PlanMaker ERF: One- and Two-Limit Gaussian Integrals
ERF evaluates the Gaussian error function, using zero as the starting point when only one limit is supplied. Compare explicit intervals, partition checks and scaled Gaussian models.
Verification status: See the evidence and test scope described in this guide. How formula examples are checked
ERF evaluates the Gaussian error function, using zero as the starting point when only one limit is supplied. Compare explicit intervals, partition checks and scaled Gaussian models.
Availability and verification
The PlanMaker 2026 manual, 2024 manual and 2021 manual were retrieved on October 15, 2026. They document this function in those manual families. They do not establish its original introduction or a last-working-version endpoint; both historical endpoints remain unverified. Support in every FreeOffice/mobile edition or individual desktop build is outside this check.
No native PlanMaker execution was performed. Expected values below are independently checked mathematical controls, not captured application results. Formulas use the English manual’s commas; verify localized names and separators in the installed product.
Exact syntax and the omitted upper limit
=ERF(LowerLimit [, UpperLimit])
LowerLimit is required; UpperLimit is optional. With two arguments PlanMaker integrates over the supplied interval. With UpperLimit omitted, the manual says the interval runs from zero to LowerLimit. Thus ERF(1) is a zero-to-one calculation, while ERF(1,2) uses one to two. The argument name alone can be misleading in the one-argument form.
The standard normalization is erf(x)=(2/√π) times the integral of exp(−t²) from 0 to x. NIST’s definition supplies the mathematical convention used for these independent checks. For two nonnegative ordered limits a and b, the expected result is erf(b)−erf(a). This is not the integral of exp(−t²/2) without rescaling.
The checked PlanMaker entry does not spell out negative-limit acceptance, reversed-limit behavior or a complete native error domain. The controls below deliberately use nonnegative limits with the first no greater than the second. General mathematical extensions are not asserted as native PlanMaker behavior.
Complete one-limit worksheet
Create GaussianControls with x in A1 and Zero_to_x in B1. Enter the numbers below in A2:A7, then fill B2 =ERF(A2) through B7. Format B to at least twelve decimal places for comparison.
| Row | A: x | Independent expected B |
|---|---|---|
| 2 | 0 | 0.000000000000 |
| 3 | 0.25 | 0.276326390168 |
| 4 | 0.5 | 0.520499877813 |
| 5 | 1 | 0.842700792950 |
| 6 | 1.5 | 0.966105146475 |
| 7 | 2 | 0.995322265019 |
In a copy change A5 from 1 to 0.75. B5 then has expected value 0.711155633654. The other rows stay unchanged. The outputs are dimensionless values of a named mathematical function; they are not percentages unless the surrounding model assigns that interpretation.
These values increase toward one for increasing nonnegative inputs. The smaller increments near two reflect the decreasing integrand, not an error in the input spacing. Twelve displayed decimals are reference rounding, not a certified native precision specification.
Complete two-limit worksheet
Create GaussianIntervals with Lower in A1, Upper in B1 and Integral in C1. Fill C2 =ERF(A2,B2) through C6 using these numeric pairs.
| Row | A: lower | B: upper | Independent expected C |
|---|---|---|---|
| 2 | 0 | 0.5 | 0.520499877813 |
| 3 | 0.5 | 1 | 0.322200915137 |
| 4 | 1 | 1.5 | 0.123404353526 |
| 5 | 1.5 | 2 | 0.029217118544 |
| 6 | 1 | 1 | 0.000000000000 |
The first four intervals partition zero through two. C8 =C2+C3+C4+C5 has mathematical expectation 0.995322265019, matching =ERF(2) after rounding. The equal-limit control has no interval width and hence zero integral.
Advanced: audit an interval two ways
In D2 use =ERF(B2)-ERF(A2) and fill through D6. Each value should agree mathematically with C on the same row. The difference of two function values provides an independent formulation, not a guarantee that both native computational paths have identical last bits.
For the interval 0.5 to 1.5, =ERF(0.5,1.5) has expected result 0.445605268662. Combining the adjacent intervals 0.5–1 and 1–1.5 gives the same quantity. If a workbook differs materially, check argument order and the one- versus two-argument form before changing formatting.
Advanced: scale a symmetric Gaussian model
Suppose a purely mathematical model has a standard normal variable Z. Its mass between −z and z for z≥0 is erf(z/√2), obtained by changing variables in the normal-density integral. Enter z values 1,2,3 in F2:F4 and G2 =ERF(F2/(2^0.5)), filled down.
Independent expected G values are 0.682689492137, 0.954499736104, 0.997300203937. Without the √2 scaling, ERF(1) is about 0.842701 and does not answer the one-standard-deviation question. The formula is conditional on the model being standard normal; it does not establish that a real dataset follows that distribution.
Errors, precision and troubleshooting
The short reference does not supply exact errors for text, blanks, arrays, extreme values or invalid intervals. No such errors are invented here. Also distinguish the mathematical erf, which has a wider domain, from the finite range and coercion behavior of the installed spreadsheet.
Use raw cell precision when adding partition results; adding displayed rounded text can introduce rounding differences. Subtracting nearly equal function values can lose relative accuracy for very small intervals. This guide does not certify PlanMaker’s numerical algorithm or extreme-tail accuracy. Reproduce the visible controls in the installed build and record platform, language and workbook format before relying on a tighter error budget.