Back To Course

Statistics 101: Principles of Statistics11 chapters | 134 lessons | 1 flashcard set

Watch short & fun videos
**Start Your Free Trial Today**

Start Your Free Trial To Continue Watching

As a member, you'll also get unlimited access to over

Your next lesson will play in
10 seconds

Lesson Transcript

Instructor:
*Kevin Newton*

Kevin has edited encyclopedias, taught middle and high school history, and has a master's degree in Islamic law.

Talking about multiples of standard deviations can get exhausting and confusing. Luckily, z-scores allow us to talk about how far a point is removed from a mean in terms of how many standard deviations away it is.

Imagine that you were dealing a data set that had a standard deviation of 2. Remember that **standard deviation** is simply a measure of how scattered a collection of data is from the mean. Obviously, much of your data will be the mean plus or minus 2. Even more will be two standard deviations away from the mean, so the mean plus or minus 4. That notation works pretty well when the standard deviation is succinct. After all, 2 is a pretty easy concept to understand.

But what if your standard deviation was a very small decimal, as it often is in engineering? Or a less-simple number, like 593, as it could be in polling. Wouldn't it be nice to have a unit that measures the value of something in terms of standard deviations away from the mean? Luckily, we have exactly such a tool: the z-score. The **z-score** is a measure of distance from the mean in terms of how many standard deviations it is removed from the mean.

But wait, how do you find the standard deviation? Chances are you may already remember, but if you don't, we'll review that quickly. First, take the mean of all the values in the set. Then, subtract the mean from each value, then square each difference. Find the mean of the sum of those differences, then take the square root of that number.

For example, let's say you wanted to know the population standard deviation of the following test scores: 98, 90, 86, 83, and 70. First, take the mean, which comes out to 85.4. Then subtract that from each score. You get the following values: 12.6, 4.6, 0.6, -2.4, and -15.4. Square each of those, giving you 158.76, 21.16, 0.36, 5.76, and 237.16. Add all those up and find the average of them, which turns out to be 84.64. Then take the square root, which is 9.2.

By far, the biggest advantage of a z-score is its brevity. When I say to include all values between z-scores of -2 and 2, you know that I want to include 95% of values. That's because within two standard deviations on either side of the mean are 95% of the data in a set.

It would be pretty easy to do that with values when the standard deviation is a highly regular number, like 2, but could you imagine doing that when the standard deviation is 0.00487? Or when it is 1,381? You could bring out a calculator and find that two standard deviations of 0.00487 is 0.00974, and the two of 1,381 is 2,762. However, that introduces a great deal of potential errors. Z-scores limit those errors by giving a concise answer. With z-scores, you can worry about other things, not making sure that 0.00974 is exactly double of 0.00487.

As calculating a z-score is simply finding something in terms of the standard deviation, it is a relatively simple formula. First, subtract the mean from the value in question. Then, divide the answer by the standard deviation. If you have a value in question of 55, a mean of 30, and a standard deviation of 20, you would take 55 and subtract 30. That leaves 25. Divide 25 by the standard deviation of 20 and you end up with 1.25. 1.25 is the z-score of that data point.

Sometimes, that number may come out negative. That's fine; negative numbers just mean that the z-score in question is less than the mean. On a normal distribution curve, that means that it is to the left of the high point of the curve. Likewise, positive numbers are always on the right of the curve.

It may seem that the math for this sort of thing is pretty simple, but let's do a few questions to make sure that we understand it. We will do one with a positive z-score and then one with a negative z-score.

First, let's say you were trying to find the z-score for a student's book collection. According to your data, the mean book collection is 50, while the standard deviation is 10. The student in question has a collection of 75 books. What is the first thing that we do? First, subtract the mean from the value in question. 75 minus 50 is 25. Then divide that quantity by the standard deviation. 25 divided by 10 is 2.5. That means that this student has an exceptionally large book collection.

But what about his roommate? Let's say that the mean stays the same at 50, while the standard deviation remains the same at 10. However, this student only has 15 books. Obviously, it's a small number, but what is its z-score? First, subtract the mean from the quantity. This means 15 minus 50, or -35. Divide -35 by 10 and you get -3.5. That is this student's z-score. It is statistically interesting because -3.5 is a number of standard deviations from the mean. Perhaps his academic advisor should have a talk with him?

In this lesson, we learn how to find the ** z-score** of a data point. The z-score is a measure of distance from the mean in terms of standard deviations. Remember that a **standard deviation** is just a measure of how scattered a collection of data is.

To find the z-score, subtract the mean from the quantity in question, and then divide by the standard deviation of the whole set. Remember that negative z-scores are on the left of the curve, while positive z-scores are on the right.

To unlock this lesson you must be a Study.com Member.

Create
your account

Already a member? Log In

BackDid you know… We have over 95 college courses that prepare you to earn credit by exam that is accepted by over 2,000 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Not sure what college you want to attend yet? Study.com has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

You are viewing lesson
Lesson
6 in chapter 6 of the course:

Back To Course

Statistics 101: Principles of Statistics11 chapters | 134 lessons | 1 flashcard set

- Go to Probability

- Graphing Probability Distributions Associated with Random Variables 6:33
- Finding & Interpreting the Expected Value of a Continuous Random Variable 5:29
- Developing Continuous Probability Distributions Theoretically & Finding Expected Values 6:12
- Probabilities as Areas of Geometric Regions: Definition & Examples 7:06
- Normal Distribution: Definition, Properties, Characteristics & Example 11:40
- Finding Z-Scores: Definition & Examples 6:30
- Estimating Population Percentages from Normal Distributions: The Empirical Rule & Examples 4:41
- Using the Normal Distribution: Practice Problems 10:32
- Using Normal Distribution to Approximate Binomial Probabilities 6:34
- How to Apply Continuous Probability Concepts to Problem Solving 5:05
- Go to Continuous Probability Distributions

- Go to Sampling

- Pennsylvania Grades 4-8 - English Language Arts Subject Concentration: Practice & Study Guide
- Praxis Reading for Virginia Educators - Elementary & Special Education: Practice & Study Guide
- Chemistry Lesson Plans & Activities
- Certified Fraud Examiner (CFE): Exam Prep & Study Guide
- PMI-PBA Certification Exam Study Guide - PMI Professional in Business Analysis
- International Marketing Basics
- PMI-PBA Certification Overview
- Business Report & Proposal Basics
- Types of Academic Essays
- Prisons in the United States
- WV Next Generation Standards for Science
- Response to Intervention (RTI) in Georgia
- The PMI-PBA Certification Process
- WIDA Can Do Descriptors for Grades 6-8
- PMI-PBA Exam: Policies, Procedures & Results
- 14 Cows for America Lesson Plan
- ELL Services in Illinois

- Implications of Business Strategy for Training
- Tragic Flaw in Macbeth: Quotes & Analysis
- Animal Farm Windmill: Symbolism & Analysis
- Supply Chain Sustainability: Environmentally Sound Choices
- Richard Feynman: Biography & Quotes
- Short Stories About Space for Kids
- Movie Review Lesson Plan for ESL Students
- Patient Care Data Set (PCDS): Purpose & Components
- Quiz & Worksheet - Women in the U.S. Revolutionary War
- Quiz & Worksheet - Lattice Energy
- Quiz & Worksheet - Lexically Ambiguous Sentences
- Quiz & Worksheet - What are Achievement Tests?
- Quiz & Worksheet - Cost Accounting & Corporate Mission
- Number Sense for Business Flashcards
- Math for Financial Analysis Flashcards

- ILTS School Counselor: Test Practice and Study Guide
- Western Civilization Since 1648: Homework Help Resource
- ASSET Numerical Skills Test: Practice & Study Guide
- CLEP History of the United States II: Study Guide & Test Prep
- Creating Organizational Culture
- Writing Revision & Skill Development: AP English Language Lesson Plans
- HiSET: The American Revolution (1775-1783)
- Quiz & Worksheet - GRE Scoring
- Quiz & Worksheet - Spain's Decline in the 17th Century
- Quiz & Worksheet - ACT English Verb Tense & Subject-Verb Agreement
- Quiz & Worksheet - Measuring an Inscribed Angle
- Quiz & Worksheet - Characteristics of Social Media and Online Communities

- Customer Relations: Definition & Concept
- What Is Syphilis? - Definition, Causes & Stages
- Poetry Activities for Kids
- 9th Grade Reading List
- ABC Games for Kids
- Space Experiments for Kids
- Sequencing Activities for 5th Grade
- What is the International Baccalaureate Middle Years Program?
- What is the EPT Test?
- Praxis Tests in Pennsylvania
- Brain Breaks for Kids in the Classroom
- How to Register for the TEAS Test

Browse by subject