Z-Score Calculator with Step-by-Step Solution
Calculate z-scores with a detailed step-by-step breakdown. See the full probability panel: percentile rank, P(Z ≤ z), percentage above, and the symmetric
Z-Score Calculator with Step-by-Step Solution
Calculator
Adjust values & calculateEnter your values below. Every result is computed in your browser — no data is sent to any server.
Formula: z = (x - μ) / σ
Worked example — z = 1.625 | 94.79th percentile
Formula
z = (x - μ) / σ
The z-score formula subtracts the mean (μ) from the raw score (x) and divides by the standard deviation (σ). This standardizes the value to the number of standard deviations from the mean, allowing comparison across different distributions and lookup in the standard normal table.
Worked Examples
Example 1: Test Score Comparison
Problem:A student scored 85 on a test where the class mean is 72 and the standard deviation is 8. What is their z-score and percentile?
Solution:z = (85 - 72) / 8 = 13 / 8 = 1.625 P(Z ≤ 1.625) ≈ 0.9479 Percentile: 94.79th The student scored better than about 94.8% of the class.
Result:z = 1.625 | 94.79th percentile
Example 2: Height Analysis
Problem:Adult male heights have a mean of 70 inches with std dev of 3 inches. What is the z-score for someone who is 64 inches tall?
Solution:z = (64 - 70) / 3 = -6 / 3 = -2.0 P(Z ≤ -2.0) ≈ 0.0228 Percentile: 2.28th Only about 2.3% of men are shorter than 64 inches.
Result:z = -2.0 | 2.28th percentile | 97.72% are taller
Frequently Asked Questions
What is a z-score and what does it mean?
A z-score (also called a standard score) measures how many standard deviations a data point is from the mean. A z-score of 0 means the value is exactly at the mean. A positive z-score means the value is above the mean, and a negative z-score means it is below. For example, a z-score of 1.5 means the value is 1.5 standard deviations above the mean. Z-scores are used to compare values from different distributions and to find probabilities in the standard normal distribution.
How do I interpret the probability from a z-score?
The probability (also called the area under the curve or cumulative probability) tells you what proportion of values in a normal distribution fall below (to the left of) the given z-score. For example, a z-score of 1.96 has a probability of 0.975, meaning 97.5% of values fall below this point. To find the probability above a z-score, subtract from 1. To find between two z-scores, subtract the smaller probability from the larger.
What is the empirical rule (68-95-99.7 rule)?
The empirical rule states that for a normal distribution: about 68% of data falls within ±1 standard deviation (z-scores between -1 and 1), about 95% within ±2 standard deviations (z-scores between -2 and 2), and about 99.7% within ±3 standard deviations (z-scores between -3 and 3). Values beyond ±3 are extremely rare and may be considered outliers.
What is the difference between a z-score and a percentile?
A z-score tells you how many standard deviations a value is from the mean, while a percentile tells you what percentage of values fall below that point. They are related through the cumulative normal distribution function. For example, a z-score of 0 corresponds to the 50th percentile (the median), and a z-score of 1.645 corresponds to approximately the 95th percentile. The z-score is a raw measure, while the percentile is its probability-based interpretation.
Can z-scores be used for non-normal distributions?
Z-scores can be calculated for any distribution using the same formula (z = (x - mean) / std dev), but the probability interpretation only applies perfectly to normal distributions. For non-normal distributions, z-scores still indicate relative position, but the exact probabilities from the z-table may not be accurate. By the Central Limit Theorem, sample means tend toward normality for large samples, so z-scores remain useful for hypothesis testing even with non-normal data.
How do you work backwards from a z-score to the original raw score?
Rearranging z equals (x minus mu) divided by sigma gives x equals mu plus z times sigma. This is how you turn a target percentile into a cut-off on the original scale. If a class has a mean of 72 and a standard deviation of 8, the score marking the 95th percentile uses z equal to 1.645, giving 72 plus 13.16, or about 85.2. A negative z simply subtracts, so z equal to minus 2 corresponds to a raw score of 56.
When should you use a t-score instead of a z-score?
Use z when the population standard deviation sigma is genuinely known, which in practice usually means it is supplied or the sample is very large. When sigma is estimated from the sample itself, that extra uncertainty makes the sampling distribution heavier in the tails, and the correct reference is Student's t with n minus 1 degrees of freedom. The t distribution converges on the standard normal as the sample grows, so with a few hundred observations the two give almost identical tail probabilities.
What z-values mark the usual confidence levels and significance cut-offs?
For a two-tailed 95 percent confidence interval the critical values are plus and minus 1.96, since each tail holds 2.5 percent of the area. A 99 percent two-tailed interval uses plus and minus 2.576, and a 90 percent interval uses plus and minus 1.645. For a one-tailed test at the 5 percent level the single cut-off is 1.645, which is why 1.645 and 1.96 are easy to confuse: the same tail area is split across two tails in one case and concentrated in one tail in the other.
What happens to the mean and spread of a dataset after every value is standardized?
Subtracting the mean from every observation shifts the centre to 0, and dividing by the standard deviation rescales the spread to exactly 1, so the standardized values always have mean 0 and standard deviation 1. What standardizing does not do is change the shape of the distribution: a skewed dataset stays skewed after conversion. This is why standardizing is a legitimate way to put variables measured in different units on a common footing, while normal-based percentiles still require the distribution to be roughly normal.
What does the probability between minus and plus the same z-value represent?
It is the central area of the standard normal curve, the chance of landing within that many standard deviations of the mean in either direction. Because the curve is symmetric, it equals twice the cumulative probability at the positive value minus 1. At z equal to 1 that gives about 68 percent, at 1.96 about 95 percent, and at 2.576 about 99 percent. Use this figure when the question is how typical a value is, and the one-sided tail probability when the question is how extreme it is in one direction.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
Related Calculators
🧮Z-Score Calculator - Quick One-Line Result
z-score calculator. Get instant, accurate results.
🧮Weighted Average Calculator (With Step-by-Step Work)
Calculate weighted average with inputs, formulas, and instant results.
🧮Equation Solver
Solve linear, quadratic, cubic, and polynomial equations with step-by-step solutions.
🧮Mean Median Mode Range Calculator
Calculate mean, median, mode, and range from a data set with step-by-step work.
🧮Related Rates Calculator
Solve related rates problems with step-by-step chain rule application.
🧮Differential Equations Solver
Solve first and second order ordinary differential equations with step-by-step work.
🧮Eigenvalue Calculator
Calculate eigenvalues and eigenvectors of a square matrix with step-by-step work.
🧮Coordinate Distance Calculator
Calculate distance between two points using the distance formula with step-by-step work.