Z Score Calculator
Z-score calculator
Compute standardized score z with percentile, tail-risk, and equivalent-score diagnostics.
Result
z = 1.75
95.9941% percentile (lower-tail) with two-tailed probability 0.08011823.
z diagnostics
Advanced options
Advanced insights
Z-score interpretation
Moderately unusual
z-score
1.75
Percentile (lower-tail)
95.994089%
T-score equivalent
67.5 (mean 50, sd 10)
IQ-equivalent / Stanine
126.25 / 8
Tail probabilities
Lower-tail P(X ≤ x)
0.959940885
Upper-tail P(X ≥ x)
0.040059115
Two-tailed probability
0.08011823
Model assumptions
- Interpretation assumes an approximately normal distribution.
- Standard deviation must be positive and in the same unit as x and mean.
- Percentile and p-values are based on standard normal approximation.
Formula
z = (x - mean) / sd Symbol legend
| Symbol | Meaning | Unit | Copy |
|---|---|---|---|
z | Standardized score | standard deviations | |
x | Observed value | data unit | |
\mu | Mean | data unit | |
\sigma | Standard deviation | data unit |
- Subtract mean from observed value.
- Divide by standard deviation.
- Interpret sign and magnitude of resulting z-score, then review percentile and tail probabilities.
Example
Worked example: x=72, mean=65, sd=4
- 1 Difference = 72 - 65 = 7
- 2 z = 7/4
- 3 z = 1.75
- 4 Percentile ≈ 96.0% and two-tailed probability ≈ 0.0801
The z-score is 1.75, which is near the 96th percentile.
How
- Choose a scenario preset for quick defaults or keep custom values.
- Enter observed value x.
- Enter dataset mean and standard deviation.
- Read z-score with positive or negative sign.
- Use percentile and two-tailed probability to assess rarity under a normal model.
Avoid
- Using variance value instead of standard deviation.
- Forgetting to subtract mean before division.
- Interpreting negative z as invalid.
- Treating percentile and two-tailed probability as exact guarantees rather than model-based estimates.
FAQ
Can z-score be negative?
Yes, negative z means the value is below the mean.
What does z equal zero mean?
z equals zero indicates the value is exactly at the mean.
Can I compare different datasets with z-score?
Yes, z-score helps compare values on different scales.
What does two-tailed probability mean in this tool?
It reports the probability of observing a value at least as far from the mean in either direction under the standard normal model.
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