Z-Score Calculator
Calculate z-score and percentile rank.
Z-Score Calculation
Must be > 0
Z-Score Results
Z-Score
0.5000
Percentile Rank
69.15%
Above mean
Interpretation
Within 1 standard deviation (68% of data)
Formula:
z = (75 - 70) / 10 = 0.5000
About This Calculator
The Z-Score Calculator helps you calculate the z-score (standard score) of a value in relation to a mean and standard deviation. Z-scores indicate how many standard deviations a value is from the mean and are used in statistics for standardization and hypothesis testing.
A positive z-score means the value is above the mean, while a negative z-score means it's below. Z-scores around 0 indicate values close to the mean, while z-scores beyond ±2 or ±3 may indicate outliers. The calculator also provides percentile rank and probability estimates.
Formula & Calculation Method
Z-Score Formula:
Z-Score:
z = (x - μ) / σ
where:
x = observed value
μ = population mean
σ = population standard deviation
Interpretation:
• z = 0: value equals the mean
• z > 0: value above the mean
• z < 0: value below the mean
• |z| < 1: within 1 SD (68% of data)
• |z| < 2: within 2 SD (95% of data)
• |z| < 3: within 3 SD (99.7% of data)
Z-scores standardize different distributions, allowing comparison across different scales. They are used in hypothesis testing, quality control, and identifying outliers in data analysis.
How to Use This Calculator
- Enter Value: Input the value you want to calculate the z-score for.
- Enter Mean: Input the population or sample mean.
- Enter Standard Deviation: Input the population or sample standard deviation (must be greater than 0).
- View Results: See the z-score, percentile rank, and interpretation.
Frequently Asked Questions
What is a z-score?
A z-score (standard score) measures how many standard deviations a value is from the mean. It standardizes values from different distributions, allowing comparison. A z-score of 1 means the value is 1 standard deviation above the mean.
What does a negative z-score mean?
A negative z-score means the value is below the mean. For example, z = -1 means the value is 1 standard deviation below the mean. The sign indicates direction, not bad or good.
What is a good z-score?
There is no "good" or "bad" z-score - it depends on context. Z-scores between -2 and +2 are common (95% of data in a normal distribution). Z-scores beyond ±3 are rare and may indicate outliers.
How is z-score used in hypothesis testing?
Z-scores are used to determine p-values and make decisions in hypothesis testing. For example, if |z| > 1.96, the result is statistically significant at the 95% confidence level in a two-tailed test.
Calculator by
Prof. Emily Watson, PhD
PhD in Applied MathematicsStatistics Certification