Z-Score: Definition, Formula, Calculation & Interpretation

Saul Mcleod, PhD

Editor-in-Chief for Simply Psychology

BSc (Hons) Psychology, MRes, PhD, University of Manchester

Saul Mcleod, Ph.D., is a qualified psychology teacher with over 18 years experience of working in further and higher education. He has been published in peer-reviewed journals, including the Journal of Clinical Psychology.

Learn about our Editorial Process

Olivia Guy-Evans, MSc

Associate Editor for Simply Psychology

BSc (Hons) Psychology, MSc Psychology of Education

Olivia Guy-Evans is a writer and associate editor for Simply Psychology. She has previously worked in healthcare and educational sectors.

On This Page:

A z-score describes the position of a raw score in terms of its distance from the mean when measured in standard deviation units. The z-score is positive if the value lies above the mean and negative if it lies below the mean.

It is also known as a standard score because it allows the comparison of scores on different kinds of variables by standardizing the distribution. A standard normal distribution (SND) is a normally shaped distribution with a mean of 0 and a standard deviation (SD) of 1 (see Fig. 1).

Gauss distribution. Standard normal distribution. Gaussian bell graph curve. Business and marketing concept. Math probability theory.

Why Are Z-Scores Important?

It is useful to standardize the values (raw scores) of a  normal distribution  by converting them into z-scores because:

  • It allows researchers to calculate the probability of a score occurring within a standard normal distribution;
  • It enables us to compare two scores from different samples (which may have different means and standard deviations).

How To Calculate

The formula for calculating a z-score is z = (x-μ)/σ, where x is the raw score, μ is the population mean, and σ is the population standard deviation.

As the formula shows, the z-score is simply the raw score minus the population mean, divided by the population standard deviation.

Z score formula

When the population mean and the population standard deviation are unknown, the standard score may be calculated using the sample mean (x̄) and sample standard deviation (s) as estimates of the population values.

Interpretation

The value of the z-score tells you how many standard deviations you are away from the mean. If a z-score is equal to 0, it is on the mean.

  • A positive z-score indicates the raw score is higher than the mean average. For example, if a z-score is equal to +1, it is 1 standard deviation above the mean.
  • A negative z-score reveals the raw score is below the mean average. For example, if a z-score is equal to -2, it is two standard deviations below the mean.

Another way to interpret z-scores is by creating a standard normal distribution, also known as the z-score distribution, or probability distribution (see Fig. 3).

Standard Normal Distribution (SND)

  • The SND (i.e., z-distribution) is always the same shape as the raw score distribution. For example, if the distribution of raw scores is normally distributed, so is the distribution of z-scores.
  • The mean of any SND always = 0.
  • The standard deviation of any SND always = 1. Therefore, one standard deviation of the raw score (whatever raw value this is) converts into 1 z-score unit.

The SND allows researchers to calculate the probability of randomly obtaining a score from the distribution (i.e., sample). For example, there is a 68% probability of randomly selecting a score between -1 and +1 standard deviations from the mean (see Fig. 3).

Proportion of a Standard Normal Distribution (SND) in %

The probability of randomly selecting a score between -1.96 and +1.96 standard deviations from the mean is 95% (see Fig. 3).

If there is less than a 5% chance of a raw score being selected randomly, then this is a statistically significant result.

Learn how to use a z-score table

Practice Problems for Z-Scores

Calculate the z-scores for the following:

Sample Questions

  • Scores on a psychological well-being scale range from 1 to 10, with an average score of 6 and a standard deviation of 2. What is the z-score for a person who scored 4?
  • On a measure of anxiety, a group of participants show a mean score of 35 with a standard deviation of 5. What is the z-score corresponding to a score of 30?
  • A depression inventory has an average score of 50 with a standard deviation of 10. What is the z-score corresponding to a score of 70?
  • In a study on sleep, participants report an average of 7 hours of sleep per night, with a standard deviation of 1 hour. What is the z-score for a person reporting 5 hours of sleep?
  • On a memory test, the average score is 100, with a standard deviation of 15. What is the z-score corresponding to a score of 85?
  • A happiness scale has an average score of 75 with a standard deviation of 10. What is the z-score corresponding to a score of 95?
  • An intelligence test has a mean score of 100 with a standard deviation of 15. What is the z-score that corresponds to a score of 130?

Answers for Sample Questions

Double-check your answers with these solutions. Remember, for each problem, you subtract the average from your value, then divide by how much values typically vary (the standard deviation).

  • Z-score = (4 – 6)/2 = -1
  • Z-score = (30 – 35)/5 = -1
  • Z-score = (70 – 50)/10 = 2
  • Z-score = (5 – 7)/1 = -2
  • Z-score = (85 – 100)/15 = -1
  • Z-score = (95 – 75)/10 = 2
  • Z-score = (130 – 100)/15 = 2

Calculating a Raw Score

Sometimes we know a z-score and want to find the corresponding raw score. The formula for calculating a z-score in a sample into a raw score is given below:

X = (z)(SD) + mean

As the formula shows, the z-score and standard deviation are multiplied together, and this figure is added to the mean.

Check your answer makes sense: If we have a negative z-score, the corresponding raw score should be less than the mean, and a positive z-score must correspond to a raw score higher than the mean.

Calculating a Z-Score using Excel

To calculate the z-score of a specific value, x, first, you must calculate the mean of the sample by using the AVERAGE formula.

For example, if the range of scores in your sample begins at cell A1 and ends at cell A20, the formula =AVERAGE(A1:A20) returns the average of those numbers.

Next, you must calculate the standard deviation of the sample by using the STDEV.S formula. For example, if the range of scores in your sample begins at cell A1 and ends at cell A20, the formula = STDEV.S (A1:A20) returns the standard deviation of those numbers.

Now to calculate the z-score, type the following formula in an empty cell: = (x – mean) / [standard deviation].

To make things easier, instead of writing the mean and SD values in the formula, you could use the cell values corresponding to these values. For example, = (A12 – B1) / [C1].

Then, to calculate the probability for a SMALLER z-score, which is the probability of observing a value less than x (the area under the curve to the LEFT of x), type the following into a blank cell: = NORMSDIST( and input the z-score you calculated).

To find the probability of LARGER z-score, which is the probability of observing a value greater than x (the area under the curve to the RIGHT of x), type: =1 – NORMSDIST (and input the z-score you calculated).

Frequently Asked Questions

Can z-scores be used with any type of data, regardless of distribution.

Z-scores are commonly used to standardize and compare data across different distributions. They are most appropriate for data that follows a roughly symmetric and bell-shaped distribution.

However, they can still provide useful insights for other types of data, as long as certain assumptions are met. Yet, for highly skewed or non-normal distributions, alternative methods may be more appropriate.

It’s important to consider the characteristics of the data and the goals of the analysis when determining whether z-scores are suitable or if other approaches should be considered.

How can understanding z-scores contribute to better research and statistical analysis in psychology?

Understanding z-scores enhances research and statistical analysis in psychology. Z-scores standardize data for meaningful comparisons, identify outliers, and assess likelihood.

They aid in interpreting practical significance, applying statistical tests, and making accurate conclusions. Z-scores provide a common metric, facilitating communication of findings.

By using z-scores, researchers improve rigor, objectivity, and clarity in their work, leading to better understanding and knowledge in psychology.

Can a z-score be used to determine the likelihood of an event occurring?

No, a z-score itself cannot directly determine the likelihood of an event occurring. However, it provides information about the relative position of a data point within a distribution.

By converting data to z-scores, researchers can assess how unusual or extreme a value is compared to the rest of the distribution. This can help estimate the probability or likelihood of obtaining a particular score or more extreme values.

So, while z-scores provide insights into the relative rarity of an event, they do not directly determine the likelihood of the event occurring on their own.

Further Information

How to Use a Z-Table (Standard Normal Table) to Calculate the Percentage of Scores Above or Below the Z-Score

Z-Score Table (for positive or negative scores)

Statistics for Psychology Book Download

Print Friendly, PDF & Email

  • PRO Courses Guides New Tech Help Pro Expert Videos About wikiHow Pro Upgrade Sign In
  • EDIT Edit this Article
  • EXPLORE Tech Help Pro About Us Random Article Quizzes Request a New Article Community Dashboard This Or That Game Popular Categories Arts and Entertainment Artwork Books Movies Computers and Electronics Computers Phone Skills Technology Hacks Health Men's Health Mental Health Women's Health Relationships Dating Love Relationship Issues Hobbies and Crafts Crafts Drawing Games Education & Communication Communication Skills Personal Development Studying Personal Care and Style Fashion Hair Care Personal Hygiene Youth Personal Care School Stuff Dating All Categories Arts and Entertainment Finance and Business Home and Garden Relationship Quizzes Cars & Other Vehicles Food and Entertaining Personal Care and Style Sports and Fitness Computers and Electronics Health Pets and Animals Travel Education & Communication Hobbies and Crafts Philosophy and Religion Work World Family Life Holidays and Traditions Relationships Youth
  • Browse Articles
  • Learn Something New
  • Quizzes Hot
  • This Or That Game New
  • Train Your Brain
  • Explore More
  • Support wikiHow
  • About wikiHow
  • Log in / Sign up
  • Education and Communications
  • Mathematics
  • Probability and Statistics

How to Calculate Z Scores

Last Updated: December 29, 2023 Fact Checked

This article was co-authored by Mario Banuelos, PhD . Mario Banuelos is an Assistant Professor of Mathematics at California State University, Fresno. With over eight years of teaching experience, Mario specializes in mathematical biology, optimization, statistical models for genome evolution, and data science. Mario holds a BA in Mathematics from California State University, Fresno, and a Ph.D. in Applied Mathematics from the University of California, Merced. Mario has taught at both the high school and collegiate levels. There are 8 references cited in this article, which can be found at the bottom of the page. This article has been fact-checked, ensuring the accuracy of any cited facts and confirming the authority of its sources. This article has been viewed 1,874,950 times.

A Z score allows you to take any given sample within a set of data and to determine how many standard deviations above or below the mean it is. [1] X Trustworthy Source Simply Psychology Popular site for evidence-based psychology information Go to source To find the Z score of a sample, you'll need to find the mean, variance and standard deviation of the sample. To calculate the z-score, you will find the difference between a value in the sample and the mean, and divide it by the standard deviation. Even though there are lots of steps to this method from start to finish, it is a fairly simple calculation.

Calculating the Mean

Step 1 Look at your data set.

  • Know how many numbers are in your sample. In the case of the sample of palm trees, there are 5 in this sample.
  • Know what the numbers represent. In our example, these numbers represent measurements of trees.
  • Look at the variation in the numbers. Does the data vary across a large range, or a small range?

Step 2 Gather all of your data.

  • The mean is the average of all the numbers in your sample.
  • To calculate this you will add all the numbers in your sample together, then divide by the sample size.
  • In mathematical notation, n represents the sample size. In the case of our sample of tree heights, n = 5 since there are 5 numbers in this sample.

Step 3 Add all the numbers in your sample together.

  • For example, using the sample of 5 palm trees, our sample consists of 7, 8, 8, 7.5, and 9.
  • 7 + 8 + 8 + 7.5 + 9 = 39.5. This is the sum of all the numbers in your sample.
  • Check your answer to make sure you did your addition correctly.

Step 4 Divide the sum by your sample size (n).

  • For example, use our sample of tree heights: 7, 8, 8, 7.5, and 9. There are 5 number in our sample so n = 5.
  • The sum of tree heights in our sample was 39.5. You would then divide this figure by 5 to figure out the mean.
  • 39.5/5 = 7.9.
  • The mean tree height is 7.9 feet. The population mean is often represented by the symbol μ, therefore μ = 7.9

Finding the Variance

Step 1 Find the variance.

  • This calculation will give you an idea about how far your data is spread out.
  • Samples with low variance have data that is clustered closely about the mean.
  • Samples with high variance have data that is spread far from the mean.
  • Variance is often used to compare the distributions between two data sets or samples.

Step 2 Subtract the mean from each of the numbers in your sample.

  • In our sample of tree heights (7, 8, 8, 7.5, and 9 feet) the mean was 7.9.
  • 7 - 7.9 = -0.9, 8 - 7.9 = 0.1, 8 - 7.9 = 0.1, 7.5 - 7.9 = -0.4, and 9 - 7.9 = 1.1.
  • Do these calculations again to check your math. It is extremely important that you have the right figures for this step.

Step 3 Square all of the answers from the subtractions you just did.

  • Remember, in our sample we subtracted the mean of 7.9 from each of our data points (7, 8, 8, 7.5, and 9) and came up with the following: -0.9, 0.1, 0.1, -0.4, and 1.1.
  • Square all of these figures: (-0.9)^2 = 0.81, (0.1)^2 = 0.01, (0.1)^2 = 0.01, (-0.4)^2 = 0.16, and (1.1)^2 = 1.21.
  • The squares from this calculation are: 0.81, 0.01, 0.01, 0.16, and 1.21.
  • Check your answers before proceeding to the next step.

Step 4 Add the squared numbers together.

  • In our sample of tree heights, the squares were as follows: 0.81, 0.01, 0.01, 0.16, and 1.21.
  • 0.81 + 0.01 + 0.01 + 0.16 + 1.21 = 2.2
  • For our example of tree heights, the sum of squares is 2.2.
  • Check your addition to make sure that you have the right figure before moving on.

Step 5 Divide the sum of squares by (n-1).

  • In our sample of tree heights (7, 8, 8, 7.5, and 9 feet), the sum of squares was 2.2.
  • There are 5 numbers in this sample. Therefore n = 5.
  • Remember the sum of squares is 2.2. To find the variance, calculate the following: 2.2 / 4.
  • 2.2 / 4 = 0.55
  • Therefore the variance for this sample of tree heights is 0.55.

Calculating the Standard Deviation

Step 1 Find your variance figure.

  • Variance is how spread out your data is from the mean or mathematical average.
  • Standard deviation is a figure that represents how spread out your data is in your sample.
  • In our sample of tree heights, the variance was 0.55.

Step 2 Take the square root of the variance.

  • √0.55 = 0.741619848709566. You will often get a very large decimal figure when you calculate this step. It is ok to round to the second or third decimal place for your standard deviation figure. In this case, you could use 0.74.
  • Using a rounded figure, the standard deviation in our sample of tree heights is 0.74

Step 3 Go through finding the mean, variance, and standard deviation again.

  • Write down all the steps you took when you did your calculations.
  • This will allow you to see where you made a mistake, if any.
  • If you come up with different figures for mean, variance, and standard deviation during your check, repeat the calculations looking at your process carefully.

Calculating Z Scores

Step 1 Use the following format to find a z-score:

  • Remember, a z-score is a measure of how many standard deviations a data point is away from the mean.
  • In the formula X represents the figure you want to examine. For example, if you wanted to find out how many standard deviations 7.5 was from the mean in our example of tree heights, you would plug in 7.5 for X in the equation.
  • In the formula, μ stands for the mean. In our sample of tree heights the mean was 7.9.
  • In the formula, σ stands for the standard deviation. In our sample of tree heights the standard deviation was 0.74.

Step 2 Start the formula by subtracting the mean from the data point you want to examine.

  • For example, in our sample of tree heights we want to find out how many standard deviations 7.5 is from the mean of 7.9.
  • Therefore, you would perform the following: 7.5 - 7.9.
  • 7.5 - 7.9 = -0.4.
  • Double check that you have the correct mean and subtraction figure before you proceed.

Step 3 Divide the subtraction figure you just completed by the standard deviation.

  • In our sample of tree heights, we want the z-score for the data point 7.5.
  • We already subtracted the mean from 7.5, and came up with a figure of -0.4.
  • Remember, the standard deviation from our sample of tree heights was 0.74.
  • - 0.4 / 0.74 = - 0.54
  • Therefore the z-score in this case is -0.54.
  • This z-score means that 7.5 is -0.54 standard deviations away from the mean in our sample of tree heights.
  • Z-scores can be both positive and negative numbers.
  • A negative z-score indicates that the data point is less than the mean, and a positive z-score indicates the data point in question is larger than the mean.

Calculator, Practice Problems, and Answers

how do you solve z score problems

Community Q&A

Community Answer

Video . By using this service, some information may be shared with YouTube.

You Might Also Like

Calculate Weighted Average

  • ↑ https://www.simplypsychology.org/z-score.html
  • ↑ https://www.omnicalculator.com/statistics/z-score
  • ↑ http://www.mathsisfun.com/data/standard-deviation.html
  • ↑ http://pirate.shu.edu/~wachsmut/Teaching/MATH1101/Descriptives/variability.html
  • ↑ https://www.mathsisfun.com/data/standard-deviation.html
  • ↑ https://www150.statcan.gc.ca/n1/edu/power-pouvoir/ch12/5214891-eng.htm
  • ↑ https://www.calculator.net/z-score-calculator.html
  • ↑ https://www.khanacademy.org/math/statistics-probability/modeling-distributions-of-data/z-scores/a/z-scores-review

About This Article

Mario Banuelos, PhD

To calculate a Z score, start by calculating the mean, or average, of your data set. Then, subtract the mean from each number in the data set, square the differences, and add them all together. Next, divide that number by n minus 1, where n equals how many numbers are in the sample, to get the variance. Once you have the variance, take the square root of it to find the standard deviation. Finally, subtract the mean from the data point you're examining, and divide the difference by the standard deviation. To learn how to calculate the mean of your sample, read on! Did this summary help you? Yes No

  • Send fan mail to authors

Reader Success Stories

Lindsey Brooks

Lindsey Brooks

Jun 21, 2017

Did this article help you?

Lorene Hudson

Lorene Hudson

Dec 16, 2018

Chad Horner

Chad Horner

Oct 6, 2018

Regina Nash

Regina Nash

Jun 17, 2017

Mari Kannan

Mari Kannan

Jan 9, 2019

Am I a Narcissist or an Empath Quiz

Featured Articles

Deal with Friendship Problems at School

Trending Articles

Everything You Need to Know to Rock the Corporate Goth Aesthetic

Watch Articles

Cook Fresh Cauliflower

  • Terms of Use
  • Privacy Policy
  • Do Not Sell or Share My Info
  • Not Selling Info

Don’t miss out! Sign up for

wikiHow’s newsletter

These lessons, with videos, examples and step-by-step solutions, help Statistics students learn how to use z-scores to work out probability based on the normal distribution.

Related Pages Normal Distribution: Probability in a given range Standard Deviation More Statistics Lessons Statistics Worksheets

Every unique pair of μ and σ defines a different normal distribution . This characteristic of the normal curve (actually a family of curves) could make analysis by the normal distribution tedious because volumes of normal curve tables – one for each different combinations of μ and σ - would be required.

Fortunately, all normal distributions can be converted into a single distribution, the standardized normal distribution or the z distribution, which has mean 0 and standard deviation 1. We write Z ∼ N(0, 1).

The conversion formula for any x value of a given normal distribution is:

A z -score is the number of standard deviations that a value, x , is above or below the mean. If the value of x is less than the mean, the z score is negative. If the value of x is more than the mean, the z score is positive. If the value of x equals the mean, the z score is zero.

This formula allows conversion of the distance of any x value form its mean into standard deviation units. A standard z score table can then be used to find probabilities for any normal distribution problem that has been converted to z scores.

Normal Distribution & Z-scores This video shows how to calculate “inside areas” and “areas in the extreme” in a normal distribution using Z-scores.

The following video gives the definition of z score based on the bell curve.

The following video gives examples of calculating z-score.

The following video shows how to solve some z-score homework problems. What is the z-score if our standard deviation was 2, mean of 49 and our specific observation is 47? What is the probability an observation is more than 49? What is the probability an observation is more than 47 and less than 49? What is the probability an observation is more than 47? What is the probability an observation is less than 47?

Mathway Calculator Widget

We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.

If you could change one thing about college, what would it be?

Graduate faster

Better quality online classes

Flexible schedule

Access to top-rated instructors

ZScore Formula HighRes- Pencil on testing sheet

Z-Score: Formula, Examples & How to Interpret It

12.20.2021 • 5 min read

Sarah Thomas

Subject Matter Expert

The Z Score Formula is easy to calculate if you know three things. Learn how to calculate & interpret a Z-Score with real-life examples using the formula.

In This Article

How To Interpret Z-Scores

What is the z-score formula, how to calculate a z-score, why z-scores are so important: z-scores and probabilities, don't overpay for college statistics.

Take Intro to Statistics Online with Outlier.org

From the co-founder of MasterClass, earn transferable college credits from the University of Pittsburgh (a top 50 global school). The world's best online college courses for 50% less than a traditional college.

Outlier Stats 1628x960 (1)

Life is full of instances where we want to compare an observed value against the norm.

Here are some examples:

The Netflix series Squid Game drew 111 million views in its first month on the streaming service. How does Squid Game’s viewership compare to other popular series?

You get offered a job out of college at a salary of $42,000 a year. Is this salary high? Low? How does it compare to the incomes of other recent college graduates?

Your niece was just born weighing 6.9 pounds. Is that normal?

Your score on the SAT was 1479. How does that compare to other students?

We can answer all of these questions in a compelling way using Z-scores.

A Z-score is a standardized number that tells you how far away a given data point is from the mean.

Let’s check out three ways to look at z-scores.

1. Z-scores are measured in standard deviation units.

For example, a Z-score of 1.2 shows that your observed value is 1.2 standard deviations from the mean. A Z-score of 2.5 means your observed value is 2.5 standard deviations from the mean and so on.

The closer your Z-score is to zero, the closer your value is to the mean. The further away your Z-score is from zero, the further away your value is from the mean. Typically, you will not see Z-scores that are more than 3 standard deviations from the mean. This is because most data points lie within 3 standard deviations of the mean. If you need a refresher, you can visit this guide on standard deviation.

2. Z-scores can be positive or negative.

A positive Z-score shows that your value lies above the mean, while a negative Z-score shows that your value lies below the mean. If I tell you your income has a Z-score of -0.8, you immediately know that your income is below average. How far below average? 0.8 standard deviations. If I tell you that an SAT score has a Z-score of 2, you know the score is above average. How far above the average? 2 standard deviations. Note that a Z-score of zero shows that your value is equal to the mean.

3. Z-Scores allow you to compare your data easily to other metrics.

Beyond telling you just how far a particular value is from the mean, Z-scores come in handy when drawing comparisons between related but distinct metrics.

For example, imagine you are a college admissions officer reviewing an application. The applicant has a 1500 on their SAT and a 3.2 GPA. It is not immediately obvious how to compare these two figures, but if you calculate a Z-score relative to the average test scores and high school GPAs of students enrolled in your college, the comparison becomes much easier. Say the applicant’s SAT Z-score is equal to 2.8 and their GPA Z-score is equal to -1.2. Immediately, you can infer that the applicant is well above average on their test scores but below average with their GPA.

Similarly, you can compare Z-scores across metrics like height and weight, household income and household debt levels, resting heart rates for men versus women, and more. The only thing to remember is that the variables being compared should have similar underlying distributions.

How to find a Z-score is a simple process where you need to know three things:

x. The value for which you want to calculate the Z-score. We sometimes call this the raw score.

𝝁. The population mean.

𝜎. The population standard deviation.

If you know these three things, calculating a Z-score is easy. In the z-score formula, you simply subtract the population mean from your raw score and divide by the population standard deviation.

Let’s go through one of the examples from above. Your niece has just been born weighing 6.9 pounds, and you want to know if this is a normal weight. Use the following information about birthweights in the US to calculate a Z-score.

1. To calculate the Z-score, start by subtracting the mean from the observed value.

2. Next, divide by the standard deviation.

3. Voila! You’ve got the Z-score for your niece’s birth weight. Her birth weight is 0.26 standard deviations below the mean.

AnnMaria De Mars talks more about z-scores and calculation correlation.

Standard Deviation Image 4

When the raw score, x, is drawn from a distribution that is approximately normal, you can use Z-scores to find probabilities. If x is normally distributed, the probability distribution of the Z-scores will be a standard normal distribution — a normal distribution with a mean equal to 0 and a standard deviation equal to 1.

Because the standard normal distribution follows the empirical rule, and because probabilities associated with the standard normal distribution are well documented, it is easy to look up probabilities based on Z-scores.

Let’s take a look at two examples.

Example 1: Calculating Probabilities using Z-Scores and the Empirical Rule

Zscore Formula Graph 1

Say that a recent college graduate named Ben has an annual income, which when compared to the incomes of other recent college graduates, has a Z-score of -1. Assuming incomes are normally distributed, you can use the empirical rule to find the percentage of recent college graduates whose incomes are above and below Ben’s. Roughly 16% of recent college graduates will have an income below Ben’s, and roughly 84% of recent graduates will have an income above Ben’s. In other words, a Z-score of -1 puts Ben at roughly the 16th percentile of the distribution.

Example 2: Calculating Probabilities using Z-Scores and a Standard Normal Table

Assume that SAT scores are normally distributed and that an SAT score of 1150 has a Z-score of 0.44. What percent of students score below 1150 and what percent of students score above 1150?

To answer questions like these, you can look up the probabilities associated with the given Z-score (in this case 0.44) in a standard normal table. If you look at a standard normal table that shows probabilities to the left of the Z-score, you will find that roughly 0.67 or 67% of the scores fall below 1150. Knowing this, we can also say that 1-0.67 or 33% of scores are above 1150.

Zscore Formula CHART 2

Explore Outlier's Award-Winning For-Credit Courses

Outlier (from the co-founder of MasterClass) has brought together some of the world's best instructors, game designers, and filmmakers to create the future of online college.

Check out these related courses:

Intro to Statistics

Intro to Statistics

How data describes our world.

Precalculus

Precalculus

Master the building blocks of Calculus.

Calculus I

The mathematics of change.

Intro to Microeconomics

Intro to Microeconomics

Why small choices have big impact.

Related Articles

Mound of letters and numbers that represent the use of sets and subsets

What Do Subsets Mean in Statistics?

This article explains what subsets are in statistics and why they are important. You’ll learn about different types of subsets with formulas and examples for each.

Outlier Blog Set Operation HighRes

Set Operations: Formulas, Properties, Examples & Exercises

Here is an overview of set operations, what they are, properties, examples, and exercises.

Outlier Blog Definite Integrals HighRes

Definite Integrals: What Are They and How to Calculate Them

Knowing how to find definite integrals is an essential skill in calculus. In this article, we’ll learn the definition of definite integrals, how to evaluate definite integrals, and practice with some examples.

Rachel McLean

Further reading, parameters vs statistic [with examples], binomial distribution: meaning & formula, understanding the normal distribution curve, understanding sampling distributions: what are they and how do they work, what is the coefficient of variation.

Have a language expert improve your writing

Run a free plagiarism check in 10 minutes, generate accurate citations for free.

  • Knowledge Base
  • The Standard Normal Distribution | Calculator, Examples & Uses

The Standard Normal Distribution | Calculator, Examples & Uses

Published on November 5, 2020 by Pritha Bhandari . Revised on June 21, 2023.

The standard normal distribution , also called the z -distribution , is a special normal distribution where the mean is 0 and the standard deviation is 1.

Any normal distribution can be standardized by converting its values into z scores. Z scores tell you how many standard deviations from the mean each value lies.

The standard normal distribution has a mean of 0 and a standard deviation of 1.

Converting a normal distribution into a z -distribution allows you to calculate the probability of certain values occurring and to compare different data sets.

Download the  z  table

Table of contents

Standard normal distribution calculator, normal distribution vs the standard normal distribution, standardizing a normal distribution, use the standard normal distribution to find probability, step-by-step example of using the z distribution, other interesting articles, frequently asked questions about the standard normal distribution, here's why students love scribbr's proofreading services.

Discover proofreading & editing

All normal distributions , like the standard normal distribution, are unimodal and symmetrically distributed with a bell-shaped curve. However, a normal distribution can take on any value as its mean and standard deviation. In the standard normal distribution, the mean and standard deviation are always fixed.

Every normal distribution is a version of the standard normal distribution that’s been stretched or squeezed and moved horizontally right or left.

The mean determines where the curve is centered. Increasing the mean moves the curve right, while decreasing it moves the curve left.

The standard deviation stretches or squeezes the curve. A small standard deviation results in a narrow curve, while a large standard deviation leads to a wide curve.

The standard normal distribution compared with other normal distributions on a graph

When you standardize a normal distribution, the mean becomes 0 and the standard deviation becomes 1. This allows you to easily calculate the probability of certain values occurring in your distribution, or to compare data sets with different means and standard deviations.

While data points are referred to as x in a normal distribution, they are called z or z scores in the z distribution. A z score is a standard score that tells you how many standard deviations away from the mean an individual value ( x ) lies:

  • A positive z score means that your x value is greater than the mean.
  • A negative z score means that your x value is less than the mean.
  • A z score of zero means that your x value is equal to the mean.

Converting a normal distribution into the standard normal distribution allows you to:

  • Compare scores on different distributions with different means and standard deviations.
  • Normalize scores for statistical decision-making (e.g., grading on a curve).
  • Find the probability of observations in a distribution falling above or below a given value.
  • Find the probability that a sample mean significantly differs from a known population mean.

How to calculate a z score

To standardize a value from a normal distribution, convert the individual value into a z -score:

  • Subtract the mean from your individual value.
  • Divide the difference by the standard deviation.

To standardize your data, you first find the z score for 1380. The z score tells you how many standard deviations away 1380 is from the mean.

The  z score for a value of 1380 is 1.53 . That means 1380 is 1.53 standard deviations from the mean of your distribution.

The standard normal distribution is a probability distribution , so the area under the curve between two points tells you the probability of variables taking on a range of values. The total area under the curve is 1 or 100%.

Every z score has an associated p value that tells you the probability of all values below or above that z score occuring. This is the area under the curve left or right of that z score.

The area under the curve in a standard normal distribution tells you the probability of values occurring.

Z tests and p values

The z score is the test statistic used in a z test . The z test is used to compare the means of two groups, or to compare the mean of a group to a set value. Its null hypothesis typically assumes no difference between groups.

The area under the curve to the right of a z score is the p value, and it’s the likelihood of your observation occurring if the null hypothesis is true.

Usually, a p value of 0.05 or less means that your results are unlikely to have arisen by chance; it indicates a statistically significant effect.

By converting a value in a normal distribution into a z score, you can easily find the p value for a z test.

How to use a z table

Once you have a z score, you can look up the corresponding probability in a z table .

In a z table, the area under the curve is reported for every z value between -4 and 4 at intervals of 0.01.

There are a few different formats for the z table. Here, we use a portion of the cumulative table. This table tells you the total area under the curve up to a given z score—this area is equal to the probability of values below that z score occurring.

The first column of a z table contains the z score up to the first decimal place. The top row of the table gives the second decimal place.

To find the corresponding area under the curve (probability) for a z score:

  • Go down to the row with the first two digits of your z score.
  • Go across to the column with the same third digit as your z  score.
  • Find the value at the intersection of the row and column from the previous steps.

Portion of the z-table

To find the shaded area, you take away 0.937 from 1, which is the total area under the curve.

Probability of x > 1380 = 1 − 0.937 = 0.063

Receive feedback on language, structure, and formatting

Professional editors proofread and edit your paper by focusing on:

  • Academic style
  • Vague sentences
  • Style consistency

See an example

how do you solve z score problems

Let’s walk through an invented research example to better understand how the standard normal distribution works.

As a sleep researcher, you’re curious about how sleep habits changed during COVID-19 lockdowns. You collect sleep duration data from a sample during a full lockdown.

Before the lockdown, the population mean was 6.5 hours of sleep. The lockdown sample mean is 7.62.

To assess whether your sample mean significantly differs from the pre-lockdown population mean, you perform a z test :

  • First, you calculate a z score for the sample mean value.
  • Then, you find the p value for your z score using a z table.

Step 1: Calculate a z -score

To compare sleep duration during and before the lockdown, you convert your lockdown sample mean into a z score using the pre-lockdown population mean and standard deviation.

A z score of 2.24 means that your sample mean is 2.24 standard deviations greater than the population mean.

Step 2: Find the  p value

To find the probability of your sample mean z score of 2.24 or less occurring, you use the  z table to find the value at the intersection of row 2.2 and column +0.04.

Finding the p-value using a z-table

The table tells you that the area under the curve up to or below your z score is 0.9874. This means that your sample’s mean sleep duration is higher than about 98.74% of the population’s mean sleep duration pre-lockdown.

Example of comparing population and sample means using a z-distribution.

To find the p value to assess whether the sample differs from the population, you calculate the area under the curve above or to the right of your z score. Since the total area under the curve is 1, you subtract the area under the curve below your z score from 1.

A p value of less than 0.05 or 5% means that the sample significantly differs from the population.

Probability of z > 2.24 = 1 − 0.9874 = 0.0126 or 1.26%

With a p value of less than 0.05, you can conclude that average sleep duration in the COVID-19 lockdown was significantly higher than the pre-lockdown average.

If you want to know more about statistics , methodology , or research bias , make sure to check out some of our other articles with explanations and examples.

  • Student’s t table
  • Student’s t distribution
  • Descriptive statistics
  • Measures of central tendency
  • Correlation coefficient

Methodology

  • Cluster sampling
  • Stratified sampling
  • Types of interviews
  • Cohort study
  • Thematic analysis

Research bias

  • Implicit bias
  • Cognitive bias
  • Survivorship bias
  • Availability heuristic
  • Nonresponse bias
  • Regression to the mean

In a normal distribution , data are symmetrically distributed with no skew. Most values cluster around a central region, with values tapering off as they go further away from the center.

The measures of central tendency (mean, mode, and median) are exactly the same in a normal distribution.

Normal distribution

The standard normal distribution , also called the z -distribution, is a special normal distribution where the mean is 0 and the standard deviation is 1.

Any normal distribution can be converted into the standard normal distribution by turning the individual values into z -scores. In a z -distribution, z -scores tell you how many standard deviations away from the mean each value lies.

The empirical rule, or the 68-95-99.7 rule, tells you where most of the values lie in a normal distribution :

  • Around 68% of values are within 1 standard deviation of the mean.
  • Around 95% of values are within 2 standard deviations of the mean.
  • Around 99.7% of values are within 3 standard deviations of the mean.

The empirical rule is a quick way to get an overview of your data and check for any outliers or extreme values that don’t follow this pattern.

The t -distribution gives more probability to observations in the tails of the distribution than the standard normal distribution (a.k.a. the z -distribution).

In this way, the t -distribution is more conservative than the standard normal distribution: to reach the same level of confidence or statistical significance , you will need to include a wider range of the data.

Cite this Scribbr article

If you want to cite this source, you can copy and paste the citation or click the “Cite this Scribbr article” button to automatically add the citation to our free Citation Generator.

Bhandari, P. (2023, June 21). The Standard Normal Distribution | Calculator, Examples & Uses. Scribbr. Retrieved February 15, 2024, from https://www.scribbr.com/statistics/standard-normal-distribution/

Is this article helpful?

Pritha Bhandari

Pritha Bhandari

Other students also liked, normal distribution | examples, formulas, & uses, t-distribution: what it is and how to use it, understanding p values | definition and examples, what is your plagiarism score.

Calculating Z-Scores in Statistics

A Sample Worksheet for Defining Normal Distribution in Statistical Analysis

  • Statistics Tutorials
  • Probability & Games
  • Descriptive Statistics
  • Inferential Statistics
  • Applications Of Statistics
  • Math Tutorials
  • Pre Algebra & Algebra
  • Exponential Decay
  • Worksheets By Grade
  • Ph.D., Mathematics, Purdue University
  • M.S., Mathematics, Purdue University
  • B.A., Mathematics, Physics, and Chemistry, Anderson University

A standard type of problem in basic statistics is to calculate the z -score of a value, given that the data is normally distributed and also given the mean and standard deviation . This z-score, or standard score, is the signed number of standard deviations by which the data points' value is above the mean value of that which is being measured.

Calculating z-scores for normal distribution in statistical analysis allows one to simplify observations of normal distributions, starting with an infinite number of distributions and working down to a standard normal deviation instead of working with each application that is encountered.

All of the following problems use the z-score formula , and for all of them assume that we are dealing with a normal distribution .

The Z-Score Formula

The formula for calculating the z-score of any particular data set is z = (x -  μ) / σ where  μ  is the mean of a population and  σ  is the standard deviation of a population. The absolute value of z represents the z-score of the population, the distance between the raw score and population mean in units of standard deviation.

It's important to remember that this formula relies not on the sample mean or deviation but on the population mean and the population standard deviation, meaning that a statistical sampling of data cannot be drawn from the population parameters, rather it must be calculated based on the entire data set.

However, it is rare that every individual in a population can be examined, so in cases where it is impossible to calculate this measurement of every population member, a statistical sampling may be used in order to help calculate the z-score.

Sample Questions

Practice using the z-score formula with these seven questions:

  • Scores on a history test have an average of 80 with a standard deviation of 6. What is the z -score for a student who earned a 75 on the test?
  • The weight of chocolate bars from a particular chocolate factory has a mean of 8 ounces with a standard deviation of .1 ounce. What is the z -score corresponding to a weight of 8.17 ounces?
  • Books in the library are found to have an average length of 350 pages with a standard deviation of 100 pages. What is the z -score corresponding to a book of length 80 pages?
  • The temperature is recorded at 60 airports in a region. The average temperature is 67 degrees Fahrenheit with a standard deviation of 5 degrees. What is the z -score for a temperature of 68 degrees?
  • A group of friends compares what they received while trick or treating. They find that the average number of pieces of candy received is 43, with a standard deviation of 2. What is the z -score corresponding to 20 pieces of candy?
  • The mean growth of the thickness of trees in a forest is found to be .5 cm/year with a standard deviation of .1 cm/year. What is the z -score corresponding to 1 cm/year?
  • A particular leg bone for dinosaur fossils has a mean length of 5 feet with a standard deviation of 3 inches. What is the z -score that corresponds to a length of 62 inches?

Answers for Sample Questions

Check your calculations with the following solutions. Remember that the process for all of these problems is similar in that you must subtract the mean from the given value then divide by the standard deviation:

  • The  z -score of (75 - 80)/6 and is equal to -0.833.
  • The  z -score for this problem is (8.17 - 8)/.1 and is equal to 1.7.
  • The  z -score for this problem is (80 - 350)/100 and is equal to -2.7.
  • Here the number of airports is information that is not necessary to solve the problem. The  z -score for this problem is (68-67)/5 and is equal to 0.2.
  • The  z -score for this problem is (20 - 43)/2 and equal to -11.5.
  • The  z -score for this problem is (1 - .5)/.1 and equal to 5.
  • Here we need to be careful that all of the units we are using are the same. There will not be as many conversions if we do our calculations with inches. Since there are 12 inches in a foot, five feet corresponds to 60 inches. The  z -score for this problem is (62 - 60)/3 and is equal to .667.

If you have answered all of these questions correctly, congratulations! You've fully grasped the concept of calculating z-score to find the value of standard deviation in a given data set!

  • Z-Scores Worksheet
  • Examples of Z-score Calculations
  • Examples of Confidence Intervals for Means
  • Calculate a Confidence Interval for a Mean When You Know Sigma
  • Standard Normal Distribution in Math Problems
  • An Example of a Hypothesis Test
  • How to Use the NORM.INV Function in Excel
  • Margin of Error Formula for Population Mean
  • The Normal Approximation to the Binomial Distribution
  • Hypothesis Test Example
  • Standard and Normal Excel Distribution Calculations
  • How to Do Hypothesis Tests With the Z.TEST Function in Excel
  • Confidence Interval for the Difference of Two Population Proportions
  • Calculating a Confidence Interval for a Mean
  • Variance and Standard Deviation
  • Example of Two Sample T Test and Confidence Interval

If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

To log in and use all the features of Khan Academy, please enable JavaScript in your browser.

AP®︎/College Statistics

Course: ap®︎/college statistics   >   unit 4.

  • Z-score introduction
  • Calculating z-scores
  • Comparing with z-scores

Z-scores-problem

  • Your answer should be
  • an integer, like 6 ‍  
  • a simplified proper fraction, like 3 / 5 ‍  
  • a simplified improper fraction, like 7 / 4 ‍  
  • a mixed number, like 1   3 / 4 ‍  
  • an exact decimal, like 0.75 ‍  
  • a multiple of pi, like 12   pi ‍   or 2 / 3   pi ‍  
  • (Choice A)   This drink has 0.8 g ‍   of sugar less than the mean of the 32 ‍   drinks. A This drink has 0.8 g ‍   of sugar less than the mean of the 32 ‍   drinks.
  • (Choice B)   This drink is 0.8 ‍   standard deviations below the mean of the 32 ‍   drinks. B This drink is 0.8 ‍   standard deviations below the mean of the 32 ‍   drinks.
  • (Choice C)   About 80 % ‍   of drinks have a lower sugar content than this drink. C About 80 % ‍   of drinks have a lower sugar content than this drink.
  • (Choice D)   About 80 % ‍   of drinks have a higher sugar content than this drink. D About 80 % ‍   of drinks have a higher sugar content than this drink.
  • (Choice A)   This drink has relatively less sugar at Jake's Java. A This drink has relatively less sugar at Jake's Java.
  • (Choice B)   This drink has relatively less sugar at Ruth's Roasts. B This drink has relatively less sugar at Ruth's Roasts.
  • (Choice C)   The relative sugar contents are equal at both locations. C The relative sugar contents are equal at both locations.
  • (Choice D)   It is impossible to tell because we do not know how many drinks are offered at Ruth's Roasts. D It is impossible to tell because we do not know how many drinks are offered at Ruth's Roasts.

Want to join the conversation?

  • Upvote Button navigates to signup page
  • Downvote Button navigates to signup page
  • Flag Button navigates to signup page

Good Answer

Statology

Statistics Made Easy

How to Find Probability from a Z-Score (With Examples)

The easiest way to find the probability from a z-score is to simply look up the probability that corresponds to the z-score in the   z table .

This tutorial explains how to use the z table to find the following probabilities:

  • The probability of a value being less than a certain z-score.
  • The probability of a value being greater than a certain z-score.
  • The probability of a value being between two certain z-scores.

Let’s jump in!

Example 1: Probability Less Than a Certain Z-Score

Suppose we would like to find the probability that a value in a given distribution has a z-score less than z = 0.25 .

To find this probability, we need to look up 0.25 in the z-table :

Example of how to read the z table

The probability that a value in a given distribution has a z-score less than z = 0.25 is approximately 0.5987 .

Note : This could also be written as 59.87% in percentage terms.

Example 2: Probability Greater Than a Certain Z-Score

Suppose we would like to find the probability that a value in a given distribution has a z-score greater than z = -0.5 .

To find this probability, we need to look up -0.5 in the z-table :

how do you solve z score problems

The probability that corresponds to a z-score of -0.5 is .3085.

However, since we want to know the probability that a value in a given distribution has a z-score greater than -0.5, we need to subtract this probability from 1.

Thus, the probability that a value in a given distribution has a z-score greater than -0.5 is: 1 – .3085 = 0.6915 .

Example 3: Probability Between Two Z-Scores

Suppose we would like to find the probability that a value in a given distribution has a z-score between z = 0.4 and z = 1 .

First, we will look up the value  0.4   in the z-table :

Example of using z table

Then, we will look up the value  1   in the z-table :

Z table example

Then we will subtract the smaller value from the larger value: 0.8413 – 0.6554 = 0.1859 .

Thus, the probability that a value in a given distribution has a z-score between z = 0.4 and z = 1 is approximately 0.1859 .

Additional Resources

The following tutorials provide additional information about z-scores:

5 Examples of Using Z-Scores in Real Life How to Convert Z-Scores to Raw Scores How to Find Z-Scores Given Area

' src=

Published by Zach

Leave a reply cancel reply.

Your email address will not be published. Required fields are marked *

Z-score Calculator

Use this calculator to compute the z-score of a normal distribution.

Z-score and Probability Converter

Please provide any one value to convert between z-score and probability. This is the equivalent of referencing a z-table.

Probability between Two Z-scores

z-score

Use this calculator to find the probability (area P in the diagram) between two z-scores.

Related Standard Deviation Calculator

What is z-score?

The z-score, also referred to as standard score, z-value, and normal score, among other things, is a dimensionless quantity that is used to indicate the signed, fractional, number of standard deviations by which an event is above the mean value being measured. Values above the mean have positive z-scores, while values below the mean have negative z-scores.

The z-score can be calculated by subtracting the population mean from the raw score, or data point in question (a test score, height, age, etc.), then dividing the difference by the population standard deviation:

where x is the raw score, μ is the population mean, and σ is the population standard deviation. For a sample, the formula is similar, except that the sample mean and population standard deviation are used instead of the population mean and population standard deviation.

The z-score has numerous applications and can be used to perform a z-test, calculate prediction intervals, process control applications, comparison of scores on different scales, and more.

A z-table, also known as a standard normal table or unit normal table, is a table that consists of standardized values that are used to determine the probability that a given statistic is below, above, or between the standard normal distribution. A z-score of 0 indicates that the given point is identical to the mean. On the graph of the standard normal distribution, z = 0 is therefore the center of the curve. A positive z-value indicates that the point lies to the right of the mean, and a negative z-value indicates that the point lies left of the mean. There are a few different types of z-tables.

The values in the table below represent the area between z = 0 and the given z-score.

How to read the z-table

In the table above,

  • the column headings define the z-score to the hundredth's place.
  • the row headings define the z-score to the tenth's place.
  • each value in the table is the area between z = 0 and the z-score of the given value, which represents the probability that a data point will lie within the referenced region in the standard normal distribution.

For example, referencing the right-tail z-table above, a data point with a z-score of 1.12 corresponds to an area of 0.36864 (row 13, column 4). This means that for a normally distributed population, there is a 36.864% chance, a data point will have a z-score between 0 and 1.12.

Because there are various z-tables, it is important to pay attention to the given z-table to know what area is being referenced.

IMAGES

  1. How to Calculate Z-Score?: Statistics

    how do you solve z score problems

  2. Z-Score Calculator (with Formulas & Steps)

    how do you solve z score problems

  3. How to Solve for and Interpret z Scores

    how do you solve z score problems

  4. How To Calculate Z

    how do you solve z score problems

  5. How to find z score on standard normal table

    how do you solve z score problems

  6. z scores to find probability

    how do you solve z score problems

VIDEO

  1. Z-SCORE

  2. Z-score problems review

  3. How to Read the Z-score Table.mp4

  4. Z test problem

  5. Calculator Technique Tutorial Series: z

  6. Z-Score

COMMENTS

  1. How to calculate Z-scores (formula review) (article)

    Example 1 The grades on a history midterm at Almond have a mean of μ = 85 and a standard deviation of σ = 2 . Michael scored 86 on the exam. Find the z-score for Michael's exam grade. z = his grade − mean grade standard deviation z = 86 − 85 2 z = 1 2 = 0.5 Michael's z-score is 0.5 . His grade was half of a standard deviation above the mean.

  2. Z-Score: Definition, Formula, Calculation & Interpretation

    The formula for calculating a z-score is z = (x-μ)/σ, where x is the raw score, μ is the population mean, and σ is the population standard deviation. As the formula shows, the z-score is simply the raw score minus the population mean, divided by the population standard deviation. Figure 2. The Z-score formula in a population.

  3. Practice Problems for Z-Scores

    One standard type of problem from an introductory statistics course is to calculate the z -score of a particular value. This is a very basic calculation, but is one that is quite important. The reason for this is that it allows us to wade through the infinite number of normal distributions .

  4. How to Calculate Z Scores: 15 Steps (with Pictures)

    1 Look at your data set. You will need certain key pieces of information to calculate the mean or mathematical average from your sample. [2] Know how many numbers are in your sample. In the case of the sample of palm trees, there are 5 in this sample. Know what the numbers represent. In our example, these numbers represent measurements of trees.

  5. Z-Score (examples, solutions, formulas, videos)

    The following video shows how to solve some z-score homework problems. What is the z-score if our standard deviation was 2, mean of 49 and our specific observation is 47? What is the probability an observation is more than 49? What is the probability an observation is more than 47 and less than 49?

  6. Z-score: Definition, Formula, and Uses

    A z-score measures the distance between a data point and the mean using standard deviations. Z-scores can be positive or negative. The sign tells you whether the observation is above or below the mean. For example, a z-score of +2 indicates that the data point falls two standard deviations above the mean, while a -2 signifies it is two standard ...

  7. Normal distribution problem: z-scores (from ck12.org)

    Chris Yun 12 years ago Positive means that it's that many standard deviations above the mean. Negative means that it's that many standard deviations below the mean. Zero states that it's equal to the mean. ( 148 votes) Upvote Flag Show more...

  8. Z-Score: Definition, Formula and Calculation

    1. What is a Z-Score? Watch the video to learn what a z-score is. Simply put, a z-score (also called a standard score) gives you an idea of how far from the mean a data point is. But more technically it's a measure of how many standard deviations below or above the population mean a raw score is.

  9. Z Test: Uses, Formula & Examples

    Use the 1-sample analysis to determine whether a population mean is different from a hypothesized value. Or use the 2-sample version to determine whether two population means differ. A Z test is a form of inferential statistics. It uses samples to draw conclusions about populations. For example, use Z tests to assess the following:

  10. Z-Score: Formula, Examples & How to Interpret It

    How To Interpret Z-Scores. Let's check out three ways to look at z-scores. 1. Z-scores are measured in standard deviation units. For example, a Z-score of 1.2 shows that your observed value is 1.2 standard deviations from the mean. A Z-score of 2.5 means your observed value is 2.5 standard deviations from the mean and so on.

  11. Standard Normal Distribution Tables, Z Scores, Probability ...

    The Organic Chemistry Tutor 7.32M subscribers Join Subscribe Subscribed 28K 2M views 4 years ago Statistics This statistics video tutorial provides a basic introduction into standard normal...

  12. ck12.org normal distribution problems: z-score

    Courses on Khan Academy are always 100% free. Start practicing—and saving your progress—now: https://www.khanacademy.org/math/statistics-probability/modeling...

  13. The Standard Normal Distribution

    You want to find the probability that SAT scores in your sample exceed 1380. To standardize your data, you first find the z score for 1380. The z score tells you how many standard deviations away 1380 is from the mean. Step 1: Subtract the mean from the x value. x = 1380. M = 1150. x - M = 1380 − 1150 = 230.

  14. Z-score introduction (video)

    A z-score is an example of a standardized score. A z-score measures how many standard deviations a data point is from the mean in a distribution. Questions Tips & Thanks Want to join the conversation? Sort by: Top Voted Ryan Giglio 5 years ago When I paused to calculate the standard deviation myself, I came up with 1.83, not 1.69.

  15. Statistics Worksheet: Calculating Z-Scores

    The formula for calculating the z-score of any particular data set is z = (x - μ) / σ where μ is the mean of a population and σ is the standard deviation of a population. The absolute value of z represents the z-score of the population, the distance between the raw score and population mean in units of standard deviation.

  16. Statistics

    This example shows how to find the z-score for a data point. Remember that the z-score tells you how many standard deviations away from the mean a particula...

  17. Working with z- Scores and Values of X

    Answer: 24.7 The question gives you a z- score and asks for its corresponding x- value. The z- formula contains both x and z, so as long as you know one of them you can always find the other: You know that z = 2.2, and so you just plug these numbers into the z- formula and then solve for x:

  18. How to Interpret Z-Scores (With Examples)

    We can use the following steps to calculate the z-score: The mean is μ = 80 The standard deviation is σ = 4 The individual value we're interested in is X = 75 Thus, z = (X - μ) / σ = (75 - 80) /4 = -1.25. This tells us that an exam score of 75 lies 1.25 standard deviations below the mean. Question 3: Find the z-score for an exam score of 80.

  19. Z-scores-problem (article)

    problem 1 A Grande Mocha Cappuccino at Jake's Java contains 14 g of sugar. Calculate the standardized score (z-score) for the Grande Mocha Cappuccino. z = problem 2 What is the best interpretation of the z-score from the previous problem? Choose 1 answer: This drink has 0.8 g of sugar less than the mean of the 32 drinks. A This drink has 0.8 g

  20. 5 Examples of Using Z-Scores in Real Life

    We use the following formula to calculate a z-score for a given value: z = (x - μ) / σ where: x: Individual data value μ: Mean of population σ: Standard deviation of population The following examples show how z-scores are used in real life in different scenarios. Example 1: Exam Scores

  21. How to Find Probability from a Z-Score (With Examples)

    Suppose we would like to find the probability that a value in a given distribution has a z-score between z = 0.4 and z = 1. First, we will look up the value 0.4 in the z-table: Then, we will look up the value 1 in the z-table: Then we will subtract the smaller value from the larger value: 0.8413 - 0.6554 = 0.1859.

  22. Z-score Calculator

    The z-score can be calculated by subtracting the population mean from the raw score, or data point in question (a test score, height, age, etc.), then dividing the difference by the population standard deviation: where x is the raw score, μ is the population mean, and σ is the population standard deviation. For a sample, the formula is ...