Z-Score Calculator with Step-by-Step Solution
Calculate z-scores and find probabilities from the standard normal distribution table. Enter values for instant results with step-by-step formulas.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator
Z-Score Calculator with Step-by-Step Solution
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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.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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