Z-Score Calculator
Calculate the Z-score (standard score) of a raw value using Z = (X − μ) / σ, and see where it falls on the standard normal distribution.
Standard Normal Distribution
Your raw score of 85 is a moderate distance from the mean (1–2 standard deviations), and is higher than approximately 89.435% of values in a normal distribution with this mean and standard deviation.
Step-by-Step Solution
Here's exactly how this answer was calculated, one step at a time.
- 1
Write down the given values
X = 85, μ = 75, σ = 8
X is the raw score you're standardizing, μ (mu) is the mean of the data set, and σ (sigma) is the standard deviation of the data set.
- 2
Subtract the mean from the raw score
X − μ = 85 − 75 = 10
This tells you how far the raw score is from the mean, in the original units of the data — positive if above the mean, negative if below.
- 3
Divide by the standard deviation
Z = (X − μ) / σ = 10 / 8 = 1.25
Dividing by σ rescales the difference into 'number of standard deviations' — this is the Z-score, and it lets you compare values from any normal distribution on the same standard scale.
- 4
Convert the Z-score to a percentile
Φ(1.25) = 0.8944 → 89.435%
Φ(Z), the standard normal cumulative distribution function, gives the proportion of values that fall below this Z-score — exactly what a printed Z-table would give you, computed instantly.
- 5
Interpret the result
Z = 1.25 → a moderate distance from the mean (1–2 standard deviations)
A score of 85 is above the mean, and higher than approximately 89.435% of values in this distribution.
✓ Final Answer: Z = 1.25 (≈ 89.435th percentile)
Free Z-Score Calculator
This Z-score calculator (also called a standard score calculator) tells you how many standard deviations a raw score is from the mean of its data set. Enter a raw score, the mean, and the standard deviation, and get the Z-score instantly, along with the approximate percentile that score falls at in a normal distribution.
Z-scores are widely used in statistics, standardized testing, quality control, finance, and research to compare values from different data sets on a common scale.
Z-Score Formula
The Z-score formula is: Z = (X − μ) / σ, where X is the raw score, μ (mu) is the mean of the data set, and σ (sigma) is the standard deviation of the data set. A positive Z-score means the value is above the mean; a negative Z-score means it's below the mean.
How to Interpret a Z-Score
A Z-score tells you exactly how unusual or typical a value is relative to the rest of its data set, assuming the data is roughly normally distributed.
- Z = 0 means the value is exactly equal to the mean.
- Z between −1 and 1 covers about 68% of values in a normal distribution.
- Z between −2 and 2 covers about 95% of values.
- Z between −3 and 3 covers about 99.7% of values (this is the basis of the '68-95-99.7 rule').
- A Z-score beyond ±3 is considered a rare, extreme value in most real-world data sets.
Worked Example
A student scores 85 on a test where the class mean is 75 and the standard deviation is 8. Z = (85 − 75) / 8 = 1.25. This means the student's score is 1.25 standard deviations above the mean, placing them at roughly the 89th percentile of the class.
Frequently Asked Questions
What is a Z-score used for?
A Z-score standardizes a raw value so it can be compared across different data sets or distributions. It's commonly used in standardized testing, statistics, finance (to spot outliers or risk), and quality control (to detect values outside acceptable limits).
What does a negative Z-score mean?
A negative Z-score means the raw value is below the mean of its data set. The more negative the Z-score, the further below average the value is.
How do you convert a Z-score to a percentile?
A Z-score is converted to a percentile using the cumulative distribution function (CDF) of the standard normal distribution, which gives the proportion of values that fall below that Z-score. This calculator does that conversion automatically.
What is considered a 'good' Z-score?
There's no universal 'good' Z-score — it depends on context. In most cases, a Z-score close to 0 means a value is typical, while a Z-score beyond ±2 or ±3 indicates the value is unusually high or low compared to the rest of the data set.