how do I calculate the probability of a z-score?

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!

how do you calculate the mean when you are only given the z-scores?

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)

Is it necessary to assume the distribution is normal?

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.

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?

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.

How do you find the data when you have the mean, the z-score, and the standard deviation?

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!

Main content

Statistics and probability

Course: Statistics and probability > Unit 4

Lesson 2: Z-scores

Z-scores review

Google Classroom

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:

z, equals, start fraction, start text, d, a, t, a, space, p, o, i, n, t, end text, minus, start text, m, e, a, n, end text, divided by, start text, s, t, a, n, d, a, r, d, space, d, e, v, i, a, t, i, o, n, end text, end fraction

Here's the same formula written with symbols:

z, equals, start fraction, x, minus, mu, divided by, sigma, end fraction

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.
[Really?]

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.

\begin{aligned}z&=\dfrac{\text{his grade}-\text{mean grade}}{\text{standard deviation}}\\ \\ z&=\dfrac{86-85}{2}\\ \\ z&=\dfrac{1}{2}=0.5\end{aligned}

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.

\begin{aligned}z&=\dfrac{\text{his grade}-\text{mean grade}}{\text{standard deviation}}\\ \\ z&=\dfrac{74-82}{4}\\ \\ z&=\dfrac{-8}{4}=-2\end{aligned}

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

Practice problems

Problem A

Current

The grades on a geometry midterm at Oak have a mean of mu, equals, 74 and a standard deviation of sigma, equals, 4, point, 0.

Nadia scored 70 on the exam.

Find the z-score for Nadia's exam grade. Round to two decimal places.

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

Want to join the conversation?

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😊
Posted 7 years ago. Direct link to 😊's post “how do I calculate the pr...”
how do I calculate the probability of a z-score?
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(19 votes)
Answer
- Piquan
  Posted 7 years ago. Direct link to Piquan's post “That's a great question! ...”
  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!
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  (25 votes)
Nick Leshuk
Posted 7 years ago. Direct link to Nick Leshuk's post “how do you calculate the ...”
how do you calculate the mean when you are only given the z-scores?
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(5 votes)
Answer
- RacheLee
  Posted 7 years ago. Direct link to RacheLee's post “To calculate the mean, yo...”
  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)
  Button navigates to signup page
  (14 votes)
Tou
Posted 5 years ago. Direct link to Tou's post “how do you calculate this...”
how do you calculate this: µ=10 and σ =1, P(X>10)
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(6 votes)
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Shaghayegh
Posted 3 years ago. Direct link to Shaghayegh's post “Is it necessary to assume...”
Is it necessary to assume the distribution is normal?
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(3 votes)
Answer
- loumast17
  Posted 3 years ago. Direct link to loumast17's post “to use z scores. If not,...”
  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.
  Button navigates to signup page
  (6 votes)
1315031658
Posted 6 years ago. Direct link to 1315031658's post “How do you find the data ...”
How do you find the data when you have the mean, the z-score, and the standard deviation?
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(2 votes)
Answer
- Ian Pulizzotto
  Posted 6 years ago. Direct link to Ian Pulizzotto's post “Let x represent the data ...”
  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!
  Comment on Ian Pulizzotto's post “Let x represent the data ...”
  (7 votes)
sebastian grez
Posted 7 years ago. Direct link to sebastian grez's post “what happens when you get...”
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
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(4 votes)
Answer
- psthman
  Posted 7 years ago. Direct link to psthman's post “You could try to find a m...”
  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.
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  (1 vote)
Bryan
Posted 3 years ago. Direct link to Bryan's post “Are z-scores only applica...”
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?
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(3 votes)
Answer
- Rebecca
  Posted 3 years ago. Direct link to Rebecca's post “The z-score could be appl...”
  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.
  Button navigates to signup page
  (2 votes)
Samael Pena
Posted 4 years ago. Direct link to Samael Pena's post “What if I don't have the ...”
What if I don't have the score but only the Z score. How can I calculate what the score is
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(3 votes)
Answer
muhammad fajar rizqi
Posted 6 years ago. Direct link to muhammad fajar rizqi's post “Hello Everybody, I want t...”
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
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(3 votes)
Answer
Rosivette Andrade
Posted 4 years ago. Direct link to Rosivette Andrade's post “Would anyone mind explain...”
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
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(2 votes)
Answer