Statistics
Z Score Table and Calculator
The Z score calculator turns a raw value into a standardized score and shows its standard-normal probability areas. You can also start with z or work backwards from a probability.
The sign shows which side of the mean you are on. A normal model is needed for the percentile and tail-probability interpretation.
Calculate a Z Score
Probability Areas
Click a card to shade that area on the curve.
Standard Normal Curve
Z-Table Lookup
Probabilities use a standard normal distribution. The calculator supports z from -6 to 6; the highlighted table covers -3.49 to 3.49.
Z Score Formula
A z-score tells you how far a value is from the mean, measured in standard deviations. You subtract the mean first, then divide by the standard deviation.
- x: the value you want to standardize.
- μ: the population mean.
- σ: the population standard deviation, which must be greater than 0.
- Sample data: the analogous calculation uses the sample mean and sample standard deviation. You should use the quantities appropriate to your problem.
How to Interpret a Z Score
The sign tells you which side of the mean the value falls on. The magnitude tells you its distance from the mean.
- z = 0: the value equals the mean.
- z = 1: the value is 1 standard deviation above the mean.
- z = -2: the value is 2 standard deviations below the mean.
A z-score can be calculated for data that aren’t normally distributed. Turning it into a standard-normal percentile or tail probability needs a normal model. Standardizing a dataset doesn’t make its shape normal.
For a normal distribution, the approximate central coverage is:
- Within ±1 standard deviation: 68.27%.
- Within ±2 standard deviations: 95.45%.
- Within ±3 standard deviations: 99.73%.
The NIST normal-distribution reference explains the probability model behind these areas.
Worked Example
Suppose exam scores follow a normal distribution with a mean of 70 and a standard deviation of 12. You want the proportion of students who score above 85.
- Left-tail probability: Φ(1.25) ≈ 0.89435, so about 89.44% score below 85.
- Right-tail probability: 1 – Φ(1.25) ≈ 0.10565, so about 10.56% score above 85.
- Interpretation: 85 is 1.25 standard deviations above the mean. The percentage estimate depends on the normal-distribution assumption.
The calculator starts with these values. You can select the right-tail result to shade the area above the score.
How to Read a Z Score Table
The downloadable table gives the area to the left of z. For z = 1.25, you read row 1.2 and column .05 to find 0.89435.
- Choose the sign. Use the positive table for positive z and the negative table for negative z.
- Find the row. It contains the whole number and first decimal place.
- Find the column. It supplies the second decimal place.
- Check the table’s convention. A left-tail table gives Φ(z). Some tables give the area between 0 and z instead.
For z = -1.25, row -1.2 and column .05 give about 0.10565 in the negative table. That is the area to the left of -1.25.
The NIST table uses the area from 0 to z, so its values need a different interpretation. I recommend checking the table’s heading before doing any subtraction.
Frequently Asked Questions
Does a Z Score Require Normally Distributed Data?
No. You can standardize a value whenever a suitable mean and positive standard deviation are available. A standard-normal table gives the correct probability interpretation only when a normal model is appropriate.
What If the Standard Deviation Is 0?
The z-score is undefined because its formula divides by the standard deviation. A standard deviation of 0 means the values have no spread, so this standardization can’t be used.
How Do I Find z From a Probability?
You can select the probability input mode. Enter a value strictly between 0 and 1, then choose a left-tail, right-tail or central area. A central area of 0.95 corresponds to limits of about -1.96 and +1.96.
Why Is the Table Rounded Differently From the Calculator?
A printed table typically uses z to 2 decimal places and probabilities to 5 decimal places. The calculator uses the entered value before rounding its display. Small differences are expected when your z-score has more decimal places.
Further Reading
You should keep the tail direction clear when working by hand. A worked statistics problem helps you check the interpretation as well as the arithmetic.
- Statistics textbooks and workbooks should include exercises on standardization, distributions and sampling assumptions.
- Scientific calculators can help with offline arithmetic. You should check which models your course or exam permits.
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Browse the other calculators for another problem.