Z-Score Calculator: Standardize and Find Probabilities

A z-score measures how many standard deviations a value is from the mean. It is the foundation of statistical inference, allowing you to convert any normal distribution into a standardized scale where you can find probabilities using a universal normal table or calculator.

The Formula

z = (x - μ) / σ

Where:

• x = the value you're evaluating
• μ (mu) = population mean
• σ (sigma) = population standard deviation

A z-score of 0 means the value equals the mean. Positive z-scores are above the mean; negative z-scores are below. The magnitude tells you the distance in standard deviations.

Worked Example

A college entrance exam has mean 500 and standard deviation 100. You score 650. What is your z-score?

z = (650 - 500) / 100 = 150 / 100 = 1.5

Your score is 1.5 standard deviations above the mean. Using the standard normal table, P(z ≤ 1.5) ≈ 0.9332, meaning about 93.32% of test-takers scored below you.

Using Z-Score Tables

After calculating z, you look up its probability in a standard normal table, which gives cumulative probabilities P(Z ≤ z). Tables show:

• One-tailed probabilities: P(Z ≤ z) or P(Z ≥ z)
• Two-tailed probabilities: useful for confidence intervals and hypothesis tests

For example, z = 1.96 corresponds to P(Z ≤ 1.96) ≈ 0.975. The area in both tails beyond z = ±1.96 is 0.05, which is why 1.96 is the critical value for 95% confidence intervals.

Common Z-Score Cutoffs

When to Use

Z-scores are essential for:

• Comparing values from different distributions
• Finding probabilities using the normal distribution
• Identifying outliers (usually |z| > 3)
• Hypothesis testing and confidence intervals
• Standardizing test scores

Tips

Z-scores only work for normally distributed data. If your distribution is severely skewed or has heavy tails, z-scores will be misleading. Also, remember the difference between z (population parameter) and t (sample statistic) — use z when σ is known, t when you estimate it from the sample.

Use our Z-Score Calculator to convert scores to z-scores and find probabilities instantly.