Lognormal Distribution Calculator With Sample

Lognormal Distribution Parameters:

\[ \mu = \text{mean}(\ln x), \quad \sigma = \text{sd}(\ln x) \]

μ (mean of ln x):

σ (sd of ln x):

PDF at x = ?:

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1. What is Lognormal Distribution?

A lognormal distribution is a probability distribution of a random variable whose logarithm is normally distributed. It's used to model data that are positively skewed, such as income distributions, stock prices, or particle sizes.

2. How Does the Calculator Work?

The calculator estimates the parameters of a lognormal distribution from sample data:

\[ \mu = \text{mean}(\ln x), \quad \sigma = \text{sd}(\ln x) \]

Where:

\( \mu \) — Mean of the natural logarithms of the data
\( \sigma \) — Standard deviation of the natural logarithms
\( x \) — Input value for probability density function (PDF)

Explanation: The calculator transforms the sample data by taking natural logarithms, then calculates the mean and standard deviation of these log values.

3. Importance of Lognormal Parameters

Details: The μ and σ parameters completely define a lognormal distribution. They allow calculation of probabilities, percentiles, and other statistical measures for lognormally distributed data.

4. Using the Calculator

Tips: Enter comma-separated positive values in the sample field. All values must be positive. The calculator will compute μ and σ, and evaluate the PDF at your specified x value.

5. Frequently Asked Questions (FAQ)

Q1: What types of data follow lognormal distribution?
A: Many naturally occurring phenomena like income distributions, city sizes, particle sizes in aerosols, and latency periods of diseases often follow lognormal distributions.

Q2: How is lognormal different from normal distribution?
A: While normal distribution is symmetric, lognormal is skewed to the right and only defined for positive values.

Q3: What does the PDF value represent?
A: The probability density function (PDF) value at x gives the relative likelihood of the random variable taking that value.

Q4: Can I use this for small sample sizes?
A: The calculator works with any sample size ≥2, but estimates become more reliable with larger samples.

Q5: What if my data contains zeros or negative values?
A: Lognormal distribution requires strictly positive values. The calculator automatically filters out non-positive values.