What Is a Z-Score?
A z-score says how many standard deviations a value sits above or below the mean. It turns any score into a common scale so that unlike things can be compared: z = (x − μ) ÷ σ. A z of +2 is two standard deviations above average, −1.5 is one and a half below, and 0 is exactly average. Because the scale is unit-free, you can compare a height with a weight or an exam mark with a lap time.
How to Calculate a Z-Score (Worked Example)
- Take the raw score x. Example: an IQ score of 130.
- Subtract the mean μ (IQ mean 100): 130 − 100 = 30.
- Divide by the standard deviation σ (15): 30 ÷ 15 = 2.
So 130 is exactly two standard deviations above average. On a normal curve about 97.72% of people score lower, which is the 97.7th percentile. If you only have raw data and no μ or σ, use the “Data list” tab: it computes both from your numbers and then every z-score.
Z-Score to Percentile (and Back): Reference Table
| z | Percentile (area left) | Area right | Typical use |
|---|---|---|---|
| 0 | 50.00% | 50.00% | The mean |
| 1 | 84.13% | 15.87% | One SD above |
| 1.282 | 90.00% | 10.00% | 90th percentile |
| 1.645 | 95.00% | 5.00% | One-tailed 95% critical value |
| 1.960 | 97.50% | 2.50% | Two-tailed 95% critical value |
| 2 | 97.72% | 2.28% | “Unusual” threshold |
| 2.326 | 99.00% | 1.00% | One-tailed 99% |
| 2.576 | 99.50% | 0.50% | Two-tailed 99% critical value |
| 3 | 99.87% | 0.13% | Outlier rule of thumb |
Percentile → z works the same way in reverse (the inverse normal). The 95th percentile has z = 1.645. For negative z, mirror the table: the percentile of −1 is 100% − 84.13% = 15.87%.
Z-Score to P-Value: Left, Right and Two-Tailed
| Test | p-value from z | Example (z = 2) |
|---|---|---|
| Left-tailed | Φ(z) | 0.9772 |
| Right-tailed | 1 − Φ(z) | 0.0228 |
| Two-tailed | 2 × (1 − Φ(|z|)) | 0.0455 |
Φ(z) is the cumulative area under the standard normal curve up to z. The two-tailed value counts extreme results on both sides, which is why z = 1.96 gives exactly 0.05. The shading option under the calculator shows the area each p-value uses.
How to Read a Z-Table
Printed tables list the cumulative area to the left of z. For z = 1.37, find row 1.3 and column 0.07 to get 0.9147. Some tables list the area from 0 to z instead, so always check the picture at the top of the table before using it. Our view: z-tables are overrated now that every spreadsheet has a function (=NORM.S.DIST(z, TRUE)), but they remain worth learning once, because they teach you what the number represents.
Z-Score for a Sample Mean (Z-Test)
When your value is a sample mean x̄ of n observations, the spread of x̄ is the standard error σ ÷ √n, so z = (x̄ − μ) ÷ (σ ÷ √n). A sample of 36 IQ scores with mean 104 gives z = 4 ÷ (15 ÷ 6) = 1.6, a two-tailed p of 0.1096. This needs σ known. If you only have the sample standard deviation and a small n, use a t-score instead; for large n (roughly 30 or more) the two nearly coincide.
Interpreting a Z-Score: What Is Unusual?
Rule of thumb: |z| under 1 is ordinary, 1 to 2 is somewhat high or low, above 2 is unusual and above 3 is very rare. In a simulation of 100,000 normal values, 4,525 had |z| above 2 (4.5%) and 278 had |z| above 3 (0.28%). That is where the “z > 3 is an outlier” rule comes from. Our caution: that rule is strictly a statement about normal data. On skewed data, such as incomes or response times, many ordinary values have a z above 3, and the percentile read from the table is only an approximation.
Comparing Scores From Different Tests
Maya scored 78 on a maths test (mean 70, SD 8) and 82 on a history test (mean 76, SD 12). Raw scores suggest history was better, but z-scores tell another story: maths z = 1.00, history z = 0.50. Relative to her classmates, the maths result is stronger. The calculator’s first tab gives each of these in seconds.
Z-Scores in Excel, TI-84 and Python
| Task | Excel / Sheets | TI-84 | Python (SciPy) |
|---|---|---|---|
| z from a value | =STANDARDIZE(x, μ, σ) | (x − μ) ÷ σ | (x − mu) / sigma |
| Percentile | =NORM.S.DIST(z, TRUE) | normalcdf(-1E99, z) | norm.cdf(z) |
| z from percentile | =NORM.S.INV(p) | invNorm(p) | norm.ppf(p) |
Need the spread first? The standard deviation calculator gives σ and s, and the probability calculator covers general events.
Common Mistakes
- Using the sample standard deviation s when the problem gives σ, or vice versa.
- Reading the “0 to z” table as if it were cumulative (off by 0.5).
- Forgetting to double the tail for a two-tailed test.
- Interpreting a negative z as “bad”. It only means below the mean.
- Trusting z-based percentiles for clearly non-normal data.
Why Students Get Stuck on Z-Scores, and What a Good Calculator Should Do
Searching for help with z-scores usually starts late at night with one problem that will not work out. The formula is short, but the table and the meaning of the result are not. A good calculator fixes that by showing each step, checking its own answer and explaining the idea in plain words. This section covers what students struggle with, what they want, and how this page responds.
In short: students need to convert a value to a z-score and read it as a percentile, not just see a final number, and a calculator earns trust only when its steps can be checked.
Understand the pain points
What students get stuck on
Most mistakes here are not about arithmetic. They come from one rule applied the wrong way round, which is why seeing every step matters.
- Dividing by the wrong spread valueZ-score of 85 when the mean is 70 and the standard deviation is 10↗
- Mixing up the sample and population versionsZ-score of 62 when the mean is 75 and the standard deviation is 8↗
- Reading the table in the wrong directionZ-score of 100 when the mean is 100 and the standard deviation is 15↗
- Not knowing what a z-score says in plain wordsZ-score of 130 when the mean is 100 and the standard deviation is 15↗
Understand the frustrations and desires
What they feel, and what they want
Each frustration points to a wish: a clearer step, a built-in check, an explanation in plain words.
- Wants the formula filled in
- Wants the percentile and the p-value
- Wants a plain-language reading
Build the page around their intent
Each intent has a place on this page
The page follows intent, not features, so each need has a clear starting point.
- I need an answer now: type the problem into the calculator at the top of this page.Z-score of 85 when the mean is 70 and the standard deviation is 10↗
- I need to understand: the sections below explain the idea in plain words and list the common mistakes.
- I need to check my work: every step is shown, so the first line that differs from yours is the one to study.
- I need practice: open the solver, tap + and choose Create Practice Test.
- I need it explained aloud: in the solver, tap + and choose Create Video.
Our View: What Is Overrated, Underrated and Wrong About Z-Scores
Overrated. Looking up the z-table by hand. The table only converts one number; what matters is knowing what that number means. Most lost marks come before the table, in the subtraction or in choosing which spread to divide by.
Underrated. The 68-95-99.7 rule as a sanity check. For roughly bell-shaped data, about 68% of values fall within one standard deviation of the mean, about 95% within two and about 99.7% within three. A z-score of 4.2 for an ordinary test mark means the data are unusual or an input is wrong. Check the sign as well: a value below the mean must give a negative z.
Plain wrong. “A z-score of 2 means a 2% chance.” A z-score only counts standard deviations from the mean; it is not a probability. Turning it into a percentile needs the normal curve (a z of 2 sits near the 97.7th percentile) and that holds only when the data are roughly normal.
Test it: a mark of 130 with mean 100 and standard deviation 15 should give z = 2, and a percentile near 97.7.