Main content

## Statistics and probability

# Z-scores review

## What are z-scores?

A z-score measures exactly how many standard deviations above or below the mean a data point is.

Here's the formula for calculating a z-score:

Here's the same formula written with symbols:

Here are some important facts about z-scores:

- A positive z-score says the data point is above average.
- A negative z-score says the data point is below average.
- A z-score close to 0 says the data point is close to average.
- A data point can be considered unusual if its z-score is above 3 or below minus, 3.

*Want to learn more about z-scores? Check out this video.*

### Example 1

The grades on a history midterm at Almond have a mean of mu, equals, 85 and a standard deviation of sigma, equals, 2.

Michael scored 86 on the exam.

**Find the z-score for Michael's exam grade.**

Michael's z-score is 0, point, 5. His grade was half of a standard deviation above the mean.

### Example 2

The grades on a geometry midterm at Almond have a mean of mu, equals, 82 and a standard deviation of sigma, equals, 4.

Michael scored 74 on the exam.

**Find the z-score for Michael's exam grade.**

Michael's z-score is minus, 2. His grade was two standard deviations below the mean.

### Practice problems

*Want to practice more problems like these? Check out this exercise.*

## Want to join the conversation?

- how do I calculate the probability of a z-score?(19 votes)
- That's a great question! Fortunately, the next set of lessons, at https://www.khanacademy.org/math/statistics-probability/modeling-distributions-of-data/normal-distributions-library/v/ck12-org-normal-distribution-problems-qualitative-sense-of-normal-distributions , discuss exactly that. It's great that you're thinking ahead!(25 votes)

- how do you calculate the mean when you are only given the z-scores?(5 votes)
- To calculate the mean, you need to know z-scores, the data, and the standard deviation.

z-score=(data-mean)/standard deviation

data-mean=(z-score)(standard deviation)

mean=(data)-(z-score)(standard deviation)(13 votes)

- how do you calculate this: µ=10 and σ =1, P(X>10)(6 votes)
- How do you find the data when you have the mean, the z-score, and the standard deviation?(2 votes)
- Let x represent the data value, mu represent the mean, sigma represent the standard deviation, and z represent the z-score.

Since the z-score is the number of standard deviations above the mean, z = (x - mu)/sigma.

Solving for the data value, x, gives the formula x = z*sigma + mu.

So the data value equals the z-score times the standard deviation, plus the mean.

Have a blessed, wonderful day!(7 votes)

- what happens when you get the number of X-U/standar desviation ahd you get a number above 3, that number will not be in the tabla of Z(4 votes)
- You could try to find a more extensive Z table, for example here: http://www.intmath.com/counting-probability/z-table.php

But at a certain point the difference in probality at changing z values gets vanishingly small, so it doesn't really matter what the actual value at that point is. If you are faced with this situation, in most cases P > P(Z=3) is a useful and sufficient answer.(1 vote)

- Are z-scores only applicable for normal distributions? You could describe how many standard deviations far a data point is from the mean for any distribution right? But is the term z-score only for normal dists?(3 votes)
- The z-score could be applied to any standard distribution or data set. It definition only depends on the (arithmetic) mean and standard deviation, and no other qualitative properties of the nature of the data set.(2 votes)

- What if I don't have the score but only the Z score. How can I calculate what the score is(3 votes)
- Hello Everybody, I want to ask, how to calculate that has z-score is more than 3 or -3? because sometimes i calculate the z-score and can't get the result because it's out of range from the z-table. Thank you(3 votes)
- Is it necessary to assume the distribution is normal?(2 votes)
- to use z scores. If not, or you do not know the population standard deviation you would use a different kind of score called the t score

For z scoreyou need both the population standard deviation and for the sample to be greater than 30.

There are other aspects in statistics where having a normal distribution is necessary.(1 vote)

- Would anyone mind explaining this ?

For a normal distribution of 1000 cases, how many cases will be farther away from the mean than + 3 standard deviations?

A. about 2

B. at least 500

C. it is impossible to estimate

D. 327(1 vote)