Z-Score Calculator — Free Online Standard Score Tool

How to Use the Z-Score Calculator

1. Enter the observed value (x): This is the raw data point you want to standardize. It can be any real number — a test score, measurement, financial figure, or any other quantitative value from a normally distributed population.
2. Enter the population mean (μ): Input the known or estimated average of the population. For standardized tests, this is often provided (e.g., SAT mean = 1060). For custom datasets, calculate it from your data or use our statistics calculator.
3. Enter the standard deviation (σ): Input the population standard deviation. This must be a positive number. The standard deviation measures how spread out the values are around the mean.
4. Interpret the results: The z-score appears prominently at the top, followed by the formula showing the calculation step. Below that, three probability values show the left tail (percentile), right tail, and between (proportion within ±z) for a standard normal distribution.

Results update instantly as you change any input. The probability calculations use a high-precision approximation of the standard normal cumulative distribution function, accurate to six decimal places for practical use.

Z-Score Formula and Explanation

Variables Explained

• z: The z-score (standard score). It represents how many standard deviations the value is from the mean. Positive z means above the mean; negative z means below.
• x: The raw observed value that you want to convert to a z-score.
• μ (mu): The population mean, representing the center of the distribution.
• σ (sigma): The population standard deviation, measuring the typical spread of values around the mean.
• n: The sample size, used only when calculating z-scores for sample means.

Step-by-Step Example

A student scores 78 on an exam where the class mean is 65 and the standard deviation is 10. Find the z-score and percentile.

1. Identify the values: x = 78, μ = 65, σ = 10
2. Apply the formula: z = (78 - 65) / 10 = 13 / 10 = 1.3
3. Look up P(Z < 1.3): Using the normal CDF, P(Z < 1.3) = 0.9032
4. Convert to percentile: The student is at the 90.32th percentile
5. Interpretation: The student scored higher than approximately 90% of the class

The z-score of 1.3 means the student scored 1.3 standard deviations above the class average. In a normal distribution, about 90.3% of students scored lower. This is a strong performance, well above average but not exceptionally rare.

Practical Examples

Example 1: Kevin's SAT Score Comparison

Kevin scored 1280 on the SAT (mean = 1060, standard deviation = 217) and 29 on the ACT (mean = 21.0, standard deviation = 5.8). He wants to know which score is relatively better.

• SAT z-score: (1280 - 1060) / 217 = 220 / 217 = 1.014
• ACT z-score: (29 - 21.0) / 5.8 = 8.0 / 5.8 = 1.379
• SAT percentile: 84.5th
• ACT percentile: 91.6th

Kevin's ACT z-score (1.379) is higher than his SAT z-score (1.014), meaning his ACT performance is relatively stronger compared to other test-takers. He scored in the 91.6th percentile on the ACT versus the 84.5th percentile on the SAT. This analysis helps Kevin decide which score to emphasize on college applications.

Example 2: Nicole's Quality Control Check

Nicole works in manufacturing where bolts must have a diameter of 10.00 mm with a standard deviation of 0.05 mm. She measures a bolt at 10.13 mm and needs to determine if it should be rejected. The company rejects any bolt with a z-score beyond ±2.

• Z-score: (10.13 - 10.00) / 0.05 = 0.13 / 0.05 = 2.6
• Percentile: 99.53rd
• Probability of being this extreme: P(Z > 2.6) = 0.47%

The z-score of 2.6 exceeds the rejection threshold of ±2. This bolt is 2.6 standard deviations above the target diameter, meaning fewer than 0.5% of properly manufactured bolts would be this far from specification. Nicole correctly flags it for rejection. This approach is fundamental to statistical quality control and six-sigma methodologies.

Example 3: Dr. Park's Clinical Study

Dr. Park is studying blood pressure in adults. The population mean systolic blood pressure is 120 mmHg with a standard deviation of 15 mmHg. A patient has a reading of 152 mmHg. Dr. Park wants to assess how unusual this is.

• Z-score: (152 - 120) / 15 = 32 / 15 = 2.133
• Percentile: 98.36th
• Only 1.64% of the population has a higher reading

A z-score of 2.133 places this patient in the top 1.64% for systolic blood pressure. This reading is statistically unusual (beyond 2 standard deviations) and clinically significant, warranting further monitoring and potential intervention. Z-scores provide an objective framework for identifying abnormal clinical values. For more probability analysis, see our p-value calculator.

Z-Score to Probability Reference Table

Tips and Complete Guide

Understanding the Standard Normal Distribution

The standard normal distribution (Z-distribution) has a mean of 0 and a standard deviation of 1. When you convert a raw value to a z-score, you are essentially mapping it onto this universal scale. This transformation preserves relative positions: a value at the 90th percentile of any normal distribution will always have the same z-score (approximately 1.282). This universality makes z-scores one of the most powerful tools in statistics.

Z-Scores in Hypothesis Testing

In hypothesis testing, z-scores help determine whether observed results are statistically significant. The z-test compares a sample statistic to a hypothesized population parameter. Critical z-values define rejection regions: for a two-tailed test at α = 0.05, reject the null hypothesis if |z| > 1.96. For one-tailed tests, use z > 1.645 (right tail) or z < -1.645 (left tail). Our p-value calculator complements this by computing exact p-values from test statistics.

The Central Limit Theorem Connection

The Central Limit Theorem states that the distribution of sample means approaches a normal distribution as sample size increases, regardless of the shape of the original population distribution. This is why z-scores work well with sample means even for non-normal populations. For sample sizes of 30 or more, the z-score is a reliable measure. For smaller samples from unknown distributions, the t-distribution provides a more appropriate framework.

Common Mistakes to Avoid

• Using sample standard deviation as if it were population standard deviation: If you only have a sample, your z-score is approximate. For small samples, use a t-score instead, which accounts for the added uncertainty from estimating the population standard deviation.
• Assuming normality without checking: Z-score probability interpretations assume a normal distribution. For heavily skewed data (income, prices, time durations), the percentile interpretation may be inaccurate. Always verify that your data is approximately normal before relying on z-score probabilities.
• Forgetting the sign matters: A z-score of -2.5 is just as extreme as +2.5 but in the opposite direction. When testing for significance, pay attention to whether you need a one-tailed or two-tailed interpretation.
• Confusing standard deviation with standard error: When standardizing a sample mean, divide by the standard error (σ/√n), not the standard deviation (σ). This distinction becomes critical for sample sizes greater than 1.
• Rounding too early: When looking up probabilities from z-scores, round the z-score only at the final step. Rounding from z = 1.965 to z = 2.0 changes the probability from 0.9753 to 0.9772, which could affect a hypothesis test conclusion.

Frequently Asked Questions