TTEST in Apple Numbers: Paired and Two-Sample t-Tests

A t-test asks how surprising a mean difference would be under a specified null hypothesis and statistical model. TTEST returns a p-value, not a percentage difference, proof that the groups are the same, an effect size, or the probability that the null hypothesis is true. Preselect one or two tails according to your question before interpreting the result. The two independent-sample variants make different assumptions about group variances; the paired variant analyzes within-pair differences. The function does not verify random sampling, outliers, approximate normality, independence or genuine pairing for you. [A1, C1]

=TTEST(sample-1-values, sample-2-values, tails, test-type)

All four positions are shown in Apple’s reference. The help page does not state implicit defaults, exact minimum sample lengths or numerical error-message text. For statistically meaningful work, supply at least two numbers per sample. A paired test needs corresponding observations in identical order and matching pair counts; type 2 and 3 concern independent samples and need not be artificially paired. Do not silently mix blank/text rows or combine groups with unrecorded missing values. [A1, C1]

Create a Numbers table named Experiment. Keep the first row as headers. In A2 put Before, in A3 After, and place these six values across columns B–G. Enter each formula in an empty cell outside the input ranges, such as I2, and switch the last two arguments to compare modes.

The first five expected outputs are Apple’s published examples; the sixth was independently computed using Welch’s two-sample formula. For the paired calculation, the six within-pair differences are −30, 10, 21, 3, 4, 1. Independent SciPy computations reproduce Apple’s paired and independent results to ordinary floating-point precision. These are expected results, not observed Numbers application outputs. [A1, C1]

Paired (type 1). Use for repeated measures on the same units or genuinely matched pairs, with row/column order preserved. Two observations from different people do not become paired merely by appearing in adjacent cells. Mathematically, with paired differences d, the test statistic is AVERAGE(d)/(STDEV(d)/SQRT(n)) using sample standard deviation and n valid pairs. This explanation is a mathematical cross-check, not an Apple-documented alternative-function implementation test. [C1]

Independent equal-variance (type 2). Use only when the common-variance assumption is justified. The pooled variance combines the groups’ sample variances; the test uses n₁+n₂−2 degrees of freedom. [A1, C1]

Independent unequal-variance (type 3). Use the Welch-type comparison if variances need not be equal. Degrees of freedom are determined from group variances and sample sizes rather than simply n₁+n₂−2. The Apple modal label calls this two-sample unequal; the Welch mathematical interpretation and numerical outputs were cross-checked independently. [A1, C1]

To label a previously chosen two-tailed paired test with a 5% threshold, put the fixed threshold 0.05 in J2 and use:

=IF(TTEST(B2:G2,B3:G3,2,1)<$J$2,"Below 0.05","At least 0.05")

The expected label on Apple’s dataset is At least 0.05. This label describes a p-value decision under the analysis model; it does not establish that two samples are equivalent. The formula uses standard IF, absolute reference and TTEST syntax, all documented at the level of generic function and cell-reference operations, but this combined formula was not executed in Numbers. A rigorous study should additionally report effect sizes, confidence intervals, study design and assumption checks. [A1, A3, C1]

For a different illustrative design, suppose one group is {12,11,13,12,12,14} and another {12,10,15,12,13,11}. Independent calculations give p≈0.8429938072 for an equal-variance two-tailed test, ≈0.8438805547 for unequal-variance, and ≈0.8220441884 when the numbers are deliberately treated as paired observations. These probabilities are similar here, but that never justifies choosing the test after inspecting p-values. [C1]

The calculation is not appropriate for unrelated unpaired rows passed as test-type=1, for strongly dependent subjects treated as independent, or for a claim of equivalence based only on p>0.05. Where assumptions are doubtful, consult a statistician rather than treating an automatic p-value as a study design validator. [A1, C1]

• Apple ZTEST — one-sample, one-tailed normal-model probability, a different test and not interchangeable with a two-group t-test.
• Apple PROB — probabilities from known discrete event weights, not a sampling-test p-value.
• Apple functions overview for argument punctuation and required-argument notation.

These instructions follow Apple’s English Formulas and Functions Help, version 15.4, reviewed 13 October 2026. Apple describes the Functions Browser for Numbers on Mac and on iPhone/iPad, and in relevant iWork apps on iCloud.com, but the documentation version is not proof that every function exists in every historical app release or platform build. There was no native Numbers execution for this guide. Do not represent the expected outputs as measured application results. [A1, A4]

The examples use English function names, a period as decimal point and a comma between arguments. Apple explicitly says that a Numbers setup using a comma for decimals requires a semicolon between function arguments when typing formulas. Spreadsheet language/region settings also affect numeric punctuation; select the appropriate localized function spelling in your own Functions Browser as translated names may differ. [A2, A5]

Inside a Numbers table, B2:C6 identifies the rectangular range and $D$2 is an absolute reference. A cross-table reference can be written as Table Name::B2 or Sheet Name::Table Name::B2 where required, and Numbers can insert the appropriate reference when you select a source cell. The formulas below assume all inputs and the result are in the same named table; do not copy an ambiguous short reference across multiple tables. [A3]

• Official documentation check: Apple’s named function help, argument/type and cell-reference instructions, and the functions-overview regional-syntax rule were read on 13 October 2026. The source links and claim map are preserved below.
• Independent calculations: Expected numbers were recomputed with Python/SciPy or exact arithmetic; these checks establish the mathematics and consistency with Apple’s published examples, not the installed application’s behaviour.
• Not performed: native Apple Numbers execution, screenshots, localized behavior or historical-platform regression tests.

• [A1] Function specification: TTEST — Apple Formulas and Functions Help 15.4. Defines syntax and supported argument descriptions; the examples explicitly attributed to Apple come from this page.
• [A2] Formula syntax/locale: Apple: Functions overview. Confirms comma separators in the displayed English syntax and semicolons if the system uses a comma decimal separator.
• [A3] Cell and table references: Apple: Refer to cells in formulas. Documents range notation, absolute references and Table::Cell.
• [A4] Collections/value types and platforms: Apple: Types of arguments and values; Apple: Formulas and Functions Help welcome. Specifies collection, number and modal types and the general device context.
• [A5] Document language and region: Apple: Format a spreadsheet for another language in Numbers on Mac. Describes regional formatting changes, not an exhaustive translation/alias map.
• [C1] Independent mathematical calculations: Python/SciPy calculations reproduced the stated examples; these are not native Numbers tests.