Statistics Tool
Z-Score Calculator
Calculate the Standard Score (Z-Score) for a normal distribution.
Z-Score
2.0000
Above Average
Σ The Formula
Z = (x - μ) / σ
Real World Examples
Example
x=85, μ=75, σ=5 → Z=2.0
# About This Calculator
A Z-Score (or Standard Score) describes a value's relationship to the mean of a group of values. It is measured in terms of standard deviations from the mean. If a Z-score is 0, it indicates that the data point's score is identical to the mean score. A Z-score of 1.0 would indicate a value that is one standard deviation from the mean.
Z-scores may be positive or negative, with a positive value indicating the score is above the mean and a negative score indicating it is below the mean.
Interpreting Z-Scores
- Z = 0: The score is exactly the average.
- Z = +1.0: The score is 1 standard deviation above the mean (approx 84th percentile).
- Z = +2.0: The score is 2 standard deviations above the mean (approx 98th percentile).
- Z = -1.0: The score is 1 standard deviation below the mean (approx 16th percentile).
How To Use
- Enter the **Raw Score (x)** you want to evaluate.
- Enter the **Mean (μ)** of the population/dataset.
- Enter the **Standard Deviation (σ)**.
- The calculator computes the Z-Score instantly.
Frequently Asked Questions
What is a 'good' Z-score?+
It depends! In finding outliers, any Z-score beyond +3 or -3 is usually considered an outlier. In standardized testing (like IQ), a Z-score of +2.0 (IQ 130) is considered 'gifted' (top 2%). A Z-score of 0 is perfectly average.
How do I convert Z-score to percentile?+
You need a Z-table or a calculator (like this one). Roughly: Z=0 is 50th percentile, Z=1 is 84th, Z=2 is 97.7th, Z=-1 is 16th. This relationship holds true for any normal distribution.
Why do we standardize scores?+
To compare apples to oranges! If you got 80/100 on a Math test and 90/100 on History, which is better? You can't say without knowing the class average and spread. If Math Mean=70 (Z=1.0) and History Mean=95 (Z=-0.5), you actually did much better in Math relative to your peers.
Can I calculated probability from Z-score?+
Yes. The Z-score corresponds to the area under the standard normal curve. Detailed probability tables (Z-tables) map every Z-score to the probability of randomly picking a value less than (or greater than) that score.
What if standard deviation is zero?+
Then the Z-score is undefined (division by zero). If SD is 0, it means every single number in the dataset is identical, so there is no variation and no 'score' relative to a spread.
Is Z-Score Calculator free to use?+
Yes, Z-Score Calculator on Matheric is completely free to use. We believe in accessible education and utility for everyone.