = 9.60 and 263.9375 ÷ 6.2621 = 42.15. Use the 0.005 and 0.995 columns for 99%.

12.3 How wide is a 95% CI for the variance? (as a multiple of s²)

Conclusion: a variance from 5 values could be anything from about a third to eight times the true value. You need around 50 values before the interval shrinks to roughly ±50%. The last column shows the population variance shrinks towards the sample variance as n grows.

12.4 Two-sided F critical values (α = 0.05) for comparing two variances

Conclusion: put the larger variance on top. If the ratio is bigger than the table value, the variances differ at the 0.05 level (two-sided). With 10 values per group (df 9 and 9) the ratio must exceed 4.03, which shows how hard it is to detect unequal variances in small samples.

12.5 How one outlier inflates the variance

Conclusion: changing one value from 12 to 60 multiplied the variance by more than 200. Squaring makes the variance the least robust common statistic.

12.6 Coefficient of variation: rough interpretation bands

Conclusion: these bands are a guide, not a law. Judge your CV against values published in your own field, and never use CV for data that can be negative.

12.7 How to find variance on a calculator

Conclusion: most handheld calculators give the standard deviation, not the variance. Square the right one: Sx (or sx) for a sample, σx for a population.

Eight worked examples show what typical, clear, borderline, null and problem results look like, and how to write each one up.

Study design: variance assumes a set of independent measurements of one quantity on an interval or ratio scale. To compare variances, each group must be measured with the same instrument and the same method.

1. Decide why you need the variance. Describing one group, checking an assumption before a t-test or ANOVA, and testing whether one group is more consistent are different aims. Write the aim down first.
2. Define the population and the unit. Decide whether you are sampling from a larger group (sample variance) or measuring every member (population variance).
3. Fix the instrument, units and resolution. Measurement error adds directly to the variance. A ruler marked only in centimetres adds spread a millimetre scale would not.
4. Calibrate before you start and record it. Drift during a study inflates the variance for reasons that have nothing to do with the subject.
5. Randomise the selection and the order of measurement. Measuring all of one group first can mix time effects into group differences in spread.
6. Keep units independent. Repeat readings of the same plant or person are not independent. Average them first, or use a mixed model, otherwise the variance looks smaller than it really is.
7. Plan the sample size for the precision you need. Use Table 12.3: at least 30 per group if you want the variance within roughly ±50%, and at least 20 per group to have a fair chance of detecting a two-fold difference in variance.
8. Record missing values as blanks, never as zero. A zero creates a large false squared deviation.
9. Write the outlier rule in advance. Because variance squares distances, decide before you see the data how you will check and handle extreme values.
10. Store data one column per group, as shown 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 groups can have different sizes.

Minimum sample: the sample variance needs at least 2 values, but anything under 10 per group gives a very rough estimate. Equal-variance tests need at least 5 per group to be meaningful.

Pre-registration: state the variance formula (n − 1 or n), the equal-variance test and its alpha, the outlier rule, and what you will do if variances differ, such as switching to Welch's test.

Common collection mistakes:

• Recording missing values as zero, which inflates the variance.
• Changing instruments or observers part way through.
• Treating repeated readings of one unit as separate observations.
• Rounding too much at collection, which hides real variation.
• Pooling two different populations into one column, which inflates the variance.

This free variance calculator is designed for students, teachers, researchers and analysts who need the variance and standard deviation of one or more groups, with the working shown.

Use it when:

• ✓ You need the sample or population variance of numeric data.
• ✓ You want to check equal variances before a t-test or ANOVA.
• ✓ You want to know which group, machine, method or site is more consistent.
• ✓ You need a confidence interval for a variance or SD.
• ✓ You need to show the step-by-step calculation for homework or a report.
• ✓ You want to convert between SD and variance, or find a percent variance.

Do not use it when:

• ✗ The data are categories or ranks. Use frequencies or the median and IQR.
• ✗ The data are heavily skewed and you want a typical spread. Use the IQR or MAD.
• ✗ You want to compare group means. Use a t-test or an ANOVA calculator (analysis of variance compares means, not variances).
• ✗ The data are repeated measures or clustered. Use a mixed model to split the variance into its parts.
• ✗ You need a two-way analysis of variance table. That is a separate design with two factors.

Real-world examples:

1. Manufacturing: comparing the variance of fill volume between machines to decide which needs servicing.
2. Finance: the variance of monthly returns as a measure of risk for a portfolio.
3. Ecology: comparing the variance of tree size between natural and planted stands.
4. Education: checking whether a new method makes student results more consistent.

Design guidance: if your main question is about spread, plan the sample size around the variance, not the mean, because variances need larger samples to estimate well.

Decision tree:

Assumptions

1. Numeric data on an interval or ratio scale. Check: the numbers measure an amount. If violated, for example with ranks or codes, the variance has no clear meaning.
2. Independent observations. Check: each value comes from a different unit. If violated, the variance is usually underestimated, because repeated readings of one unit are similar.
3. A random, representative sample. Check: the sampling method. If violated, the variance can be biased in either direction, often too small when the sample is more uniform than the population.
4. Normal data, for the confidence interval and Bartlett's test. Check: the skewness and kurtosis in the checks table and Chart 4. If violated with heavy tails, the CI is too narrow and Bartlett's test gives too many false positives.
5. No gross errors or unchecked outliers. Check: the largest share of SS. If violated, the variance is inflated, sometimes many times over.
6. Same measurement method in every group, for comparisons. Check: the protocol. If violated, a group measured less precisely shows an inflated variance.

Limitations

• The variance is not robust: one extreme value can dominate it. Use the IQR, the MAD or a trimmed variance for messy data.
• It is in squared units, which are hard to explain. Report the SD for descriptive purposes.
• The chi-square interval depends strongly on normality. For skewed data, use a bootstrap interval in R or Python (Section 11.5).
• The F-test is not robust to non-normal data. Prefer Levene's (Brown-Forsythe) test, which this page shows first.
• Equal-variance tests have low power with small groups. A non-significant result does not prove the variances are equal.
• This tool treats data as a simple random sample. Survey designs with weights or clusters need design-based variance estimators.
• It does not compute ANOVA tables. For comparing means, use a t-test or ANOVA tool, choosing Welch's version when variances differ.

A variance calculator answers a simple question: how far, on average, do the values sit from their mean? The steps are always the same. Find the mean, subtract it from every value, square the differences, add them up and divide. The only real choice is the divisor: n − 1 for a sample, which is almost every research situation, or n for a complete population.

The number itself is in squared units, so in most writing the standard deviation is the friendlier way to describe spread. The variance earns its place elsewhere. It is what statistical tests and models work with, it adds up neatly across independent sources of variation, and comparing variances tells you which group, method or machine is more consistent.

Three habits make variance results trustworthy. First, always say whether you used the sample or population formula and give n. Second, look at the squared deviations, because one extreme value can make up most of the variance. Third, remember that a variance from a small sample is uncertain: the confidence interval shows how much, and it is often wider than people expect.

When you compare groups, start with Levene's test because it works on real, imperfect data, and use the result to choose the right test of means, such as Welch's t-test or Welch's ANOVA. A difference in spread can also be a finding in its own right.

Paste your numbers into the calculator above, check the step-by-step table and the charts, copy a reporting template from Section 5, and come back to this variance calculator whenever you need the sample variance, population variance or standard deviation with every step shown.

The following peer-reviewed references support the methods used in this variance calculator: computing the variance accurately, confidence intervals, tests for equal variances, robust alternatives and good reporting practice.

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3. Box, G. E. P. (1953). Non-normality and tests on variances. Biometrika, 40(3/4), 318–335. https://doi.org/10.1093/biomet/40.3-4.318
4. Brown, M. B., & Forsythe, A. B. (1974). Robust tests for the equality of variances. Journal of the American Statistical Association, 69(346), 364–367. https://doi.org/10.1080/01621459.1974.10482955
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6. Conover, W. J., Johnson, M. E., & Johnson, M. M. (1981). A comparative study of tests for homogeneity of variances, with applications to the outer continental shelf bidding data. Technometrics, 23(4), 351–361. https://doi.org/10.1080/00401706.1981.10487680
7. Cumming, G., Fidler, F., & Vaux, D. L. (2007). Error bars in experimental biology. Journal of Cell Biology, 177(1), 7–11. https://doi.org/10.1083/jcb.200611141
8. Gastwirth, J. L., Gel, Y. R., & Miao, W. (2009). The impact of Levene's test of equality of variances on statistical theory and practice. Statistical Science, 24(3), 343–360. https://doi.org/10.1214/09-STS301
9. Hartley, H. O. (1950). The maximum F-ratio as a short-cut test for heterogeneity of variance. Biometrika, 37(3/4), 308–312. https://doi.org/10.1093/biomet/37.3-4.308
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. 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
12. Lim, T.-S., & Loh, W.-Y. (1996). A comparison of tests of equality of variances. Computational Statistics & Data Analysis, 22(3), 287–301. https://doi.org/10.1016/0167-9473(95)00054-2
13. Ling, R. F. (1974). Comparison of several algorithms for computing sample means and variances. Journal of the American Statistical Association, 69(348), 859–866. https://doi.org/10.1080/01621459.1974.10480219
14. Markowitz, H. (1952). Portfolio selection. The Journal of Finance, 7(1), 77–91. https://doi.org/10.1111/j.1540-6261.1952.tb01525.x
15. Nakagawa, S., & Cuthill, I. C. (2007). Effect size, confidence interval and statistical significance: A practical guide for biologists. Biological Reviews, 82(4), 591–605. https://doi.org/10.1111/j.1469-185X.2007.00027.x
16. Reed, G. F., Lynn, F., & Meade, B. D. (2002). Use of coefficient of variation in assessing variability of quantitative assays. Clinical and Diagnostic Laboratory Immunology, 9(6), 1235–1239. https://doi.org/10.1128/CDLI.9.6.1235-1239.2002
17. 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
18. 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
19. 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
20. Welford, B. P. (1962). Note on a method for calculating corrected sums of squares and products. Technometrics, 4(3), 419–420. https://doi.org/10.1080/00401706.1962.10490022