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VARP Function in Apple Numbers: Population Variance and Standard Deviation

Statistical Intermediate Apple Numbers

Calculate true population variance in Apple Numbers with source-checked class data and clear sample-vs-population guidance.

Verification status: Official documentation referenced; example outcomes independently reasoned or arithmetically checked where stated. Not executed in Apple Numbers. How formula examples are checked

Quick answer: =VARP(B2:B5) calculates the population (true) variance of a complete group of four numbers. For 2,4,6,8 the independently verified expected value is 5. A variance measures average squared distance from the mean, so its unit is squared (for example, dollars²), whereas the related STDEVP standard deviation is expressed in the original unit. [A1, C1]

Exact Apple signature and documented types

=VARP(value, value…)
Apple argument Required? Apple-documented accepted inputs
value Yes A number, date/time value, or collection of those types. At least two total values are required.
value… No One or more additional compatible values or collections.

All supplied values must be of the same type. Do not infer Duration support for VARP from STDEVP: Apple’s VARP page lists numbers and dates/times but does not explicitly include duration values; STDEVP’s page does. A function accepting a different type in a companion guide does not establish VARP’s type support. The repeated argument symbol … denotes optional additional values, not a default parameter; Apple does not document an optional divisor or mode flag. [A1, A2]

For numeric values x₁...xₙ, VARP = Σ(xᵢ−μ)² / n, with μ = Σx/n. If only a sample of a larger population is being used to estimate variance, Apple’s distinct function VAR divides by n−1 for the sample variance. Thus VARP and VAR are not synonyms. [A1, A2, C1]

Numbers-specific setup, release scope, references and localisation

This guide uses Apple’s Formulas and Functions Help, edition 15.4, checked 2026-10-13, and its documented function signature. The help edition is not evidence of a minimum Numbers version on Mac, iPhone, iPad or iCloud.com. Check the specific application’s release notes and test an older installation instead of extrapolating availability. Apple’s Mac Numbers release history is linked below. No native application run was conducted for this review. [R1, R4]

The reproducible examples use one Numbers table, with its first row configured as a header. A range such as B2:B5 identifies four values in that table; put all formulas in otherwise empty cells outside the input ranges to avoid circular references. Numbers can use Table 2::B5 for another table on the same sheet, and Sheet 2::Table 2::B5 for a different sheet and table. Use the formula editor to construct qualified references; Excel’s Sheet1!B5 notation is not Numbers syntax. Header-derived names can be substituted by the editor, so verify the range endpoints. [R2]

English examples use period decimals and comma argument separators, e.g. =AVERAGE(2,4). Apple says that if your decimal separator is a comma, manually typed function arguments require semicolons (for example =AVERAGE(2;4)). The range-colon B2:B5 does not change. Formula-name case is not significant. The optional ellipsis … in Apple’s syntax notation is not typed into the formula. [R1]

Evidence labels: Apple-documented means the exact result appears in official help; independently calculated means reproduced with separate fractional/decimal arithmetic or Python’s standard statistics tools, not a Numbers output. Do not equate either with native Mac/iOS/iPadOS/iCloud execution or screenshots. Error conditions described below are guidance from documented constraints; unless Apple’s reference specifies an exact error code, this guide does not invent one. [R3]

Reproduce VARP with a complete four-reading population

In Numbers create table ShiftSummary, mark row 1 a header, and enter the four readings in B2:B5. Put the result in D2, outside the input range. Enter digits as numbers rather than text with units.

Row A: Time B: Output
2 Day 1 2
3 Day 2 4
4 Day 3 6
5 Day 4 8 
=VARP(B2:B5)

Independently calculated expected result: 5. The four values average 5; their squared deviations sum to 9+1+1+9=20, and 20/4=5. This assumes those four readings are the whole population of interest; whether they truly are depends on your definition of that population. It is not an observation from Apple Numbers. [A1, C1]

Useful reproducibility and formula comparisons

Formula Independent expected result What it establishes
=VARP(2,4,6,8) 5 Direct numeric arguments agree with range calculation.
=VARP(B2:B5) 5 Four-number full-population variance.
=VAR(B2:B5) 6.666666666666667 n−1 sample variance with the same observations.
=STDEVP(B2:B5) 2.23606797749979 Square root of VARP for numeric input.
=VARP(5,5,5,5) 0 Constant population has zero variance.
=VARP(4,8,12,16) 20 Doubling the original values multiplies variance by 4.
=VARP(102,104,106,108) 5 Adding 100 to every value leaves variance unchanged.

In a comma-decimal locale, the four direct arguments are separated by semicolons, =VARP(2;4;6;8). Numeric results might be displayed at fewer decimals, but cell formatting doesn’t change the underlying mathematical answer. [R1, C1]

Reproduce Apple’s published five-test class example

Apple documents five students and five tests. For direct reproduction, use a separate table with no header row and place these five numeric rows in A1:E5 (the displayed header labels below are explanatory, not spreadsheet cells). For a table with a header row, place values in A2:E6 and adjust the range references accordingly. [A1]

Student Test A Test B Test C Test D Test E
Student 1 75 82 90 78 84
Student 2 100 90 95 88 90
Student 3 40 80 78 90 85
Student 4 80 35 95 98 92
Student 5 90 98 75 97 88

In Apple’s no-header-row example, these five formulas have the following results. Both Apple’s published numbers and an independent population-variance calculation agree. [A1, C1]

Formula Apple-documented result Corresponding numeric standard deviation
=VARP(A1:A5) 416 √416 ≈ 20.3960780543711
=VARP(B1:B5) 481.6 √481.6 ≈ 21.9453867589523
=VARP(C1:C5) 72.24 √72.24 ≈ 8.49941174435031
=VARP(D1:D5) 52.16 √52.16 ≈ 7.22218803410711
=VARP(E1:E5) 8.96 √8.96 ≈ 2.99332590941915

Test B has the most variation by variance; Test E has the least. This is not automatically a judgement on educational merit or test fairness. Square roots match the separately documented STDEVP outcomes for this numeric dataset; the equivalence was independently calculated, not executed in Numbers. [A1, A2, C1]

Advanced: compare batches, not just formula sizes

Suppose you record every unit produced in one short batch. You can use VARP for that complete batch and compare to the VARP of a later complete batch only when measurement definitions and units match. Larger variance says results are more spread out around their own mean; it does not necessarily indicate poorer quality if batches have different targets or tolerances.

For a sample of an ongoing production process, using VARP as if the sampled observations were the entire process population yields a different estimator; Apple’s VAR is meant for sample variance. Record which measurements were excluded or filtered and whether each batch is complete. [A1, A2]

Date/time arguments: Apple explicitly allows date/time values when consistent with the other values; this guide demonstrates numeric values only. Before making physical interpretations from date/time variances, test the chosen Numbers release and output formatting. Duration arguments: excluded from the stated VARP types in the examined Apple specification; don’t promote them to verified support by analogy. [A1]

Why squared units matter: population standard deviation STDEVP is better for explaining dispersion to non-statisticians because it restores the input’s unit. This doesn’t make one statistic more statistically correct than the other; use the one relevant to the analysis. [A1, A2]

Errors, limitations, and checking input ranges

  1. Fewer than two included values: Apple requires a minimum of two same-type values. Verify the actual number of eligible data points; the presence of two visible cells doesn’t guarantee that both are numeric. [A1]
  2. Mixing numeric and duration data: Apple documents VARP numbers/date-times and same-type values; it doesn’t expressly list durations. Do not assume text or Boolean coercion either. [A1]
  3. Confusing population and sample: VARP(B2:B5) gives 5, while VAR(B2:B5) gives 6.666666… for these four numbers. The distinction is denominator n versus n−1, not two syntactic spellings for the same calculation. [A1, A2]
  4. Including a header, subtotal, or the result itself: Ensure the selected range contains exactly the observations. Output cells inside an input range may introduce circular references. [R2, R3]
  5. Numbers table location: Use :: for qualified table/sheet refs when required. Do not substitute Excel’s exclamation-mark syntax. [R2]
  6. Surprising 0: A truly constant dataset has zero variance, but unintended repeated values (such as copied formulas that reference the same cell) can also produce zero. Inspect values individually.
  7. Decimal separators and precision: Switch commas to semicolons between typed arguments if decimal-comma locale is set; increase displayed decimals if verifying a square-root relationship. [R1]
  8. Errors and historical support: Use Apple’s built-in error indicators and the specific application’s release notes. This review does not invent the exact error message a particular iPhone/iPad/Mac version will return. [R3, R4]

Read VAR for sample variance, STDEVP for population standard deviation, STDEV for sample standard deviation, and AVERAGE for the mean. [A2]

Apple documentation and calculation method

Native-validation limitations: These formulas have not been executed in Numbers. Verify relevant blank/error/type behavior and displayed precision on the intended version and locale.