STANDARDIZE Function in Apple Numbers: Z-Scores and Examples

=STANDARDIZE(num, average, stdev)

Apple’s official page says all three arguments are number values. There are no optional arguments, no built-in selector for STDEV versus STDEVP, and no defaults. STANDARDIZE does not itself calculate the reference average or standard deviation from a table; provide them directly or use separately computed AVERAGE and STDEVP formulas. [A1, A2]

For a value x, known mean μ, and positive population deviation σ, the mathematical operation is z=(x−μ)/σ. Units cancel, so the result is dimensionless and directly comparable across measurements when each uses a suitably matched reference distribution. Standardizing does not automatically turn skewed or non-normal observations into a normal distribution, and it is not a percentage conversion. [C1]

These examples target the current English Apple Formulas and Functions Help, edition 15.4. In Numbers, make a named table, type numeric observations into cells, select the formula result cell outside its inputs, and use the formula editor to select the source cells. A range such as B2:B6 is a reference inside the same table. For data in a different table, select that table’s cells and let Numbers insert a qualified reference such as Measurements::B2:B6. Avoid Excel’s Sheet1!B2:B6: its exclamation-mark sheet notation is not Numbers’ reference convention. Apple explains table references in Refer to cells in formulas. [R2]

Apple’s English function syntax uses commas to separate arguments, for example =AVERAGE(2,4). When the document’s settings use a comma as the decimal separator, Apple’s Functions overview tells users to separate manually typed arguments with a semicolon—for example =AVERAGE(2;4). A range’s colon remains a colon. Enter values with the decimal symbol configured for the file. The spellings shown below are the canonical English function names; Apple’s overview says capitalization does not affect manually entered names. [R1]

Apple documents these functions in Formulas and Functions Help 15.4 and makes its formulas help available to iWork apps. That help edition does not prove the function first appeared in Numbers 15.4 or establish a minimum release for Mac, iPhone, iPad or iCloud.com. Apple’s Mac release-history page separately identifies Numbers 14.4 as adding over 30 advanced functions such as LET, LAMBDA and FILTER; it does not specify a PEARSON/RSQ/STANDARDIZE introduction date. Release-specific behaviour, display formatting and parity across Mac, iPhone, iPad and iCloud.com are not natively tested here. [R3, R4]

Verification key: “Apple-documented” means a statement appears in the cited Apple Help article. “Independently calculated” means the arithmetic was reproduced without the Numbers engine (exact rational or standard-library mathematics). Neither is an on-device formula execution. No native screenshots or instrumented Numbers execution is claimed.

=STANDARDIZE(6,15,2.1)

Apple’s published result is approximately −4.28571428571429. Independently, (6−15)/2.1 = −9/2.1 = −30/7 ≈ −4.285714285714286; the difference in the last few displayed digits is rounding. The negative sign indicates that 6 lies over four population standard deviations below the specified average of 15, assuming that mean and standard deviation genuinely describe the reference distribution. [A1, C1]

All expected results were calculated arithmetically outside Numbers; the first four are original examples, not outcomes supplied by Apple. [A1, C1]

Create a Numbers table named ExamResults. Put headers in A1=Student and B1=Score. Then enter the following five numeric score cells exactly.

Put a standardized result in an empty cell, for example C2:

=STANDARDIZE(B2,80,10)

With B2=60, the independent expected result is −2. Copy the analogous formula to later rows, changing only the input cell, and the expected sequence is −2, −1, 0, 1, 2. These examples assume a reference population standard deviation of 10, which is an externally supplied parameter; it is not the actual standard deviation of the five visible scores. This distinction prevents a common misleading analysis. [C1]

Advanced: calculate the actual population parameters from the table

The exact five-score table above has mean 80 and population standard deviation √200 ≈ 14.1421356237 (the sample standard deviation would be different). To standardize against those five scores as the complete population, calculate both parameters rather than supplying 10:

=STANDARDIZE(B2,AVERAGE(B2:B6),STDEVP(B2:B6))

Independently calculated expected result at B2=60: −√2 ≈ −1.4142135623730951. At B4=80 the analogous formula returns 0; at B6=100 it returns +√2 ≈ 1.4142135623730951. STDEVP’s population result follows from squared deviations 400+100+0+100+400=1000 divided by five = 200. Apple’s STANDARDIZE article requires a known population deviation, and its STDEVP article documents the corresponding population summary function. [A1, A2, C2]

For a practical calculation column, use a second reference table holding fixed mean and standard deviation values and select those cells in the Numbers formula editor. This makes the reference parameters auditable. When filling down formulas that compute AVERAGE(B2:B6) or STDEVP(B2:B6), lock the source row range appropriately (for example B$2:B$6) before autofill so the population does not silently shift with every destination row. Native autofill and qualification across different table names still require testing. [R2]

Comparing units: A z-score can compare how unusual a reading is relative to its own group without mixing original units, e.g. a temperature anomaly and a test score. +2 means two reference standard deviations above the reference mean; it does not mean 2°C, 2%, or “twice as good.” [C1]

Population versus sample: Apple explicitly calls for population standard deviation. If you feed an estimate from a sample instead, it may be an intentionally chosen modeling approximation, but the result is not based on the known entire-population σ described by the official syntax. The distinction affects both the denominator and interpretation. [A1, A2]

Outliers and distributions: A high |z| may flag an unusual point, but STANDARDIZE alone cannot decide if the point is erroneous, rare, dangerous or statistically significant. Interpret it in the context of the measurement process and distribution shape. [C1]

• Zero or negative stdev: Apple explicitly requires stdev > 0. =STANDARDIZE(95,80,0) and =STANDARDIZE(95,80,-10) violate the documented condition; the specific error wording was not natively checked. [A1]
• Wrong parameter order: =STANDARDIZE(95,10,80) is not the same as =STANDARDIZE(95,80,10). Check the browser’s num, average, stdev ordering before troubleshooting arithmetic. [A1]
• Text or date/time instead of numbers: Apple’s article specifies number values for all three arguments. Use Number-formatted cells rather than copying labels such as "95 points", and don’t infer unsupported date coercion. [A1]
• Unexpected large magnitude: Confirm scale and units. An average measured in kilometres and deviation in metres makes an incoherent standardized comparison even when each cell is numeric. [C1]
• Formula gives a shifted result after copying: Check if the mean/standard-deviation reference range moved in the fill operation; use stable ranges or a parameter table. [R2]
• All observations identical: An independently calculated STDEVP for a constant numeric population equals zero, so passing that result to STANDARDIZE violates the documented positive-stdev condition. Treat the standardized score as undefined rather than inventing a zero. [A1, A2, C2]
• Rounding differences: An exact formula may display fewer decimals due to cell formatting. Use a reasonable number format and distinguish an approximate on-screen value from mathematical rounding. [C1]
• Confusion with standard normal probability: STANDARDIZE returns a z-score, not a cumulative probability. Apple’s related NORMDIST help links to probability-distribution functions for a different question; this guide makes no untested claim about their arguments. [A1]

Standard scores are mathematically portable when the same x, mean, and positive population σ are supplied. That does not prove every spreadsheet product handles nonnumeric values, text coercion, error cells, time formats, locale separators or cross-sheet references the same way. This page documents Apple Numbers only. Test formulas after export/import; do not paste Excel’s Sheet1!B2 notation into Numbers as though it were native. [A1, R2]

• STDEVP — calculates population standard deviation for a numeric reference collection.
• AVERAGE — finds the arithmetic mean used as STANDARDIZE’s second argument.
• VARP — produces population variance; its square root matches population STDEVP for numeric data.
• STDEV — sample standard deviation, generally not the population parameter required by Apple’s STANDARDIZE definition.

• [A1] Official Apple STANDARDIZE help, syntax, stdev positivity and published example, 15.4.
• [A2] Official Apple STDEVP help, population/sampling meaning, 15.4.
• [R1] Apple Functions overview, syntax, capitalization and locale separators.
• [R2] Apple Refer to cells in formulas, table/cell references and ranges.
• [R3] Apple Formulas and Functions Help, general editor and app scope.
• [R4] Apple What’s new in Numbers for Mac, version-specific history (not proof of first STANDARDIZE release).
• [C1] Direct independent arithmetic of (num−average)/stdev in integer and rational form, 2026-10-13.
• [C2] Five-value population mean, variance and standard-deviation calculations checked independently with Python’s standard library, 2026-10-13.

Native-validation limitation: No Numbers execution or screenshots were performed; verify relevant input and locale edge cases on your intended version.