Free calculator
Z-score calculator
Enter a raw value, the mean μ, and the standard deviation σ (same units as your data). You get the z-score z = (x − μ) / σ—how many standard deviations the value sits above or below the mean—and standard normal tail probabilities (P(Z ≤ z), P(Z > z), two-sided). Use the Google Sheets & Excel section for STANDARDIZE and NORM.S.DIST patterns. This is textbook normal-model math, not pediatric growth charts.
When to use this calculator
Quick standardization checks before you drop the same idea into Sheets or Excel—transparent formulas, not a full statistics suite.
- Turn a raw measurement into how unusual it is vs a stated mean and spread, in σ units.
- Match
STANDARDIZEor manual (x−μ)/σ cells against this page when auditing a template row. - Read off standard normal tail masses for the computed z (symmetric bell curve N(0,1) story).
- Need σ from pasted data? Use this site’s standard deviation calculator first, then paste μ and σ here.
The z-score (or standard score) measures how far a value x sits from the mean μ in units of the standard deviation σ: z = (x − μ) / σ. It needs a positive σ.
Compute z
Subtract the mean from your raw value, then divide by σ. A positive z means above the mean; negative means below; z = 0 means equal to the mean.
Standard normal tails
Assuming Z ~ N(0,1), P(Z ≤ z) is the left-tail area under the bell curve up to z. The right tail is 1 − P(Z ≤ z). The two-sided tail shown here is twice the smaller one-sided tail at |z|—a common symmetric two-tail summary.
Where does σ come from?
This page does not estimate σ from raw rows. If your σ comes from a sample on the same scale as x, use the sample standard deviation from your data (see the standard deviation tool for STDEV.S vs STDEV.P alignment).
We do not compute inverse z from p-values, areas between two z’s, two-sample tests, or proportion z statistics in this v1—add those in a dedicated workflow when you need them.
To get σ (and μ) from a pasted list, open the standard deviation calculator for **mean** and **SD** with explicit **sample vs population** control.
To inspect variance and sum of squares on the raw list before you choose σ, open the variance calculator.
FAQs on this page cover σ = 0, sample SD as an estimate of σ, growth-chart z-scores (out of scope), and graphing calculator pointers—without turning the page into exam software.
Google Sheets & Excel
In Google Sheets and Microsoft Excel (English / US-UK function names), STANDARDIZE(x, mean, standard_dev) returns the same z as this page when you pass the same three numbers. NORM.S.DIST(z, TRUE) is the standard normal CDF P(Z ≤ z) for that z. Replace cell references with your values. If your Excel is not in English, use Formulas → Insert function to find the same patterns under your language pack.
=STANDARDIZE(A2, B2, C2)A2 = raw x, B2 = μ, C2 = σ (must be > 0). In Google Sheets the function name is the same.
=NORM.S.DIST(D2, TRUE)Put the z you computed (or a STANDARDIZE cell) in D2. TRUE means cumulative = P(Z ≤ z) for Z ~ N(0,1).
=1-NORM.S.DIST(D2, TRUE)Complement of the left tail for the same z in D2.
Frequently asked questions
What is a z-score?
A z-score tells you how many standard deviations a value x is from the mean μ on a normal model: z = (x − μ) / σ. σ must be positive.
What is the formula for a z-score?
z = (x − μ) / σ, where x is your observation, μ is the mean you compare against, and σ is the standard deviation that sets the scale. If x = μ, then z = 0.
Why can’t σ be zero?
Dividing by σ measures distance in spread units. When σ = 0, every value in that reference model sits on the mean, so (x − μ)/σ is not defined—the tool stops with an error instead of inventing a number.
Can I use a sample standard deviation for σ?
In practice people often plug in a sample SD as an estimate of the population scale—especially with large n—but the interpretation depends on your methods course or team rules. Use the standard deviation calculator’s sample vs population modes to match STDEV.S vs STDEV.P before you paste σ here.
How do the tail probabilities relate to percentiles?
For the standard normal curve, P(Z ≤ z) is the cumulative probability up to z—sometimes read as a percentile of the N(0,1) distribution after multiplying by 100. The page labels tails explicitly so you do not confuse left vs right.
Is this for pediatric growth z-scores?
No. Growth charts use reference populations and LMS/Lookup tables that are not the same as typing a generic μ and σ here. This tool is introductory textbook z on a normal model only.
Which Google Sheets or English Excel functions match this page?
STANDARDIZE(x, mean, standard_dev) reproduces z. NORM.S.DIST(z, TRUE) gives P(Z ≤ z) for Z ~ N(0,1). The copy cards show example cell layouts.
What are the German Excel names for STANDARDIZE and NORM.S.DIST?
Typical Excel (Deutsch) names are STANDARDISIERUNG for STANDARDIZE and NORM.S.VERT(z;WAHR) for the standard normal CDF—confirm in your function library; list separators may be semicolons.
What are the French Excel names for STANDARDIZE and NORM.S.DIST?
Typical Excel (français) names include CENTREE.REDUITE (or legacy STANDARDIZE) and LOI.NORMALE.STANDARD(z;VRAI) for the standard normal CDF—verify on your install.
Can I type a probability and get z (inverse normal)?
Not in this v1. Inverse lookups (p → z) need NORM.S.INV / NORM.INV patterns in a sheet or a dedicated calculator mode—planned as a later enhancement if demand is clear.
Does this replace a TI-84 statistics menu?
No. Handheld menus bundle many distribution modes. This page is a transparent web check for one z from x, μ, and σ, plus N(0,1) tail readouts—use your course’s calculator steps for exams.
Is this professional statistics advice?
No. It is a free educational calculator. For regulated reporting, research protocols, or medical decisions, follow qualified professionals and institutional methods.