ATANH in Zoho Sheet: Inverse Hyperbolic Tangent and Transformations

=ATANH(number)

Zoho’s official ATANH help documents one mandatory argument and the strict range. Its own example outputs include =ATANH(-0.3) → −0.309519604, =ATANH(0.9) → 1.47221949, and =ATANH(1/3) → 0.34657359. Zoho states #NUM! when the number is not between −1 and 1. This guide records those vendor claims, but has not executed the formulas inside the Zoho Sheet product.

There is no optional mode or base argument. The result is negative for negative valid input, zero for input zero, and positive for positive input. The function becomes increasingly steep near ±1, so even a small input difference near a boundary can matter considerably. Those are mathematical properties, not engine-specific precision guarantees.

Enter the input values below in D2:D9. In E2 type =ATANH(D2) and fill the formula through E9. The independently calculated values below display nine decimal places.

Steps to try in your own spreadsheet: (1) Inspect imported D2:D9 and confirm all values are numeric. (2) Enter =ATANH(D2) in a free column E2. (3) Fill downward to E9, keeping the raw inputs visible. (4) Compare negative and positive pairs; the output should change sign symmetrically. (5) Check the exact zero row (D6=0), which should produce mathematical zero. (6) Create separate test cells for ATANH(1) and ATANH(-1) and check the application’s actual error display; those were not natively tested in this preparation.

Each expected output was calculated with Python’s math.atanh and verified a second way using the logarithmic identity shown later. Inverting it with math.tanh returned the source value to within numerical tolerance. These checks validate the mathematics, not Zoho’s input coercion, formatting or floating-point implementation.

If a probability p is strictly between 0 and 1, its log odds (also called the logit) is:

logit(p) = LN(p/(1-p)) = 2 * ATANH(2*p-1)

Suppose a model assigns probability 0.80 to an event. Put 0.8 in B2 on a separate probability worksheet, then use:

=2*ATANH(2*B2-1)

The independently calculated log odds are 1.386294361. Another formula for the same mathematical transformation is =LN(B2/(1-B2)). The ATANH version is particularly interesting because it connects a probability scale to a symmetric interval centered at 0.5. When B2=0.5, the transformed input is zero and log odds are zero; probabilities above 0.5 give positive log odds, and probabilities below 0.5 give negative log odds.

To recover a probability from the log odds in C2, the mathematical inverse is:

=(TANH(C2/2)+1)/2

For C2=1.386294361, the independently expected probability is 0.8 to displayed precision. The identity is valid only for 0<p<1, and the corresponding log-odds input may be any finite real value. Avoid using probabilities expressed as whole percentages (80 rather than 0.8): 80 is not a valid probability for this formula.

A guarded, illustrative Zoho-style expression is:

=IF(OR(B2<=0;B2>=1);"CHECK PROBABILITY";2*ATANH(2*B2-1))

The intended result for B2=0.8 is 1.386294361 and for B2=0 or 1 is CHECK PROBABILITY. This combined formula, including its semicolon separators and treatment of text cells, needs native Zoho validation. Reject invalid probabilities explicitly rather than replacing 0 or 1 with arbitrary small numbers just to force a finite logit.

ATANH also forms the basis of the Fisher z transformation: for a sample Pearson correlation r with −1<r<1, set z=ATANH(r). This maps a bounded correlation coefficient onto the real line and can make uncertainty calculations more tractable under the appropriate statistical assumptions. It does not transform a correlation into a probability or prove causation.

For a purely illustrative sample with r=0.5 and sample size n=30, the transformed value is z=0.549306144. An approximate 95% interval in z-space for an appropriate independent, approximately bivariate-normal sample is:

lower_z = ATANH(r) - 1.96/SQRT(n-3)
upper_z = ATANH(r) + 1.96/SQRT(n-3)

This conventional approximation is described in SAS’s Fisher transformation documentation. Back-transform by applying TANH to each endpoint. Independent calculations yield approximately 0.170 to 0.729 on the correlation scale. That is an approximate conventional construction, not a native Zoho result or a substitute for a domain-appropriate statistical analysis. It requires n>3, makes assumptions about the sampling process, and is not appropriate without adjustment for every data design.

In a worksheet with the correlation in B2 and n in C2, proposed formulas are =TANH(ATANH(B2)-1.96/SQRT(C2-3)) and =TANH(ATANH(B2)+1.96/SQRT(C2-3)). Validate ABS(B2)<1 and C2>3 first, and confirm that the function names and dependencies are available in the target editor.

For valid −1<x<1:

ATANH(x) = 0.5 * LN((1+x)/(1-x))
TANH(ATANH(x)) = x
ATANH(-x) = -ATANH(x)

For x=0.5, the log identity evaluates LN(3)/2 = 0.549306144. For x=0.9 it yields LN(19)/2 = 1.4722194896. This helps with an independent arithmetic audit when a migration produces a suspicious value. The inverse relationship is a mathematical identity: in finite-precision spreadsheet engines, TANH(ATANH(x)) may differ from x by a tiny rounding error. Do not compare computed floating-point numbers with exact equality unless the application and purpose warrant it.

The mathematical derivative is 1/(1−x²), which becomes large near both endpoints. If x moves from 0.9 to 0.99, the result jumps substantially. That steepness is expected; it is not necessarily evidence of a broken formula. However, it makes upstream measurement precision and how values were rounded essential when interpreting output.

Zoho’s help page lists generic #REF! and #N/A! guidance as well. That does not establish those as direct ATANH arithmetic errors for valid numeric inputs. If the argument cell itself contains an upstream error or deleted reference, that error may propagate. Error propagation order, very-near-boundary rounding, import behavior and platform/account differences require native testing.

ATANH differs from TANH, which takes a real hyperbolic parameter and returns a value between −1 and 1; ATAN, which inverts ordinary tangent and gives a principal circular angle; and ACOTH, which requires absolute input greater than 1 instead. For nonzero real x with |x|>1, ACOTH(x)=ATANH(1/x) is a useful domain-transform identity. ASINH and ACOSH are other hyperbolic inverses with different domains.

Microsoft’s Excel ATANH page and Google’s Sheets ATANH help both describe a strict input between −1 and +1. This verifies a documented mathematical overlap, not identical coercion or numeric precision. Excel pages often use comma-separated examples, while Zoho multiargument examples frequently show semicolons; this one-argument ATANH syntax itself has no delimiter ambiguity. Check import/export locale behavior instead of assuming a direct copy of a compound formula will parse unchanged.

• Zoho vendor source: ATANH—Zoho Sheet Help, directly checked 2026-10-11; exactly one argument, strict range, official sample outputs and vendor #NUM! statement.
• Cross-software primary sources: ATANH—Excel and ATANH—Google Sheets, checked 2026-10-11. No Excel or Google program executions were performed either.
• Independent calculations: Python evaluated eight input rows, logarithmic identities, TANH inverse tests, probability/logit example and an illustrative Fisher z interval. Native Zoho Sheet executions: zero. Precision, parser, data-type errors and cross-platform rollout remain PENDING_NATIVE_VALIDATION. No first-supported version is claimed from current help content.