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LOGNORMDIST in Apple Numbers: Lognormal CDF and threshold probability

Statistical Intermediate Apple Numbers

Calculate a positive-variable lognormal cumulative probability in Numbers, using the logarithmic mean and population SD.

Verification status: Official documentation referenced; example outcomes independently reasoned or arithmetically checked where stated. Not executed in Apple Numbers. How formula examples are checked

Quick answer: =LOGNORMDIST(0.78,1.7,2.2) returns approximately 0.187899237956868 in Apple’s example. The result is the modelled probability P(X ≤ 0.78) when LN(X) is normally distributed with mean 1.7 and population standard deviation 2.2; the parameters are not the mean and SD of X itself. [A1, C1]

Purpose and when to use it

Use LOGNORMDIST for strictly positive measurements where it is reasonable to model the natural logarithm of the measurement as normal, for example a positively skewed elapsed time or a multiplicative growth factor. The formula evaluates the cumulative distribution at a threshold: it does not fit the distribution to observed records, count data rows, or return the lognormal density. Compare an upper-tail probability via 1-LOGNORMDIST(...); for a threshold at a prescribed percentile use LOGINV. [A1, A2, C1]

Exact syntax, argument order, types and defaults

=LOGNORMDIST(pos-x-value, mean, stdev)
Position Apple’s argument Type Required Meaning / bounds / default
1 pos-x-value Number value Yes Positive threshold x > 0; no default.
2 mean Number value Yes Mean of the natural logs LN(X); not the arithmetic mean of original X. No default.
3 stdev Number value Yes Positive population SD of LN(X) (stdev > 0); no default.

All three arguments are required in Apple’s displayed signature. Unlike GAMMADIST or NORMDIST, this function has no function-type switch; its returned quantity is a CDF, between 0 and 1 for valid finite inputs. The inverse is LOGINV(probability,mean,stdev), not 1/LOGNORMDIST(x,...). [A1, A2]

Mathematical check, not an execution claim: If μ=mean, σ=stdev, then LOGNORMDIST(x,μ,σ) = Φ((ln(x)−μ)/σ) for x>0, where Φ is the standard normal CDF. This identity explains why changing μ or σ on the log scale shifts/skews the original-scale distribution. The result is a probability rather than a density height. [C1]

Reproducible Numbers example: time to complete an online task

Create a Numbers table named TaskTimes, with B1 Threshold x, C1 Log mean, D1 Log population SD, and E1 CDF. Enter these numbers in rows 2–6, then enter the row-2 formula in E2 and fill down. One observation per row is a threshold to model, not a row of sample data used to estimate parameters. For this illustration μ=1, σ=0.5 and the x values may be interpreted as minutes. [A1, A5, C1]

Row B: threshold x C: μ D: σ E: formula Expected CDF
2 0.5 1 0.5 =LOGNORMDIST(B2,C2,D2) 0.000354216744666
3 1 1 0.5 =LOGNORMDIST(B3,C3,D3) 0.0227501319482
4 2 1 0.5 =LOGNORMDIST(B4,C4,D4) 0.269704930735
5 4 1 0.5 =LOGNORMDIST(B5,C5,D5) 0.780117089512
6 8 1 0.5 =LOGNORMDIST(B6,C6,D6) 0.984570380702

At a 4-minute threshold the model predicts about 78.01% complete by that time; it does not claim 78.01% of a measured sample completed. A time of 0 is outside Apple’s documented x domain, even though the mathematical limiting CDF as x approaches zero is 0. [A1, C1]

Apple’s original published example, independently reproduced

=LOGNORMDIST(0.78,1.7,2.2)

Apple states 0.187899237956868. Independent normal-CDF evaluation of z = (LN(0.78)−1.7)/2.2 yields 0.187899237956869. The independent computation supports the published rounding but does not prove that a particular Numbers installation produces exactly the same last digit. [A1, C1]

Advanced examples and realistic interpretations

Upper tail: tasks taking longer than a cutoff

=1-LOGNORMDIST(4,1,0.5)

Expected 0.219882910488, or about 21.99%. This is P(X>4) under the model. For extreme right-tail thresholds numerical cancellation can occur if a CDF is rounded to exactly 1; the actual Numbers precision limit at extremes was not tested. [C1]

Interval probability between 2 and 4 minutes

=LOGNORMDIST(4,1,0.5)-LOGNORMDIST(2,1,0.5)

Expected 0.510412158777. Subtract two cumulative probabilities, not the two thresholds and not two unrelated populations. Inclusive or exclusive endpoints have zero probability mass for this continuous mathematical model. [C1]

Confirm a model median, with the correct scale

For a lognormal model, the median of X is EXP(μ). With μ=1, the median is EXP(1) ≈ 2.71828182846 in the original positive units. Thus:

=LOGNORMDIST(EXP(1),1,0.5)

The mathematical expectation is 0.5. By contrast, mean=1 is not the mean of X, which is EXP(μ+σ^2/2) ≈ 3.08021684892 for this model. Both relationships are external mathematical checks, not additional undocumented Apple function features. [C1]

Validate a cutoff with the inverse (for review, not an alias)

=LOGINV(0.9,1,0.5)
=LOGNORMDIST(LOGINV(0.9,1,0.5),1,0.5)

The first has expected positive value 5.15917035562 and the second returns 0.9 up to calculation precision. These are separate Apple functions with different argument positions and unique guide URLs; do not create a second page for a translated alias. [A2, C1]

Locale-specific function spelling, one canonical page

English-language examples use the identifier LOGNORMDIST, decimal periods and comma-separated arguments. Apple’s Portuguese-Brazil help calls the corresponding function DIST.LOGNORMAL and shows locale syntax of the form =DIST.LOGNORMAL(0,78; 1,7; 2,2)—semicolons between arguments, decimal commas within numbers. This is a localization of Apple’s same function help identifier. Always use the formula editor and language settings of the installed Numbers version. [A3]

Numbers-specific references and formula entry

Build each example as a Numbers table, not an assumed Excel worksheet: enter row-1 headings, type the numeric values in the listed cells, enter each formula beginning with =, and fill down only where instructed. B2 addresses a cell in the current table; copied relative references change by row. A locked reference such as $C$2 keeps a shared parameter fixed. Apple documents that a sheet can contain multiple named tables; if the source is in another table named Model, select the actual cell in the Formula Editor or use a table-qualified reference such as Model::B2. Do not assume a single global sheet grid. [A5]

For LOGINV and GAMMAINV, keep probability inputs as numeric values strictly between 0 and 1, not 90 for a 90% target. You may format cumulative probability results as Percent, but formatting does not alter a mathematical value. The LOGNORMDIST and GAMMADIST first arguments are observation thresholds, not probabilities. Keep parameters numeric and do not append units inside formula input cells. If copied formulas point to the wrong table or a moving shared parameter, inspect the reference tokens in Numbers. [A5]

Errors, input limits and troubleshooting

Apple’s individual function help specifies permitted argument domains; it does not document a stable displayed error token for every invalid input, device, locale, or cell-type conversion. Consequently the following are predicted invalid inputs or user misunderstandings, not observed Numbers errors. No #... token or screenshot is invented. [A1, A4]

Symptom / input Reason Correction
=LOGNORMDIST(0,1,0.5) Apple requires x strictly positive. Use a positive threshold; do not substitute zero automatically.
=LOGNORMDIST(-2,1,0.5) A negative original-scale observation is outside the lognormal model. Fix the data or use an appropriate alternative model.
=LOGNORMDIST(2,1,0) or (...,-0.5) Log-scale population SD must be >0. Calculate a positive log-scale SD.
Using mean/SD of raw X in columns C and D Parameters describe LN(X), not X. Transform sample values to logs for parameter estimation, or use properly derived log-scale parameters.
CDF mistaken for density height LOGNORMDIST only exposes the cumulative distribution. Interpret as P(X≤threshold), not as a density.
=LOGNORMDIST(2,1) or text in numeric cells A required numeric argument is missing / mismatched. Supply all three numeric arguments in the documented order.
English formula fails under Brazilian locale Separators and translated function spelling may differ. Use the localized Functions Browser and separators.
A displayed value looks like 0.78 instead of 0.780117… Number formatting may hide digits. Increase decimals before comparison.

Compatibility, support and evidence boundaries

The function’s English signature and examples are from Apple Formulas and Functions Help 15.4, inspected 2026-10-13 UTC. Apple also documents formula editing and references for Numbers; the Help navigation covers Apple’s iWork formula ecosystem, but the existence of a current documentation article is not proof of native execution on Mac, iPad, iPhone, iCloud, or older versions. Apple publishes app release documentation, which must not be read as evidence that this particular function debuted in that release. Its earliest supported Numbers version was not independently established. [A1, A4, A5, A6]

Do not transfer Excel, Google Sheets or other spreadsheet engines’ default arguments, error codes, statistical parameter definitions, locale punctuation, or precision guarantees to Numbers. Finite-precision corner cases, text/blank coercion, large-parameter stability, and exact error messages require native confirmation. These guides were checked against documentation and by independent numerical computation only; there was no Apple Numbers execution, import/export test, screenshot or historical-platform regression test. Intermediate expected results below are rounded for readability; a Numbers cell’s number format may show fewer digits. [A1, C1]

  • LOGNORMDIST — Apple’s exact English arguments and published numeric example. [A1]
  • LOGINV — inverse: obtain x for a target cumulative probability. [A2]
  • LN — natural log needed to interpret the model’s parameters. [A2]
  • EXP — undo the log transform for lognormal median calculations. [A2]
  • NORMDIST — normal CDF or density of logarithmic standardized values; different X-scale distribution. [C1]

Claim-level official references and validation sources