## Introduction

In general, a statistical analysis of univariate data starts with a histogram. If the histogram shows a pattern with only positive values and is right skewed, 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 based on the normal distribution for 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

The construction of the lognormal probability plot starts with a simple x-y scatter plot. The horizontal and vertical plotting coordinates need to be determined for each data value. The horizontal coordinate is the natural logarithm of the data. The vertical coordinate, the cumulative probability is calculated for each data value using the median rank and the plotting coordinate is based on the inverse of the standard normal. The process is:

- 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 $-F(i)=norm.inv(P(i))-$

If the data is lognormally distributed, the scatter plot will tend to follow a straight-line, figure 5.

Figure 7

There are some deviations from linearity at the ends and random variation about the trend line. However, the bulk of the data follows a linear trend.

Figure 5, which is basically a scatter plot, is rather difficult to interpret. But, using a logarithmic axis for the horizontal axis displays the original data values. The vertical axis is the inverse of the cumulative standard normal. Improve readability by calculating tick marks and labels based on the inverse of the cumulative standard normal for $-p = 0.01, 0.05, 0.1,\ldots, 0.9, 0.95, 0.99-$,and replacing the original y-scale.

Relabeling scales did not change the plotting coordinates of the sample data. Only the appearance of the axes are changed. The plot is now a lognormal probability plot, shown in figure 6.

Figure 8

Generally, probability plots are enhanced by adding a best fit trend line and confidence limits. Sometimes basic statistics, including a goodness of fit using numerical methods. There remains the question of how to treat incomplete (censored) data sets. For example, some items in a reliability test may have been terminated early for reasons other than part failure or test completion. Each of these topics may employ different methods so deserves a separate article. They are not discussed here.

## Conclusions:

A lognormal probability plot is a scatter plot that uses a logarithmic horizontal scale and a standard normal inverse of the cumulative probability for the vertical axis. Data, that is lognormally distributed and plotted on lognormal probability paper, will tend to follow a straight line. The trend allows one to project the cumulative probabilities.

Dennis Craggs, Consultant

Quality, Reliability and Analytics Services

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