COVAR Function in Apple Numbers: Population Covariance and Paired Data

=COVAR(dataset-1, dataset-2)

Apple requires the two collections to have the same number of data points. Its help says included strings and booleans are ignored; it does not fully document how missing/non-numeric entries in different rows are paired. For dependable calculations, use entirely numeric, aligned rows and remove or correct missing pairs explicitly. No third argument, denominator switch, default, or sample-covariance option is documented for COVAR. [A1]

Mathematical meaning: for n aligned values, COVAR(x,y) = Σ[(xᵢ−x̄)(yᵢ−ȳ)] / n. This is population covariance, not the sample estimate using n−1. In a simple linear relationship, if x rises with y, covariance tends to be positive; if one rises while the other falls, negative. Zero covariance means no linear co-movement in the selected observations, not proof of independence. [A1, C1]

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]

1. Add a Numbers table named Shipments with at least five rows and header row 1. Enter only numbers in columns B and C. Keep the result formula outside columns B:C, such as E2.
2. Copy the exact values below; matching rows describe the same shipment observation.
3. In E2 enter =COVAR(B2:B5,C2:C5) and show several decimals if needed.

=COVAR(B2:B5,C2:C5)

Independently calculated expected result: 25 unit-minutes. The B mean is 5 and the C mean is 25. The deviations are B [-3,-1,1,3] and C [-15,-5,5,15]; products are [45,5,5,45]. Their sum is 100; population covariance is 100 / 4 = 25. This is independently computed, not a Numbers runtime result. [A1, C1]

Additional calculated checks on the same data

When using comma-decimal conventions, the two arguments are entered with a semicolon, =COVAR(B2:B5;C2:C5). This separator choice comes from Apple’s overview, not a second calculation. [R1, C1]

Apple illustrates COVAR with heating-oil prices and thermostat settings. To reproduce the documented example, make a separate 11-row table with headers in A1:B1 and these pairs in rows 2–11. Do not conflate this with the Shipments table above. [A1]

=COVAR(A2:A11,B2:B11)

Apple-documented result: approximately -1.6202. The arithmetic was independently checked: average oil price 3.241, mean setting 67.2, and sum of centered cross-products -16.202; dividing by all 10 pairs gives -1.6202. The negative sign captures opposite movement across the recorded periods. Neither the data nor the result were observed in a local Apple Numbers run. [A1, C1]

For the shipment sample, perfect proportionality means CORREL(B2:B5,C2:C5) has a mathematically expected coefficient of 1. For general real-world data, covariance alone cannot establish strong correlation: the scales and units vary. Apple’s separately documented CORREL computes a unitless correlation; SLOPE, INTERCEPT and FORECAST address a best-fit linear relationship and prediction, each with distinct requirements. Do not infer forecasting reliability from covariance alone. [A1, A2, C1]

To test a reversed pattern, make B2:B5 1,2,3,4 and C2:C5 8,6,4,2. Then =COVAR(B2:B5,C2:C5) has independently calculated expected result −2.5: means 2.5 and 5; products sum −10; /4=−2.5. This demonstrates direction changes without changing the function’s argument order. [C1]

What COVAR does not do

• It does not tell you which variable causes the other to change; correlation is not causation.
• It does not automatically convert different units or normalize results to a −1..1 scale.
• It is not a sample-covariance function; an n−1 estimator is a different statistic. Do not silently replace the denominator.
• It does not document how to resolve unequal or individually missing paired cells. Choose complete numeric observations deliberately rather than relying on undocumented coercion. [A1]

1. Unequal range lengths: =COVAR(B2:B5,C2:C6) supplies four and five row positions, violating Apple’s equal-data-point requirement. Make both ranges span the exact same rows. Apple’s page does not specify a universal displayed error string. [A1]
2. Non-numeric entries: Apple says strings and booleans within collections are ignored, but the exact effect when only one partner in a row is non-numeric is not established. Clean the paired records, compare row counts, and test locally before using results. [A1]
3. Wrong alignment: The same values sorted independently into different row orders no longer describe the same pairings; covariance changes. Sort complete rows together and preserve identifiers.
4. Mixing entire population and sample: COVAR’s denominator is n. For sample inference, use a correctly documented method rather than assuming a hidden flag.
5. Ranges built across tables: In Numbers choose target table and source ranges in the formula editor; use Table::Cell references as appropriate. [R2]
6. Unreliable decimal appearance: Cells formatted to zero or two decimal places may conceal the underlying result. Increase display precision.
7. Errors or empty results: Check the referenced numbers and cell formats, formula editor warnings and source ranges. Apple documents general error indicators; this review does not claim a specific COVAR error code. [R3]

Consult the official help: CORREL, VARP, SLOPE, FORECAST. [A2]

• [A1] Function definition, argument types, restrictions and Apple examples: Apple COVAR reference; checked 2026-10-13, documentation edition 15.4.
• [A2] Related statistical function definitions: CORREL, VARP, SLOPE, FORECAST; checked 2026-10-13.
• [R1] Function syntax and locale rules: Apple Functions overview; checked 2026-10-13.
• [R2] Table/range/cross-table references: Apple Refer to cells in formulas; checked 2026-10-13.
• [R3] Formula-error troubleshooting: Apple Review formula errors and follow best practices and Types of arguments and values; checked 2026-10-13.
• [R4] Compatibility limitation: Apple Formulas and Functions Help and Apple Numbers Mac release notes; checked 2026-10-13. Neither proves universal historical per-function availability.
• [C1] Mathematical verification: calculations are independently reproduced using precise rational/decimal arithmetic and/or Python standard-library numerical functions; using the exact input sets displayed above. This is not a native Numbers test.

Native-validation limitations: No Numbers execution or screenshots were performed. Check important mixed-type, blank and locale edge cases on the intended device.