T.TEST

Microsoft publishes =T.TEST(A2:A10,B2:B10,2,1) = 0.196016 ("Probability associated with a Student's paired t-Test, with a two-tailed distribution"). DERIVATION (nothing here comes from a statistics package's t-test routine): the paired t-statistic is dbar/(sd/sqrt(n)) over the nine pairwise differences -3,-15,2,6,-5,-3,-3,-13,4, whose mean is -30/9 and whose sample standard deviation uses the n-1 divisor; that gives t = -1.4106912317171967 on 8 degrees of freedom. The p-value is then 2*(1 - F(|t|)) evaluated on this batch's doubly-cross-checked Student t CDF -- the regularized incomplete beta AND an independent numerical quadrature of the t density, agreeing to 25 digits -- giving 0.1960157849252821063357. scipy.stats.ttest_rel reproduces both the statistic and the p-value to double precision as a third check. That is 0.196016 at the published precision. Microsoft's page was read live on 2026-08-31 at https://support.microsoft.com/en-us/excel/functions/t-test-function. FETCH NOTE FOR THIS BATCH: batch F recorded every /en-us/office/<name>-function-<guid> URL returning Microsoft's 'Sorry, the page you're looking for can't be found' body; today BOTH paths serve the article again -- the GUID form redirects to /en-us/excel/functions/<slug>-function -- and every page cited in this batch came back complete (217-223 KB) in one or two attempts, with no 87 KB stub responses. Nothing here is sourced from a search snippet or a mirror.

Asserted at twelve places from the derivation: 0.196015784925.

The page publishes it outright: "The value returned by T.TEST when tails=2 is double that returned when tails=1." A structural assertion with no derived constant. For the record the one-tailed value derives to 0.09800789246264105316783, exactly half the headline figure.

MICROSOFT PUBLISHES ONLY THE type = 1 ROW; this value is entirely ours, derived so that the `type` argument is actually pinned down rather than assumed. Type 2 is documented as "Two-sample equal variance (homoscedastic)": the pooled variance over both samples on n1+n2-2 = 16 degrees of freedom gives t = -1.3622298275610 and a two-tailed p of 0.1919958867603962174402, i.e. 0.19199588676 at twelve places. It sits within 0.005 of the paired figure, which is precisely why a case is needed: an engine that silently ran the wrong type would still look plausible against the published number.

Also ours, not Microsoft's. Type 3 is documented as "Two-sample unequal variance (heteroscedastic)", i.e. Welch's test: the same t = -1.3622298275610 as type 2 (the two samples are the same size, so the statistic coincides) but on the Welch-Satterthwaite degrees of freedom 10.255208532, giving 0.2022939233686778922367. THE t STATISTICS BEING EQUAL WHILE THE p VALUES DIFFER IS THE WHOLE CONTENT OF THIS CASE: it isolates the degrees-of-freedom calculation, which is the only thing separating types 2 and 3 on equal-sized samples, and which no published Microsoft figure exercises anywhere.

The Remarks publish: "If array1 and array2 have a different number of data points, and type = 1 (paired), T.TEST returns the #N/A error value." The restriction is specific to the paired type -- types 2 and 3 accept unequal sample sizes -- which is why this case names type 1.

The Remarks publish: "If tails is any value other than 1 or 2, T.TEST returns the #NUM! error value."

The Remarks publish: "If tails or type is nonnumeric, T.TEST returns the #VALUE! error value." Note the pair of rules together: a non-numeric tails is #VALUE! while a numeric-but-invalid tails is #NUM!, and this corpus asserts both because engines routinely collapse them into one code.

Microsoft publishes =T.TEST(A2:A10,B2:B10,2,1) = 0.196016 ("Probability associated with a Student's paired t-Test, with a two-tailed distribution"). DERIVATION (nothing here comes from a statistics package's t-test routine): the paired t-statistic is dbar/(sd/sqrt(n)) over the nine pairwise differences -3,-15,2,6,-5,-3,-3,-13,4, whose mean is -30/9 and whose sample standard deviation uses the n-1 divisor; that gives t = -1.4106912317171967 on 8 degrees of freedom. The p-value is then 2*(1 - F(|t|)) evaluated on this batch's doubly-cross-checked Student t CDF -- the regularized incomplete beta AND an independent numerical quadrature of the t density, agreeing to 25 digits -- giving 0.1960157849252821063357. scipy.stats.ttest_rel reproduces both the statistic and the p-value to double precision as a third check. That is 0.196016 at the published precision. Microsoft's page was read live on 2026-08-31 at https://support.microsoft.com/en-us/excel/functions/t-test-function. FETCH NOTE FOR THIS BATCH: batch F recorded every /en-us/office/<name>-function-<guid> URL returning Microsoft's 'Sorry, the page you're looking for can't be found' body; today BOTH paths serve the article again -- the GUID form redirects to /en-us/excel/functions/<slug>-function -- and every page cited in this batch came back complete (217-223 KB) in one or two attempts, with no 87 KB stub responses. Nothing here is sourced from a search snippet or a mirror.

Asserted at twelve places from the derivation: 0.196015784925.

The page publishes it outright: "The value returned by T.TEST when tails=2 is double that returned when tails=1." A structural assertion with no derived constant. For the record the one-tailed value derives to 0.09800789246264105316783, exactly half the headline figure.

MICROSOFT PUBLISHES ONLY THE type = 1 ROW; this value is entirely ours, derived so that the `type` argument is actually pinned down rather than assumed. Type 2 is documented as "Two-sample equal variance (homoscedastic)": the pooled variance over both samples on n1+n2-2 = 16 degrees of freedom gives t = -1.3622298275610 and a two-tailed p of 0.1919958867603962174402, i.e. 0.19199588676 at twelve places. It sits within 0.005 of the paired figure, which is precisely why a case is needed: an engine that silently ran the wrong type would still look plausible against the published number.

Also ours, not Microsoft's. Type 3 is documented as "Two-sample unequal variance (heteroscedastic)", i.e. Welch's test: the same t = -1.3622298275610 as type 2 (the two samples are the same size, so the statistic coincides) but on the Welch-Satterthwaite degrees of freedom 10.255208532, giving 0.2022939233686778922367. THE t STATISTICS BEING EQUAL WHILE THE p VALUES DIFFER IS THE WHOLE CONTENT OF THIS CASE: it isolates the degrees-of-freedom calculation, which is the only thing separating types 2 and 3 on equal-sized samples, and which no published Microsoft figure exercises anywhere.

The Remarks publish: "If array1 and array2 have a different number of data points, and type = 1 (paired), T.TEST returns the #N/A error value." The restriction is specific to the paired type -- types 2 and 3 accept unequal sample sizes -- which is why this case names type 1.

The Remarks publish: "If tails is any value other than 1 or 2, T.TEST returns the #NUM! error value."

The Remarks publish: "If tails or type is nonnumeric, T.TEST returns the #VALUE! error value." Note the pair of rules together: a non-numeric tails is #VALUE! while a numeric-but-invalid tails is #NUM!, and this corpus asserts both because engines routinely collapse them into one code.

Microsoft publishes =T.TEST(A2:A10,B2:B10,2,1) = 0.196016 ("Probability associated with a Student's paired t-Test, with a two-tailed distribution"). DERIVATION (nothing here comes from a statistics package's t-test routine): the paired t-statistic is dbar/(sd/sqrt(n)) over the nine pairwise differences -3,-15,2,6,-5,-3,-3,-13,4, whose mean is -30/9 and whose sample standard deviation uses the n-1 divisor; that gives t = -1.4106912317171967 on 8 degrees of freedom. The p-value is then 2*(1 - F(|t|)) evaluated on this batch's doubly-cross-checked Student t CDF -- the regularized incomplete beta AND an independent numerical quadrature of the t density, agreeing to 25 digits -- giving 0.1960157849252821063357. scipy.stats.ttest_rel reproduces both the statistic and the p-value to double precision as a third check. That is 0.196016 at the published precision. Microsoft's page was read live on 2026-08-31 at https://support.microsoft.com/en-us/excel/functions/t-test-function. FETCH NOTE FOR THIS BATCH: batch F recorded every /en-us/office/<name>-function-<guid> URL returning Microsoft's 'Sorry, the page you're looking for can't be found' body; today BOTH paths serve the article again -- the GUID form redirects to /en-us/excel/functions/<slug>-function -- and every page cited in this batch came back complete (217-223 KB) in one or two attempts, with no 87 KB stub responses. Nothing here is sourced from a search snippet or a mirror.

Asserted at twelve places from the derivation: 0.196015784925.

The page publishes it outright: "The value returned by T.TEST when tails=2 is double that returned when tails=1." A structural assertion with no derived constant. For the record the one-tailed value derives to 0.09800789246264105316783, exactly half the headline figure.

MICROSOFT PUBLISHES ONLY THE type = 1 ROW; this value is entirely ours, derived so that the `type` argument is actually pinned down rather than assumed. Type 2 is documented as "Two-sample equal variance (homoscedastic)": the pooled variance over both samples on n1+n2-2 = 16 degrees of freedom gives t = -1.3622298275610 and a two-tailed p of 0.1919958867603962174402, i.e. 0.19199588676 at twelve places. It sits within 0.005 of the paired figure, which is precisely why a case is needed: an engine that silently ran the wrong type would still look plausible against the published number.

Also ours, not Microsoft's. Type 3 is documented as "Two-sample unequal variance (heteroscedastic)", i.e. Welch's test: the same t = -1.3622298275610 as type 2 (the two samples are the same size, so the statistic coincides) but on the Welch-Satterthwaite degrees of freedom 10.255208532, giving 0.2022939233686778922367. THE t STATISTICS BEING EQUAL WHILE THE p VALUES DIFFER IS THE WHOLE CONTENT OF THIS CASE: it isolates the degrees-of-freedom calculation, which is the only thing separating types 2 and 3 on equal-sized samples, and which no published Microsoft figure exercises anywhere.

The Remarks publish: "If array1 and array2 have a different number of data points, and type = 1 (paired), T.TEST returns the #N/A error value." The restriction is specific to the paired type -- types 2 and 3 accept unequal sample sizes -- which is why this case names type 1.

The Remarks publish: "If tails is any value other than 1 or 2, T.TEST returns the #NUM! error value."

The Remarks publish: "If tails or type is nonnumeric, T.TEST returns the #VALUE! error value." Note the pair of rules together: a non-numeric tails is #VALUE! while a numeric-but-invalid tails is #NUM!, and this corpus asserts both because engines routinely collapse them into one code.