Transforming, averaging and transforming back can produce a different summary from averaging the original values.
The scale changes the statistical quantity
For positive data, exponentiating the average of natural logarithms gives the geometric mean. That differs in general from the arithmetic mean of the original observations. In a lognormal model, the parameter corresponding to the mean on the log scale relates to the original-scale median, while the original-scale mean requires additional distribution information.
A hypothetical two-value calculation
Take positive illustrative values 2 and 8. Their arithmetic mean is 5. Exponentiating the average of their natural logarithms gives the square root of 16, which is 4. Both computations are correct; they produce different summaries.
The example does not assume that two values establish a lognormal distribution. It also gives no HM product-property range or acceptance requirement. Its purpose is to expose a spreadsheet-label mistake before transformed results enter an equipment comparison.
Label the analysis and reported quantity
Retain the original data and units, transformation definition, any reference or shift used, the log base and the quantity being estimated. Discuss how zero, negative or censored observations are handled; they should not be silently replaced merely to permit a logarithm.
Ask whether the buyer’s requirement concerns a typical multiplicative value, an arithmetic average or another specified characteristic. Return the appropriate named estimate with uncertainty and assumptions. A transformation can be useful to an analysis without making every original-scale summary interchangeable.
Trial discussion worksheet
This blank worksheet is for your own project. It contains no H M machine trial result.
| Review question | Reference or observation to retain |
|---|---|
| What transformation was used? | Base and any reference or shift |
| Which original-scale quantity is reported? | Geometric mean, median or arithmetic mean |
| Were all original values retained? | Identified handling of nonpositive or censored observations |
Customer Questions
Is the exponential of a log mean always the arithmetic mean?
No. For positive unshifted observations it gives the geometric mean.
Do two positive observations prove lognormality?
No. They illustrate arithmetic, not a distribution assessment.
Can zeros be changed without explanation?
No. Preserve original values and document the analytical treatment.
Primary References
These references support the technical principles discussed in this guide. The worked examples and review questions are educational.
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