Calculate Mean And Variance Of Lognormal Distribution

Use this ultra-premium calculator to compute the arithmetic mean, variance, standard deviation, median, mode, and distribution curve for a lognormal variable. Enter either the normal-space parameters of ln(X) or a mean/variance pair in real space and visualize the resulting shape instantly.

Lognormal Distribution Calculator

Choose whether your known parameters belong to log-space or original-space.

Higher resolution creates a smoother probability density curve.

This is the mean of the underlying normal distribution.

This is the standard deviation of the underlying normal distribution.

How to Calculate Mean and Variance of Lognormal Distribution

When people search for how to calculate mean and variance of lognormal distribution, they are usually working with data that are strictly positive, right-skewed, and multiplicative in nature. The lognormal model appears in finance, hydrology, biology, air pollution analysis, reliability engineering, insurance severity modeling, and many other fields where values cannot go below zero and where occasional large observations are not just possible but expected. Understanding the mean and variance of the lognormal distribution is essential because those two moments summarize both the typical magnitude of the data and the spread around that magnitude in original measurement units.

A random variable X is lognormally distributed if its natural logarithm, ln(X), is normally distributed. That means if ln(X) ~ N(μ, σ²), then X follows a lognormal distribution with parameters μ and σ. This distinction matters. In practice, many mistakes come from confusing the parameters of the normal distribution in log-space with the mean and standard deviation in real space. They are not the same, and the conversion formulas are nonlinear.

Core formulas for the lognormal mean and variance

If ln(X) ~ N(μ, σ²), then the arithmetic mean of X is:

Mean = E[X] = exp(μ + σ² / 2)

The variance of X is:

Variance = Var(X) = (exp(σ²) – 1) × exp(2μ + σ²)

The standard deviation is simply the square root of the variance:

SD = sqrt(Var(X))

Two other useful quantities are the median and mode:

Median = exp(μ)     Mode = exp(μ – σ²)

These formulas reveal an important property of the lognormal distribution: as σ grows, the distribution becomes increasingly skewed to the right. The mean moves farther above the median, and the variance can increase very rapidly. That is why even a modest amount of variability in log-space can create a very wide spread in the original data.

Why the lognormal distribution matters

The reason analysts often need to calculate mean and variance of lognormal distribution is that many real-world systems behave multiplicatively rather than additively. If growth occurs by percent changes, if wear accumulates over time, or if several independent positive factors multiply together, the resulting variable often resembles a lognormal pattern more than a symmetric normal one. This is common in:

• Finance: asset prices, gross returns over time horizons, and some forms of volatility-adjusted modeling.
• Reliability engineering: time-to-failure or fatigue-life measurements for components with multiplicative degradation effects.
• Environmental science: concentrations of contaminants, rainfall intensity, and airborne particulate measurements.
• Biostatistics: biomarker levels, exposure data, and dosage-response variables that remain positive.
• Operations and economics: income components, waiting times, or production quantities with long right tails.

In all of these examples, the arithmetic mean still has practical importance because it corresponds to expected value in original units. If you want average cost, average concentration, average payout, or average lifetime in the observed scale, you need the lognormal mean, not just the mean in log-space.

Step-by-step method to calculate mean and variance of lognormal distribution

Method 1: You already know μ and σ of ln(X)

This is the most direct case. Suppose your transformed variable ln(X) has mean μ and standard deviation σ. Then:

• Compute σ².
• Find the mean using exp(μ + σ²/2).
• Find the variance using (exp(σ²) – 1) × exp(2μ + σ²).
• If needed, calculate the standard deviation as the square root of the variance.

For example, if μ = 1 and σ = 0.5, then σ² = 0.25. The mean becomes exp(1 + 0.125) = exp(1.125), and the variance becomes (exp(0.25) – 1) × exp(2 + 0.25). This yields a mean of approximately 3.0802 and a variance of approximately 2.6948. Notice how the mean exceeds the median exp(1) = 2.7183, reflecting right skew.

Method 2: You know the real-space mean and variance and need μ and σ

Sometimes you know the arithmetic mean and variance of X, but you need the underlying log-space parameters. In that case, let:

m = E[X],   v = Var(X)

Then the conversion back to μ and σ is:

σ² = ln(1 + v / m²)

μ = ln(m) – σ² / 2

This reverse calculation is especially useful when model documentation, published studies, or engineering specifications provide a mean and variance directly in original units while simulation software requires μ and σ in log-space.

Known Inputs	What to Calculate	Formula	Interpretation
μ and σ of ln(X)	Mean of X	exp(μ + σ²/2)	Average value in original units
μ and σ of ln(X)	Variance of X	(exp(σ²)-1) × exp(2μ + σ²)	Spread in original units squared
Mean m and variance v of X	σ²	ln(1 + v/m²)	Log-space variance
Mean m and variance v of X	μ	ln(m) – σ²/2	Log-space mean

Interpreting the output correctly

One of the biggest practical challenges is interpretation. If your data are lognormal, the arithmetic mean can be heavily influenced by a few very large observations. That does not make the mean incorrect; it simply means the distribution is asymmetric. In fact, for a lognormal distribution, you will usually observe the ordering:

Mode < Median < Mean

This is a defining feature of right-skewed distributions. The mode indicates the peak of the density, the median indicates the 50th percentile, and the mean captures the long-right-tail contribution. Depending on your application, each may answer a different question:

• Mean: best for expected totals, budgeting, exposure, or average aggregate outcomes.
• Median: best for a typical central observation under skewness.
• Variance: best for quantifying uncertainty and dispersion.
• Mode: best for understanding where the density peaks.

Worked example: mean and variance of a lognormal distribution

Imagine that the natural log of a service time is normally distributed with μ = 2.2 and σ = 0.8. You want to calculate the mean and variance of the original service time variable X.

• σ² = 0.64
• Mean = exp(2.2 + 0.64/2) = exp(2.52) ≈ 12.4286
• Variance = (exp(0.64) – 1) × exp(4.4 + 0.64)
• Since exp(0.64) ≈ 1.8965, the variance is approximately 0.8965 × exp(5.04)
• exp(5.04) ≈ 154.47, so variance ≈ 138.49

This tells you that the expected service time is around 12.43 units, but the spread is very large. That large variance is exactly what a right-skewed model often implies: many moderate values and a small number of very large ones.

Common mistakes when calculating mean and variance of lognormal distribution

• Confusing μ with the arithmetic mean: μ is the mean of ln(X), not the mean of X.
• Confusing σ with the original standard deviation: σ belongs to log-space unless otherwise stated.
• Using base-10 logs without adjustment: the standard formulas assume natural logarithms.
• Ignoring positivity: the lognormal model only applies when X is strictly positive.
• Reporting median as mean: exp(μ) is the median, not the arithmetic mean.
• Underestimating tail behavior: a larger σ can produce extremely large means and variances.

Quick comparison of key lognormal measures

Measure	Formula	What it tells you
Mean	exp(μ + σ²/2)	The expected value in original units
Variance	(exp(σ²)-1) × exp(2μ + σ²)	The spread around the mean in squared units
Standard deviation	sqrt(Variance)	The dispersion in original units
Median	exp(μ)	The middle value of the distribution
Mode	exp(μ – σ²)	The most likely value near the density peak

When should you use a lognormal calculator?

A calculator like the one above is valuable when you need fast, accurate conversions and immediate interpretation. Instead of manually typing formulas into a spreadsheet and risking a parameter mismatch, you can enter your values once and instantly get the full summary. The chart also helps you develop intuition. When σ is small, the curve looks relatively concentrated. As σ increases, the peak shifts left and the right tail extends farther, which can dramatically raise the mean and variance.

Practical use cases

• Estimating expected claim sizes and volatility in actuarial work.
• Modeling particulate concentration data in environmental monitoring.
• Converting simulation assumptions between log-space and real-space inputs.
• Analyzing laboratory assay results that are positively skewed.
• Understanding uncertainty in project costs, lead times, or biological measurements.

Trusted academic and government references

For readers who want authoritative statistical background, these resources provide strong supplementary context:

• NIST Engineering Statistics Handbook for probability distributions, estimation concepts, and statistical modeling guidance.
• Penn State STAT Online for rigorous university-level explanations of distributions, transformations, and applied inference.
• CDC for examples of positively skewed environmental and public health data where log transformations are often useful.

Final takeaway

If you need to calculate mean and variance of lognormal distribution, the most important thing to remember is that the underlying parameters μ and σ usually belong to the normal distribution of ln(X), not to X itself. Once that is clear, the formulas become straightforward: the mean is exp(μ + σ²/2), and the variance is (exp(σ²) – 1) × exp(2μ + σ²). With those relationships, you can move confidently between theoretical modeling, applied analysis, and practical decision-making in any field where positive skewed data appear.