Log Transformation and Back-Transformation: How to Report the Geometric Mean
Eight serum samples come back at 12, 18, 25, 40, 62, 95, 210, and 480 pg/mL. The arithmetic mean is 117.75 pg/mL, which sits above six of the eight values you measured. Shapiro–Wilk on the raw numbers gives p = .003. A single sample forty times larger than the smallest is doing most of the work, and a log transformation is the standard response.
The part that trips people up is the return trip rather than the transformation itself. Back-transforming the mean of the logs does not give you back the arithmetic mean. It gives you the geometric mean, and reporting it correctly means understanding why the confidence interval comes back lopsided.
Why Concentration Data Arrives Log-Normal
Many biological quantities are the product of several factors rather than their sum. Antibody titres, cytokine concentrations, bacterial counts, and expression ratios all accumulate multiplicatively — a doubling here, a threefold induction there. Sums of random effects tend toward a normal distribution; products of them tend toward a log-normal one. Right skew in a concentration series is usually the data behaving as its underlying biology dictates, not a contamination artefact.
That has a practical implication worth stating plainly: the log scale is often the scale on which the biology is additive. A treatment that raises concentrations by 50% raises them by 50% at 20 pg/mL and at 400 pg/mL, which is a constant shift in logs and a wildly non-constant shift in raw units. Transforming is not a workaround for an inconvenient distribution so much as a move onto the scale where the effect is constant.
The Transformation, Step by Step
Natural log is the convention in most biology; log₁₀ is common in pharmacology and clinical chemistry. The choice does not change any p-value or confidence interval after back-transformation — it changes only the intermediate numbers. Here it is with natural log.
| Value (pg/mL) | ln(value) |
|---|---|
| 12 | 2.4849 |
| 18 | 2.8904 |
| 25 | 3.2189 |
| 40 | 3.6889 |
| 62 | 4.1271 |
| 95 | 4.5539 |
| 210 | 5.3471 |
| 480 | 6.1738 |
The eight log values sum to 32.485, so the mean on the log scale is 32.485 / 8 = 4.0606. Their standard deviation is 1.2574, which gives a standard error of
$SE = \frac{1.2574}{\sqrt{8}} = 0.4446$Shapiro–Wilk on the log values returns p = .85. The transformation worked: the same data that failed a normality test on the raw scale passes comfortably once logged. Note what was tested here — the within-group values. For a regression or ANOVA the assumption applies to the model residuals rather than the raw outcome, a distinction that a large share of published papers gets wrong, and one we cover in our guide to normality testing before ANOVA.
Back-Transformation Gives You the Geometric Mean
Exponentiating the mean of the logs:
That 58.0 is the geometric mean. You can confirm it a second way without touching a logarithm: the geometric mean is the eighth root of the product of the eight values, and multiplying all eight and taking the eighth root also gives 58.0 pg/mL. The two routes agree because that is what the log-mean-exponentiate sequence computes.
Now compare the three candidate summaries of the same eight numbers:
| Summary | Value (pg/mL) | What it does with the 480 |
|---|---|---|
| Arithmetic mean | 117.75 | Lets it dominate; result exceeds 6 of 8 observations |
| Median | 51.0 | Ignores its magnitude entirely |
| Geometric mean | 58.0 | Counts it, on the scale where it is one step up |
The geometric mean is the useful middle position. It uses every observation, which the median does not, while refusing to let a single large value drag the centre above most of the data, which the arithmetic mean does. For right-skewed concentration data it is generally the better measure of central tendency, and it is what the standard treatment of log transformation and back-transformation reports.
Reporting “mean = 58.0 pg/mL.” It is not the mean of your data; the mean is 117.75. Call it the geometric mean, and say so in the Methods section. A reader who computes the arithmetic mean from your supplementary table and gets a different number will assume an error.
The Confidence Interval Comes Back Asymmetric, and That Is Correct
On the log scale the interval is built the ordinary way. With df = 7, the critical value is t = 2.365, so the half-width is 2.365 × 0.4446 = 1.0515 and the log-scale interval is 4.0606 ± 1.0515, or [3.0091, 5.1121].
Exponentiating both ends gives 20.3 to 166.0 pg/mL. The geometric mean of 58.0 does not sit in the middle of that: it is 37.7 units above the lower bound and 108.0 units below the upper one, nearly three times further from the top than the bottom.
Nothing has gone wrong. Exponentiation is not linear, so it cannot preserve equal spacing. What it preserves is ratios, and that is where the symmetry actually lives. Since e1.0515 = 2.862, the interval is the geometric mean divided and multiplied by the same factor:
- 58.0 ÷ 2.862 = 20.3 pg/mL
- 58.0 × 2.862 = 166.0 pg/mL
The interval is multiplicatively symmetric around the estimate even though it is additively lopsided. This is worth stating in the text when you report it, because it pre-empts the reviewer question. It also means an additive ± presentation is not available to you, because no single ± value describes an interval whose two arms differ by a factor of nearly three. Report both bounds. The same reasoning applies to error bars on a log-scaled figure, which is why bars drawn symmetrically around a back-transformed point are usually wrong — see our guide to choosing between SEM, SD, and CI for error bars.
Where Log Transformation Breaks
Zeros. ln(0) is undefined, and the usual patch is to add a constant, most often ln(x + 1). The constant is a modelling choice with consequences: 1 is arbitrary for data measured in pg/mL, where 1 pg/mL may be a substantial fraction of the assay range, and alternatives like half the smallest non-zero value give different answers. If a handful of values sit below the limit of detection, a censored-data method is a better fit than any additive constant. If a large share of values are zero, the distribution is not log-normal at all and the honest options are a zero-inflated model or a rank-based approach, covered in our comparison of Mann–Whitney and the t-test.
Choosing the transformation after seeing the p-value. Log is one rung on Tukey’s ladder of powers, where λ = 1 leaves data untouched, λ = 0.5 is the square root, λ = 0 is the log, and λ = −1 is the reciprocal, with lower rungs compressing right tails more aggressively. Box–Cox estimates λ from the data directly, and the NIST Engineering Statistics Handbook’s Box–Cox section sets out the procedure. Either route is defensible. Trying several rungs and keeping whichever produces p < .05 is not, and the fix is to fix the rule in advance: pick the transformation from the distribution’s shape and from what is conventional for the measurement, then report that you did.
Interpreting log-scale coefficients as if they were additive. A difference of 0.5 on the natural-log scale is a ratio of e0.5 = 1.65, so a treatment effect of 0.5 log units means the treated geometric mean is 1.65 times the control geometric mean. It does not mean 0.5 pg/mL, and it does not mean a 50% increase.
Reporting It on Both Scales
The Methods section needs the transformation named, the base stated, and the handling of any zeros described. The Results section needs the back-transformed numbers, because a reader cannot interpret pg/mL on a log scale. For this sample:
“Concentrations were right-skewed (Shapiro–Wilk p = .003) and were natural-log transformed prior to analysis; the transformed values were consistent with normality (p = .85). Results are reported as geometric means with 95% confidence intervals, back-transformed to pg/mL. The geometric mean concentration was 58.0 pg/mL (95% CI 20.3 to 166.0, n = 8).”
Two conventions worth following. Give the interval as both bounds rather than a ± value, for the reason above. And keep the arithmetic mean out of the Results text if the analysis was done on logs — presenting 117.75 alongside a geometric mean of 58.0 invites the reader to wonder which one the statistics were run on. Broader notation conventions are covered in our guide to reporting statistics in a biology methods section.
Planning Sample Size on the Log Scale
If the analysis will happen on the log scale, the power analysis has to as well. The effect size is a standardised difference in log units, which means your expected effect is expressed as a fold-change: a 2-fold difference between groups is ln(2) = 0.693 log units, and dividing that by the expected standard deviation of the logs gives Cohen’s d on the log scale. With the SD of 1.2574 from this sample, a 2-fold effect corresponds to d = 0.693 / 1.2574 = 0.55.
That number goes straight into an ordinary two-sample calculation, because a t-test on logged data is still a t-test. Our free sample size and power calculator handles the independent-samples case and reports achieved power alongside n, so you can check what a plausible fold-change costs before committing samples. Pilot data is the best source for the log-scale SD; it is a quantity that varies enough between assays that a borrowed value is worth treating with suspicion.
The habit that prevents most reporting errors here is a small one: decide the scale before the analysis, and write down which quantity you will report from it. Almost every back-transformation mistake is the result of that decision being made after the numbers came out.