z-score calculator
Math & numbersHow unusual is this value?
The z-score and percentile of a value on a normal distribution, the value at a percentile, or the chance between two values.
What you enter stays on your device while you use this tool. Opening a shared link sends the settings in its address to HowToolbox to prepare the page.
z-score
2
130 lies 2 standard deviations above the mean; 97.72% of values are lower.
More about the value
- Below 130
- 97.72%
- Above 130
- 2.28%
- At least as far from the mean (two-tailed)
- 4.55%
Sponsored Links
Normal distribution
The bell curve with mean 100 and standard deviation 15; the shaded area is the share in the answer.
| Value | z-score | Share below |
|---|---|---|
| 55 | -3 | 0.13% |
| 70 | -2 | 2.28% |
| 85 | -1 | 15.87% |
| 100 | 0 | 50% |
| 115 | 1 | 84.13% |
| 130 | 2 | 97.72% |
| 145 | 3 | 99.87% |
The z-score and percentile of a value on a normal distribution, the value at a percentile, or the chance between two values.
How to use
- Choose what to find: the z-score of a value, the value at a percentile, or the chance between two values.
- Enter the mean and standard deviation of the distribution; use 0 and 1 for the standard normal.
- Read the answer and the shaded area on the curve; the table lists the values one, two and three standard deviations from the mean.
How it works
A z-score says how many standard deviations a value lies from the mean: z = (x − μ) ÷ σ, where μ is the mean and σ the standard deviation. A positive z is above the mean, a negative one below. On a normal (bell-shaped) distribution the z-score also tells what share of values are lower, its percentile.
That share is the area under the bell curve to the left of z, written Φ(z). For IQ-style scores with mean 100 and standard deviation 15, a score of 130 has z = (130 − 100) ÷ 15 = 2, and Φ(2) ≈ 97.72 %, so about 2.3 % score higher. About 68 % of values lie within one standard deviation of the mean, 95 % within two and 99.7 % within three. The definitions follow the NIST/SEMATECH e-Handbook of Statistical Methods.
The percentages hold only for data that is close to normally distributed. Incomes, waiting times and other skewed data can be far from it, and then a z-score still measures distance from the mean but not the share of values below.
Worked example
What is the z-score of 65 when the mean is 50 and standard deviation is 10?
- z = (65 − 50) ÷ 10 = 1.5.
The value is 1.5 standard deviations above the mean.
Sponsored Links
Frequently asked questions
What is a good z-score?
It depends on the question. A z-score between −2 and 2 is ordinary; beyond ±2 is unusual (about 5 % of values) and beyond ±3 rare (about 0.3 %).
How do I turn a z-score into a percentile?
Choose z-score, set the mean to 0 and the standard deviation to 1, and enter the z-score as the value. The share below it is the percentile.
What z-score is the 95th percentile?
About 1.645. The value with 97.5 % below it, z ≈ 1.96, is the one used for 95 % confidence intervals, because it leaves 2.5 % in each tail.
Can I use this for a p-value?
Yes, for a z-test: the share above a positive z is the one-tailed p-value, and the two-tailed figure is listed below the answer.
Can a z-score be negative?
Yes. A negative z-score means the value is below the mean. Zero means it equals the mean.
Next you might need
- Average calculatorMean, median, mode and range of a list of numbers, with the count, sum, smallest and largest., private, runs on your deviceMath & numbersCalculatorStatistics
- Combinations and permutations calculatorExact counts of combinations and permutations, with or without repetition, and the chance of one particular pick., private, runs on your deviceMath & numbersCalculatorStatistics
- Percentage calculatorPercent of a number, what percent one number is of another, percentage change and difference, and the number before a discount., private, runs on your deviceMath & numbersCalculatorCostsStatistics
- Sample size calculatorThe number of survey responses you need for a margin of error and confidence level, for any population size., private, runs on your deviceMath & numbersCalculatorStatistics
- Fraction calculatorAdd, subtract, multiply and divide fractions and mixed numbers, or turn a decimal into a fraction, with the answer in lowest terms., private, runs on your deviceMath & numbersCalculator
- GCD and LCM calculatorThe greatest common divisor and the least common multiple of two to ten whole numbers, with the Euclidean steps., private, runs on your deviceMath & numbersCalculatorReference
More in Math & numbers
All 24 utilities in Math & numbers- Area of shapesArea and perimeter of rectangles, circles, triangles and eight more shapes, drawn to scale., private, runs on your device
- Number base converterConvert whole numbers between binary, octal, decimal, hexadecimal and any base up to 36., private, runs on your device
- Prime number checkerTest any whole number up to 100 digits for primality, with its prime factors and the next and previous prime., private, runs on your device
- Ratio calculatorSimplify a ratio to its lowest terms, or find the missing number in an equal ratio, the rule of three., private, runs on your device
- Scientific calculatorType a calculation or use the keys., private, runs on your device
- Scientific notation converterWrite a number in scientific and engineering notation with the significant figures you choose, or turn 6.022e23 back into digits., private, runs on your device
Popular now
Sponsored Links
Sponsored Links
Sponsored Links