π Statistics & Probability
Z Score Calculator
Calculate z-score from a value, mean, and standard deviation. Free z-score calculator with probability lookup.
What This Calculator Tells You
A z-score tells you how many standard deviations a value sits above or below the average. A z-score of 0 is exactly average, +1.5 is one and a half standard deviations above, and β2 is two below.
Because z-scores put different measurements on the same scale, they let you compare things that are otherwise hard to compare β an exam score and a sales figure, or results from two different tests. The calculator also shows the percentage of a normal distribution below and above your value.
What to Enter
- Value (X) β the individual result you want to assess.
- Mean (ΞΌ) β the average of the group it belongs to.
- Std Dev (Ο) β the standard deviation of that group. Calculate it from raw data with the standard deviation calculator.
Formula: z = (X β ΞΌ) Γ· Ο. The standard deviation must be greater than zero.
Worked Examples
1. An exam score
A student scores 85 on a test where the class average is 70 and the standard deviation is 10.
- Result β z = 1.50; about 93.3% of a normal distribution falls below this score.
2. Comparing results on different scales
A score of 1,350 on a test with mean 1,050 and standard deviation 200 also gives z = 1.50.
- 85 on the first test and 1,350 on the second are equally strong relative to their groups, even though the raw numbers look nothing alike.
3. A below-average result
A store sells 42 units in a week when the weekly average is 50 and the standard deviation is 5.
- Result β z = β1.60; only about 5.5% of weeks would be this low if sales are normally distributed β worth investigating.
How It’s Calculated
- z: Z-score (standard normal)
- x: Observed value
- ΞΌ: Population mean
- Ο: Population standard deviation
How to Read Your Result
- Between β1 and +1: fairly typical β about 68% of values in a normal distribution fall here.
- Beyond Β±2: unusual β only about 5% of values fall outside this range.
- Beyond Β±3: rare β about 0.3% of values. Often worth checking for errors or special causes.
- Percentiles assume a normal distribution. If your data is skewed (incomes, house prices, order values), the percentages will be off, though the z-score itself is still valid.
- To turn a test statistic into a p-value, use the p-value calculator.
Common Mistakes to Avoid
- Using the wrong group's mean and standard deviation. Compare a value to the group it actually belongs to.
- Trusting percentiles for skewed data. The "% below" figures rely on a bell-shaped distribution.
- Mixing sample and population standard deviation. For small groups, the choice changes the z-score noticeably; be consistent.
- Reading the sign backwards. Negative means below average, not "bad" β for costs or defect rates, below average is good.
When to Use This Calculator
- Compare scores from tests with different scales.
- Flag unusual days, weeks or transactions in business data.
- Standardize metrics before combining them into a score.
- Check statistics homework.
Frequently Asked Questions
What is a good z-score?
There is no universal "good". Positive means above average; how far above matters relative to your goal. In quality control, values beyond Β±3 are often treated as signals.
Can a z-score be negative?
Yes. A negative z-score means the value is below the mean.
How do I convert a z-score to a percentile?
Look up the cumulative normal probability β the calculator does this for you. z = 1.5 corresponds to roughly the 93rd percentile.
What is the difference between a z-score and a t-score?
Both standardize values, but t-scores are used for small samples where the population standard deviation is unknown. Read z-scores and p-values explained for more.
Do you store the values I enter?
No. All calculations happen in your browser. Your data never leaves your device.
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