ICE — Lognormal (Histogram X + ln(X) PDFs, Toggle + Explanations)

1) Data (positive values)

Interpretation example for students: waiting times (minutes) at a very popular restaurant. Values are positive and right-skewed (many short waits, a few long waits).

n: — | mean (raw X): — | sd (raw X): —

μ̂, σ̂ = —

μ̂ and σ̂ are the mean and sd of ln(X) (log space). We fit on ln(X), then interpret back on the original minutes scale.

2) Histogram (raw X) + Lognormal PDF

Paste data and click Analyze to draw the histogram.

Bars are density-scaled; smooth line is Lognormal(μ̂,σ̂) PDF.

3) Why Lognormal? Show ln(X) is ~Normal

Logic: If \(Y=\ln X\) ≈ Normal(μ,σ²), then \(X\) is Lognormal(μ,σ). This explains the long right tail on the original scale.

After Analyze, we’ll draw the histogram of ln(data) + Normal(μ̂,σ̂) PDF.

Bell-shaped ln(X) supports using the lognormal model for X.

4) Math Work — Estimating μ̂ and σ̂

Step 1: Transform \(y_i=\ln(x_i)\).

Step 2 (MLEs): \(\hat\mu=\bar{y}\), \(\hat\sigma=\mathrm{sd}(y)\).

From your data: μ̂ = —, σ̂ = —

Back to X: \(E[X]=e^{\mu+\sigma^2/2}\), \(\mathrm{Median}[X]=e^{\mu}\), \(\mathrm{Mode}[X]=e^{\mu-\sigma^2}\).

On X scale: mean ≈ —, median ≈ —, mode ≈ —

5) Four Core Probabilities

μ̂, σ̂ = —

A) P(X<5)

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B) P(3<X<5)

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C) P(X>5)

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D) P(X>7)

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Summary for students:

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6) Excel — Replicate the Results

μ̂: =AVERAGE(LN(A2:A101))
σ̂: =STDEV.S(LN(A2:A101))
Mean on X: =EXP( μ̂ + (σ̂^2)/2 )
Median on X: =EXP( μ̂ )
Mode on X: =EXP( μ̂ - σ̂^2 )

P(X<5): =LOGNORM.DIST(5, μ̂, σ̂, TRUE)
P(3<X<5): =LOGNORM.DIST(5, μ̂, σ̂, TRUE) - LOGNORM.DIST(3, μ̂, σ̂, TRUE)
P(X>5): =1-LOGNORM.DIST(5, μ̂, σ̂, TRUE)
P(X>7): =1-LOGNORM.DIST(7, μ̂, σ̂, TRUE)