Z-Score Calculator
Turn x, μ and σ into a z-score and a left-tail, right-tail or two-tailed probability, or standardize every value in your dataset at once. The standard normal curve fills the same area the table reports.
- No upload
- Worked steps
- Free, no sign-up
100% Local in Browser — Your Data Never Leaves This Tab
Comma, space, tab or newline separated. Paste an Excel column directly; blank lines and labels are ignored.
Try a sample
Paste a dataset to begin
The worked steps, table, box plot and summary update as you type. Nothing is sent anywhere.
How it works
- 1
Choose the mode
Single value takes an observation x, a mean μ and a standard deviation σ. Whole dataset reuses the shared list, computes its sample mean and n − 1 SD, and standardizes every observation.
- 2
Standardize
z = (x − μ) / σ. A positive z is above the mean, a negative z is below it, and the magnitude is the distance in standard deviations. The worked steps substitute your numbers into the formula.
- 3
Read the area
The standard normal CDF Φ(z) gives P(Z < z); the right tail is 1 − Φ(z); a two-tailed p is 2 × the smaller tail. The bell curve fills exactly the area selected, using an Abramowitz & Stegun approximation to Φ.
More tools
- Median CalculatorPaste a dataset and get the median with the sorted list, the exact middle position or pair, and a box plot. Odd and even counts are handled separately, and the numbers never leave your browser.
- Mean CalculatorAdd a list of numbers and see the mean with the sum, the count and every substitution in Σx / n. The dataset stays in your browser and carries over to the median, SD, IQR and z-score pages.
- Standard Deviation CalculatorGet sample or population SD with a row-by-row deviation table, the definition formula and the computational Σx² − n·x̄² formula. Copy the APA result or switch to population variance without re-pasting the data.
- Average CalculatorFind the average of any list with the sum, count and substitution shown line by line. Also called an average finder; the page reports the arithmetic mean and compares it with the median so a skew does not fool you.
- Interquartile Range (IQR) CalculatorCompare Q1, Q2, Q3 and IQR under the TI-83/84/AP exclusive rule, Tukey’s inclusive hinges and Excel QUARTILE.INC interpolation. See the 1.5 × IQR fences, potential outliers and a live box plot.
What it won't do
- Dataset mode standardizes with the sample SD (n − 1). If your course gives a population σ, use Single value and enter that σ directly.
- The normal curve is a model. A z-score does not make skewed data normal, and a tail probability is only as meaningful as the distribution assumption behind it.
- Φ is an approximation with error around 1.5 × 10⁻⁷, far below the precision of a homework answer; very large |z| values are clamped in the drawing but the numeric CDF still uses the full formula.
- Everything is computed in the tab. The dataset is shared through sessionStorage with the other five calculators and is cleared when the tab closes.
Questions
What does a z-score of 1.96 mean?
The value is 1.96 standard deviations above the mean. For a standard normal distribution, P(Z < 1.96) ≈ 0.975, so the right tail is about 0.025 and a two-tailed p is about 0.05. Enter 1.96 as x with μ = 0 and σ = 1 to see the same numbers here.
How do I find the probability from a z-score?
Use the standard normal CDF: P(Z < z) = Φ(z). For the right tail, subtract from 1. For a two-tailed test, double the smaller tail. The page prints all three and shades the selected area on the curve.
Should the z-score use n or n − 1?
The denominator is the standard deviation you were given or the one appropriate to your data. In Whole dataset mode this page uses the sample SD with n − 1; in Single value mode you type the σ explicitly, so there is no hidden denominator choice.
Can a z-score be negative?
Yes. A negative z means the observation is below the mean. The sign is not an error; the left-tail probability for a negative z is below 0.5, and the curve shading moves to the left side.
What is the difference between a z-score and a t-score?
A z-score uses a known population σ (or treats the given SD as known). A t-score uses the sample SD and a t distribution, which has heavier tails for small samples. This page is the normal/z version.
Why does the curve shade two separate tails in two-tailed mode?
A two-tailed p-value asks how extreme a value is in either direction, so it counts both z ≤ −|z| and z ≥ |z|. The two shaded regions together equal 2 × the smaller one-tail area.