Apple Numbers VARPA: Population Variance and Mixed-Value Rules

• Introduced: Original release and introduction date are not established by the checked references.
• Availability: Currently documented in Apple’s Functions Help 15.4 documentation edition; no final supported-version endpoint is established.
• Scope: Shared iWork reference. Function-specific historical availability across Mac, iPad, iPhone and iCloud is unverified.
• Checked: October 14, 2026, against the official VARPA reference. Independent typed-data and mathematical checks; no native Numbers test.

=VARPA(value, value…)

The first value or collection is required; further values or collections are optional. Apple specifies at least two values, including for the population functions. One range argument can contain multiple values. Do not confuse the number of arguments with the number of included observations.

Text is allowed in a referenced cell but is not allowed as a directly entered string argument. For example, put pending in a cell instead of constructing =VARPA(4,6,"pending"). The latter is outside the documented argument contract; this guide does not assign it an unverified error token.

Apple’s Notes explicitly say included date/time or duration values produce an error. Keep those types out of these numeric examples, even though the argument description also mentions a general same-type rule.

Use VARPA when the included values represent the whole population you mean to describe. VARA uses the sample definition. Deciding which observations belong in that population is separate from choosing a formula.

Create a Numbers table named MixedDispersion, with eight columns A:H and ten rows, including one header row. Use Input, Type, Mapped value and Squared deviation as A1:D1. Enter:

Enter pending without quote characters. Leave A7 and C7 truly empty, rather than typing the instruction or a formula that returns empty text. The five numbers in C2:C6 are a manual reconstruction, not a claim that a pending measurement really equals zero.

In E2 enter =AVERAGEA(C2:C6) for 2.2. In D2 enter =(C2-$E$2)^2, and fill down through D6. The five squared deviations are 3.24, 14.44, 4.84, 1.44 and 4.84. Put =SUM(D2:D6) in E3 for 28.8.

Use G2:G6 for the results:

The reconstruction uses five included values, so its denominator is 5, or n. Variance has squared units. It is not a typical distance in the original measurement unit. G5 deliberately selects only the two original numbers, 4 and 6, to show a different data-selection policy. It does not silently treat all five mapped entries as measurements.

The printed decimals are rounded expected outputs. Keep unrounded formulas for later calculations; a display rounded to a few places should not become a replacement input.

Create SensorExample, with six columns and nine rows. In A1:C1 enter Day, High and Low. Use the following complete source data; enter the day labels as plain text and n/a as ordinary text without quotes, not a spreadsheet error value.

In E2 enter =VARPA(B2:B8) for approximately 867.1428571429. In E3 enter =VARP(B2:B8) for approximately 112.5555555556. These reproduce Apple’s stated comparison: the A-function includes the text marker as a seventh, zero-valued observation; the numeric-only calculation uses six recorded highs.

A missing temperature is not a measured 0 °F. The larger mixed-input result reflects that chosen conversion. Although the source example calls these days a sample, VARPA remains the population formula. Choose a population interpretation deliberately rather than importing that label into a statistical conclusion.

Create UnitCheck, with five columns and eight rows. Enter 20,40,60 in A2:A4 as millimetres. In B2 enter =A2/10 and fill through B4 to obtain 2,4,6 centimetres.

Dividing each observation by ten divides variance by one hundred. D2 is measured in square millimetres, D3 in square centimetres, and D5 in centimetres. Apple’s temperature example compares the numerical variance with the temperature range; those have different units, so that comparison is not a general interpretation rule.

Create RepeatedPopulation, with three columns and eight rows. Enter 2,4,6,2,4,6 in A2:A7. Put =VARPA(A2:A4) in C2 and =VARPA(A2:A7) in C3; both return approximately 2.6666666667. Each original value has been repeated equally, leaving the population distribution unchanged.

In C4 enter =VARA(A2:A4) for 4, and in C5 enter =VARA(A2:A7) for 3.2. The sample denominator changes from two to five. Duplicating records is not a way to collect new independent observations, even though a formula can calculate a result for them.

Create ZeroControl, with three columns and six rows. Enter numeric 0 in A2:A3, text pending in A4 and =FALSE() in A5. Put =VARPA(A2:A5) in C2 and =VARP(A2:A3) in C3; both return 0. Text and a Boolean are present even though the dispersions match.

A difference between an A-function and a numeric-only calculation can reveal an inclusion-policy difference, as in the sensor example. Equality does not prove that every input is numeric, complete or valid. Inspect types and collection rules before interpreting the statistic.

Empty cells, text markers, numeric zeros and error cells are distinct inputs. The checked references do not establish every imported numeric-string representation, formula-produced empty string, native error label or numerical limit. Keep those cases separate from the explicitly typed examples above.

VARP explains the corresponding numeric-only statistic. VARA uses the other denominator choice. AVERAGEA explains the mapped mean, and SQRT connects variance to standard deviation.