MARGINOFERROR

GOOGLE'S OWN PUBLISHED RESULT, from the Examples table in the article body: over A1:A4 = 8, 4, 3, 6 with mean 5.25, =MARGINOFERROR(A1:A4, 0.95) -> 3.528, with the confidence interval printed as [1.722, 8.778]. DERIVATION. Google's page gives the definition in TEXT, not as an image: "MARGINOFERROR(range, confidence) is equal to CONFIDENCE.T(1 - confidence, STDEV(range), COUNT(range))." STDEV is the SAMPLE standard deviation, and CONFIDENCE.T is the Student-t form, not the normal one -- the page names CONFIDENCE.NORM only as a related link. The figures below were derived TWICE along paths sharing no code: the t density integrated by numerical quadrature to 40 digits with the critical value obtained by root-finding on that CDF, and scipy's inverse t as an independent check. The two agree to 15 significant digits. For the page's data {8, 4, 3, 6}: n = 4, mean 5.25, sample standard deviation 2.2173557826083451, t(0.025, 3) = 3.1824463052837096, margin = 3.5283078589306981. Google's MARGINOFERROR page, read live on 2026-08-31 at https://support.google.com/docs/answer/12487850. BATCH PROVENANCE (batch H, the first Sheets/LibreOffice-only batch). Every function in this batch has x == false in docs/data/compat.json: it is a Google Sheets function that Microsoft does not document at all, so this corpus makes NO claim about it in that engine and nothing here is measured against that vendor's documentation. The authority is Google's own support.google.com function page, cited by full URL and by the date it was read -- Google publishes no version number for these pages, so a bare URL dates nothing. WHAT GOOGLE ACTUALLY PRINTS, WHICH IS LESS THAN IT LOOKS: most of these pages carry a 'Sample Usage' block of FORMULAS WITH NO RESULTS. Where a value below is Google's own published output the note says so; where it is derived from the page's stated semantics the note says that instead, and says from which sentence. No value in this batch was taken from a search snippet, a blog or a mirror. LIBREOFFICE: probed on all four pinned builds (24.2.0.3, 24.8.7.2, 25.2.0.3, 25.8.7.3) in five storage spellings -- plain, _xlfn., COM.MICROSOFT., ORG.OPENOFFICE. and _xlfn.ORG.OPENOFFICE. -- and #NAME? under every one of them, so the expected values below describe Google Sheets and the LibreOffice column records absence.

DERIVED, not published: the page rounds to three decimals and stops. Asserted at six places from the independent derivation, 3.5283078589306981. This is the case that would catch a normal-distribution implementation masquerading as a t one -- the z form gives 2.1727 here, which the page's own three-decimal figure already rules out. Google's MARGINOFERROR page, read live on 2026-08-31 at https://support.google.com/docs/answer/12487850.

A structural assertion with no derived constant, taken verbatim from the page's definition bullet. It is the strongest single check available here: it holds whatever the engine's t quantiles are, and it fails if MARGINOFERROR quietly uses the population standard deviation or the normal quantile. Google's MARGINOFERROR page, read live on 2026-08-31 at https://support.google.com/docs/answer/12487850.

GOOGLE'S OWN PUBLISHED FIGURE: the Examples table prints "Lower Bound (Mean - MARGINOFERROR)" as 1.722 and the upper bound as 8.778, over mean 5.25. 5.25 - 3.5283079 = 1.7216921, which rounds to 1.722. Google's MARGINOFERROR page, read live on 2026-08-31 at https://support.google.com/docs/answer/12487850.

DERIVED, not published. The page's second Sample Usage line is MARGINOFERROR(A1:C3, 0.99) with no result and no data, so this case applies its confidence level to the data the page DOES publish. t(0.005, 3) = 5.8409093097333573 and the margin is 6.4756870168140886, derived twice as above. It checks that the confidence argument actually reaches the quantile: a hard-coded 95% level would return 3.528308 here. Google's MARGINOFERROR page, read live on 2026-08-31 at https://support.google.com/docs/answer/12487850.

GOOGLE'S OWN PUBLISHED RESULT, from the Examples table in the article body: over A1:A4 = 8, 4, 3, 6 with mean 5.25, =MARGINOFERROR(A1:A4, 0.95) -> 3.528, with the confidence interval printed as [1.722, 8.778]. DERIVATION. Google's page gives the definition in TEXT, not as an image: "MARGINOFERROR(range, confidence) is equal to CONFIDENCE.T(1 - confidence, STDEV(range), COUNT(range))." STDEV is the SAMPLE standard deviation, and CONFIDENCE.T is the Student-t form, not the normal one -- the page names CONFIDENCE.NORM only as a related link. The figures below were derived TWICE along paths sharing no code: the t density integrated by numerical quadrature to 40 digits with the critical value obtained by root-finding on that CDF, and scipy's inverse t as an independent check. The two agree to 15 significant digits. For the page's data {8, 4, 3, 6}: n = 4, mean 5.25, sample standard deviation 2.2173557826083451, t(0.025, 3) = 3.1824463052837096, margin = 3.5283078589306981. Google's MARGINOFERROR page, read live on 2026-08-31 at https://support.google.com/docs/answer/12487850. BATCH PROVENANCE (batch H, the first Sheets/LibreOffice-only batch). Every function in this batch has x == false in docs/data/compat.json: it is a Google Sheets function that Microsoft does not document at all, so this corpus makes NO claim about it in that engine and nothing here is measured against that vendor's documentation. The authority is Google's own support.google.com function page, cited by full URL and by the date it was read -- Google publishes no version number for these pages, so a bare URL dates nothing. WHAT GOOGLE ACTUALLY PRINTS, WHICH IS LESS THAN IT LOOKS: most of these pages carry a 'Sample Usage' block of FORMULAS WITH NO RESULTS. Where a value below is Google's own published output the note says so; where it is derived from the page's stated semantics the note says that instead, and says from which sentence. No value in this batch was taken from a search snippet, a blog or a mirror. LIBREOFFICE: probed on all four pinned builds (24.2.0.3, 24.8.7.2, 25.2.0.3, 25.8.7.3) in five storage spellings -- plain, _xlfn., COM.MICROSOFT., ORG.OPENOFFICE. and _xlfn.ORG.OPENOFFICE. -- and #NAME? under every one of them, so the expected values below describe Google Sheets and the LibreOffice column records absence.

DERIVED, not published: the page rounds to three decimals and stops. Asserted at six places from the independent derivation, 3.5283078589306981. This is the case that would catch a normal-distribution implementation masquerading as a t one -- the z form gives 2.1727 here, which the page's own three-decimal figure already rules out. Google's MARGINOFERROR page, read live on 2026-08-31 at https://support.google.com/docs/answer/12487850.

A structural assertion with no derived constant, taken verbatim from the page's definition bullet. It is the strongest single check available here: it holds whatever the engine's t quantiles are, and it fails if MARGINOFERROR quietly uses the population standard deviation or the normal quantile. Google's MARGINOFERROR page, read live on 2026-08-31 at https://support.google.com/docs/answer/12487850.

GOOGLE'S OWN PUBLISHED FIGURE: the Examples table prints "Lower Bound (Mean - MARGINOFERROR)" as 1.722 and the upper bound as 8.778, over mean 5.25. 5.25 - 3.5283079 = 1.7216921, which rounds to 1.722. Google's MARGINOFERROR page, read live on 2026-08-31 at https://support.google.com/docs/answer/12487850.

DERIVED, not published. The page's second Sample Usage line is MARGINOFERROR(A1:C3, 0.99) with no result and no data, so this case applies its confidence level to the data the page DOES publish. t(0.005, 3) = 5.8409093097333573 and the margin is 6.4756870168140886, derived twice as above. It checks that the confidence argument actually reaches the quantile: a hard-coded 95% level would return 3.528308 here. Google's MARGINOFERROR page, read live on 2026-08-31 at https://support.google.com/docs/answer/12487850.

GOOGLE'S OWN PUBLISHED RESULT, from the Examples table in the article body: over A1:A4 = 8, 4, 3, 6 with mean 5.25, =MARGINOFERROR(A1:A4, 0.95) -> 3.528, with the confidence interval printed as [1.722, 8.778]. DERIVATION. Google's page gives the definition in TEXT, not as an image: "MARGINOFERROR(range, confidence) is equal to CONFIDENCE.T(1 - confidence, STDEV(range), COUNT(range))." STDEV is the SAMPLE standard deviation, and CONFIDENCE.T is the Student-t form, not the normal one -- the page names CONFIDENCE.NORM only as a related link. The figures below were derived TWICE along paths sharing no code: the t density integrated by numerical quadrature to 40 digits with the critical value obtained by root-finding on that CDF, and scipy's inverse t as an independent check. The two agree to 15 significant digits. For the page's data {8, 4, 3, 6}: n = 4, mean 5.25, sample standard deviation 2.2173557826083451, t(0.025, 3) = 3.1824463052837096, margin = 3.5283078589306981. Google's MARGINOFERROR page, read live on 2026-08-31 at https://support.google.com/docs/answer/12487850. BATCH PROVENANCE (batch H, the first Sheets/LibreOffice-only batch). Every function in this batch has x == false in docs/data/compat.json: it is a Google Sheets function that Microsoft does not document at all, so this corpus makes NO claim about it in that engine and nothing here is measured against that vendor's documentation. The authority is Google's own support.google.com function page, cited by full URL and by the date it was read -- Google publishes no version number for these pages, so a bare URL dates nothing. WHAT GOOGLE ACTUALLY PRINTS, WHICH IS LESS THAN IT LOOKS: most of these pages carry a 'Sample Usage' block of FORMULAS WITH NO RESULTS. Where a value below is Google's own published output the note says so; where it is derived from the page's stated semantics the note says that instead, and says from which sentence. No value in this batch was taken from a search snippet, a blog or a mirror. LIBREOFFICE: probed on all four pinned builds (24.2.0.3, 24.8.7.2, 25.2.0.3, 25.8.7.3) in five storage spellings -- plain, _xlfn., COM.MICROSOFT., ORG.OPENOFFICE. and _xlfn.ORG.OPENOFFICE. -- and #NAME? under every one of them, so the expected values below describe Google Sheets and the LibreOffice column records absence.

DERIVED, not published: the page rounds to three decimals and stops. Asserted at six places from the independent derivation, 3.5283078589306981. This is the case that would catch a normal-distribution implementation masquerading as a t one -- the z form gives 2.1727 here, which the page's own three-decimal figure already rules out. Google's MARGINOFERROR page, read live on 2026-08-31 at https://support.google.com/docs/answer/12487850.

A structural assertion with no derived constant, taken verbatim from the page's definition bullet. It is the strongest single check available here: it holds whatever the engine's t quantiles are, and it fails if MARGINOFERROR quietly uses the population standard deviation or the normal quantile. Google's MARGINOFERROR page, read live on 2026-08-31 at https://support.google.com/docs/answer/12487850.

GOOGLE'S OWN PUBLISHED FIGURE: the Examples table prints "Lower Bound (Mean - MARGINOFERROR)" as 1.722 and the upper bound as 8.778, over mean 5.25. 5.25 - 3.5283079 = 1.7216921, which rounds to 1.722. Google's MARGINOFERROR page, read live on 2026-08-31 at https://support.google.com/docs/answer/12487850.

DERIVED, not published. The page's second Sample Usage line is MARGINOFERROR(A1:C3, 0.99) with no result and no data, so this case applies its confidence level to the data the page DOES publish. t(0.005, 3) = 5.8409093097333573 and the margin is 6.4756870168140886, derived twice as above. It checks that the confidence argument actually reaches the quantile: a hard-coded 95% level would return 3.528308 here. Google's MARGINOFERROR page, read live on 2026-08-31 at https://support.google.com/docs/answer/12487850.