Turn a raw score into a z-score and its probability, look up any z-score or probability without a z-table, or find the chance between two z-scores.
Enter a value, the mean of the population and its standard deviation.
This works like a z-table. Fill in any one box and the rest are worked out.
Find the area under the normal curve between a left and a right z-score.
A z-score says how many standard deviations a value sits above or below the average. A z-score of 0 is exactly average, +1 is one standard deviation above, and −2 is two below. That makes very different things comparable. A 78 on an easy exam and a 64 on a hard one can both be turned into z-scores, and the higher z-score is the better result relative to the group. The calculation is simply (x − μ) ÷ σ.
Say a test has a mean of 70 and a standard deviation of 10, and you scored 85. Your z-score is (85 − 70) ÷ 10 = 1.5. Enter those three numbers above and the probability of scoring below 85 comes out as 0.93319, so you did better than about 93% of the group, and about 6.7% scored higher. Because the figure assumes a roughly normal bell-shaped spread of scores, it is a good guide for exams, heights and measurement errors, and a poor one for heavily skewed data like incomes.
Before calculators, people looked these probabilities up in a printed z-table. The converter does the same job in both directions. Enter a z-score to see the area to its left, to its right and in between, or enter a probability to get the z-score that goes with it. A few values are worth memorising: the 90th percentile is at z = 1.282, the 95th at 1.645, the 97.5th at 1.96 and the 99th at 2.326. They show up again and again in testing and in building ranges of likely values.
For a normal distribution, about 68% of values fall within one standard deviation of the mean, 95% within two, and 99.7% within three. You can check that with the last box: between z = −1 and 1 the area is 0.68269, between −2 and 2 it is 0.9545, and between −3 and 3 it is 0.9973. A value more than three standard deviations out is rare, about 3 in 1,000, which is why it often gets flagged as an outlier.
If you have a list of raw data instead of a known mean and standard deviation, get them first from the Statistics Calculator or the Standard Deviation Calculator, then bring the numbers back here. When you are building a range for an unknown average rather than judging one value, the same z-values (1.96 for 95%) drive the Confidence Interval Calculator. And for questions about chances of independent events rather than a bell curve, the Probability Calculator is the right tool.
Using the sample standard deviation when the population value is known, or the other way round, shifts the z-score slightly, so be clear which one you have. Another mistake is applying a z-score to a sample average without dividing the standard deviation by the square root of the sample size. Finally, remember that the probabilities assume a normal distribution, so check with a quick histogram or common sense before trusting them for strongly lopsided data.
Subtract the mean from your value and divide by the standard deviation: z = (x − μ) ÷ σ.
It depends on what you are measuring. A positive z-score means above average and a negative one below. For exam results, a z-score above 1 puts you roughly in the top 16%.
Use the converter and read P(x < Z). That probability, written as a percentage, is the percentile. A z-score of 1.5 gives 0.93319, which is about the 93rd percentile.
Yes. A negative z-score means the value is below the mean. The probability to its left is then less than 0.5.
A z-score uses the known population standard deviation. A t-score is used when you only have a small sample and must estimate the standard deviation from it, which gives slightly wider tails.
Pick another tool to jump straight to it.
Get the mean and standard deviation from your data.
LiveMean, median, mode, range and spread in one place.
LiveUse z-values to build a range for the mean.
LiveChances of single and combined events.