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### Course: Statistics and probability>Unit 4

Lesson 2: Z-scores

# 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:
$z=\frac{\text{data point}-\text{mean}}{\text{standard deviation}}$
Here's the same formula written with symbols:
$z=\frac{x-\mu }{\sigma }$
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 $-3$.

### Example 1

The grades on a history midterm at Almond have a mean of $\mu =85$ and a standard deviation of $\sigma =2$.
Michael scored $86$ on the exam.
Find the z-score for Michael's exam grade.
$\begin{array}{rl}z& =\frac{\text{his grade}-\text{mean grade}}{\text{standard deviation}}\\ \\ z& =\frac{86-85}{2}\\ \\ z& =\frac{1}{2}=0.5\end{array}$
Michael's z-score is $0.5$. 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 $\mu =82$ and a standard deviation of $\sigma =4$.
Michael scored $74$ on the exam.
Find the z-score for Michael's exam grade.
$\begin{array}{rl}z& =\frac{\text{his grade}-\text{mean grade}}{\text{standard deviation}}\\ \\ z& =\frac{74-82}{4}\\ \\ z& =\frac{-8}{4}=-2\end{array}$
Michael's z-score is $-2$. His grade was two standard deviations below the mean.

### Practice problems

Problem A
The grades on a geometry midterm at Oak have a mean of $\mu =74$ and a standard deviation of $\sigma =4.0$.
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?

• how do I calculate the probability of a z-score?
• 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)
• how do you calculate this: µ=10 and σ =1, P(X>10)
• 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.
• 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!
• 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
• 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?
• 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.