Copyright

X-Bar in Statistics: Theory & Formula

An error occurred trying to load this video.

Try refreshing the page, or contact customer support.

Coming up next: Finding the 40th Percentile

You're on a roll. Keep up the good work!

Take Quiz Watch Next Lesson
 Replay
Your next lesson will play in 10 seconds
  • 0:00 What Is the x Bar?
  • 1:40 Random Samples and…
  • 3:30 Calculating X-Bar
  • 4:09 The Sampling Distribution
  • 5:05 Lesson Summary
Save Save Save

Want to watch this again later?

Log in or sign up to add this lesson to a Custom Course.

Log in or Sign up

Timeline
Autoplay
Autoplay
Speed Speed

Recommended Lessons and Courses for You

Lesson Transcript
Instructor: Tracy Payne, Ph.D.

Tracy earned her doctorate from Vanderbilt University and has taught mathematics from preschool through graduate level statistics.

A cow visits an x-bar to give the bartender samples of milk. The bartender asks the cow, 'What is the mean of this milk?' The cow replies, 'Trying to esti-mate mu!' This lesson explains x-bars and their role in estimating parameters.

What is the x-Bar

A young boy was overheard asking his mother these questions:

  • How tall is a professional basketball player?
  • How many calories are in a scoop of chocolate chip ice cream?
  • How much money does a schoolteacher make?

All of these questions can be answered using statistics. Statistics is the science of collecting and analyzing numerical data gathered from a representative sample in order to infer the true mean or proportion of a population.

Obviously, all professional basketball players aren't the same height, every scoop of chocolate chip ice-cream contains slightly more or fewer calories than the next, and every teacher in America does not make exactly the same income.

We can take samples of data from the populations they represent and calculate a single value called a statistic. We can then use that value to estimate characteristics that are true for the whole population, called the parameter. The x-bar is the symbol (or expression) used to represent the sample mean, a statistic, and that mean is used to estimate the true population parameter, mu.

To find the average height of professional basketball players (the population), we don't need to measure every player, just some of them (the sample). How do we select which ones to measure? How many players are enough to call a sample? How a researcher makes these decisions influences the inferences that can be made. After all, an anonymous but oft cited quote about statistics is that any analysis is only as good as the data on which it is based!

Random Samples and Sample Size Matter

Accurate sample means come from samples that are randomized and include a sufficient number of people. Statistical inferencing is only appropriate for random data. The act of randomizing guarantees that the results of analyzing our data are subject to the laws of probability.

Simple random sampling (SRS) is one type of sampling method, that is to say, it is a procedure for selecting the sample that will represent the population. SRS is simple and reliable and so is most often used when selecting a sample. There are various strategies used to obtain a random sample. A researcher could place the names of every professional basketball player on slips of paper and place those in a hat and, without looking, draw out a random sample, or assign each player a number and use a random table generator to select a random sample.

In addition to how samples are selected, sample size is also important. The central limit theorem states that as sample size increases to at least 30 individual observations, the sampling distribution of statistics obtained for any random variable will be normal.

To estimate the calories in a scoop of chocolate chip ice cream, we need to ensure the sample is random and a sufficient size. We use a trucking service to randomly select 30 ice cream trucks in transport across America on some Thursday. We ask the local college to send a scientist to stop these trucks and take 1 scoop of chocolate chip ice cream from 20 different containers. The scientists measure each scoop, then place the scoop in a calorie counter thingy that measures the calories of each (now melted) scoop of ice cream. Each researcher then calls in a single statistic: the sample mean, x-bar.

x-bar-from-population

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

Register to view this lesson

Are you a student or a teacher?

Unlock Your Education

See for yourself why 30 million people use Study.com

Become a Study.com member and start learning now.
Become a Member  Back
What teachers are saying about Study.com
Try it risk-free for 30 days

Earning College Credit

Did you know… We have over 200 college courses that prepare you to earn credit by exam that is accepted by over 1,500 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

Transferring credit to the school of your choice

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.

Create an account to start this course today
Try it risk-free for 30 days!
Create an account
Support