Lognormal Probability Plots

Introduction

In general, a statistical analysis of univariate data starts with a histogram. If the histogram doesn’t show a bell shape, the data probably does not follow a normal distribution. If the logarithm of the data plots as a normal histogram, then the data is lognormally distributed. Any statistical projections and parameter estimates are based on the normal distribution of the log of the data.  This article focuses on the lognormal distribution and the lognormal probability plot.

Applications:

A normal probability distribution is defined over the range $-(-\infty,\infty)-$ for any value of $-\mu-$ and $-\sigma-$. This means that there is a finite probability of data in the range $-(\infty,0]-$. However, sometimes physical constraints allow only positive data. Then a non-normal distribution is required for reasonable probability predictions.

Some examples of lognormal data are:

• Customer usage data like vehicle trips/day, refrigerator open/close cycles, …
• Time to repair of a maintainable system.
• Particle size distributions in chemistry
• Rainfall distributions
• Biological measurements, i.e., blood pressures of males or females, size of appendages like teeth, hair, claws, …
• Human behavior, i.e., dwell time on articles/jokes, length of chess games…

Fundamentals

In the histogram, figure 1, the x-values are right skewed. For this data,  $-\bar(x)=2.99-$ and $-s=1.51-$. The average is about 2 standard deviations larger than 0 so $-Pr(x\le 0)\approx 0.15-$. The data does not appear to be normally distributed because there are not any values $-\leq 0 and being right skewed. The lognormal should be checked.  

Figure 1

Using $-y=ln(x)-$ to transform the data, the histogram of Y is

Figure 2

The y histogram shows the symmetric (bell) shape that is characteristic of a normal distribution. With this information, one can assume the y-data is normally distributed. Statistical tests for normality can be used to verify normality at a specified confidence. 

Distributions

A quick analysis of the Y data shows $-\bar(x)=0.9757-$ and $-s=0.4976-$. Assuming $-\mu=1-$ and $-\sigma=0.5-$, then the probability density plot of y is

Figure 3

The lognormal probability density for the X data with lognormal parameters $-\mu=1-$ and $-\sigma=0.5-$ is:

Figure 4

This plot approximates the shape of the x histogram.

Distributions

For y, the normal probability density f(y) is:

$$\begin{array}{cc}f(y)=\frac{1}{\sigma\sqrt{2\pi}}e^{(x-\mu)^2/2\sigma^2} & for & y \in (-\infty,\infty)\end{array}$$

(1)

And the cumulative normal probability density F(y) is:

$$\begin{array}{cc}F(y)=\int_{-\infty}^y f(\phi)d\phi & for & y \in (-\infty,\infty)\end{array}$$

(2)

In equations 1 and 2, negative values of y are allowed.

For x, the lognormal probability density f(x) is:

$$\begin{array}{cc}f(y)=\frac{1}{x\sigma\sqrt{2\pi}}e^{(ln(x)-\mu)^2/2\sigma^2} & for & x \in [0,\infty)\end{array}$$

(3)

And the cumulative normal probability density F(x) is:

$$\begin{array}{cc}F(x)=\int_{-\infty}^x f(\phi)d\phi & for & x \in [o,\infty)\end{array}$$

(4)

In equations 3 and 4, only positive values of x are allowed. In equations 2 and 4, $-\phi-$ is a dummy variable of integration.

Percentiles

Using equation 2, the cumulative probability F(y) is the area under the probability density to the left of y. Alternatively, for any desired cumulative probability $-\alpha-$, the inverse $-y^\alpha-$ may be determined. There are two basic Excel function that support these calculations:

$$\alpha = F(y) = norm.dist(y,\mu,\sigma,true)$$

(5)

$$y_\alpha = norm.inv(\alpha,\mu,\sigma)$$

(6)

Equation 5 provides the cumulative probability of y under the distribution curve, i.e., to the left of y. Equation 6 is the inverse operation. For more details, see Excel help.

On probability density plots, important probabilities are  highlighted with a vertical line at the specified variable values. For example, if the analyst is interested in the 5^th, 50^th and 95^th cumulative percentiles of the y data, then corresponding values of y are 0.18, 1.0 and 1.82:

Figure 5

The corresponding significance values for the probability density plot of x are obtained using the inverse of the natural logarithm transformation, i.e., $-x=e^{y_\alpha}-$. These x values are 1.19, 2.72, and 6.19, so vertical red bars were added to the x probability density plot.

 

Figure 6

This allows statistical statements to be made about significant x values.

Distribution estimates

The underlying population parameters are unknown, but can be estimated from data.  If a random variable  is lognormally distributed, then $-\mu-$ and $-\sigma-$ are directly estimated using

$$\mu\approx\bar{y}=\frac{1}{N}\sum_{i=1}^{i=n}ln(x)_i$$

(7)

And

$$\sigma^2\approx s_y^2=frac{1}{n-1}[ln(x_i)-\mu]^2$$

(8)

Confidence Intervals

Any sample data contains random error which limits our ability to determine the underlying population parameters. Confidence intervals that will contain the population parameters are obtained by:

• Calculate the sample mean and standard deviation, using equations 7 and 8.
• Specify the desired confidence, C. 
• Split the uncertainty equally to both tails of the distribution into a lower significance limit,

$$\alpha_{LL}=\frac{(1-C)}{2}$$

(9)

And into an upper significance limit,

$$\alpha_{UL}=\frac{(1+C)}{2}$$

(10)

• Calculate an interval containing the population mean,

$$ \bar{y}-\frac{s}{\sqrt{n}}t_{n-1,\alpha_{LL}} <\mu< \bar{y}-\frac{s}{\sqrt{n}}t_{n-1,\alpha_{UL)}} $$

(11)

• The interval containing the population standard deviation is,

$$\frac{(n-1)s^2}{\chi_{n-1,\alpha_{UL}}} <\sigma^2 \frac{(n-1)s^2}{\chi_{n-1,\alpha_{LL}}}$$

(12)

These equations can be used to select sample sizes for the analysis.

Lognormal Probability Plot

• Sort the x data from lowest to highest.
• Assign a index number to each sorted data value starting from $-i=1-$ to $-i=n-$.
• Calculate the median ranks of each point, i.e., $-MR(x_i)=(i-0.3)(n+0.4)-$. These cumulative probabilities are the median ranks $-P(x_i)=MR(i)-$.
• Calculate the vertical axis plotting coordinates using $-F(i)=norm.inv(P(i))-$

Figure 7

Figure 8

Conclusions:

Dennis Craggs, Consultant

Quality, Reliability and Analytics Services