How to Calculate X-bar

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Suppose you're a clothing manufacturer, and you want to maximize profits. One way to do this is to determine the median height of the people in your market city or country and make most of your clothing to fit people of that height. Because it's impractical to measure the height of every person, you would measure the heights of only some of the people and average the results of that sample. In statistics, this average is the x-bar, which appears as an x with a horizontal line over it. It's a simple arithmetic average, which means it's the sum of all the measurements divided by the number of measurements.

TL;DR (Too Long; Didn't Read)

Calculate x-bar for a sample by adding the measurement values and dividing by the number of measurements. In other words, x-bar is a simple arithmetic average.

Mathematic Definition

In mathematical notation, the definition of x-bar looks more sophisticated and complex than it really is. If you have a number of measurements n, and you represent each measurement by the letter x, you get x-bar by performing the following operation:

x-bar = ∑x_i_/n

This simply means that you add all all the values of xi for values of i from 0 to n and divide by the number of measurements. A familiar example demostrates how straightforward this is:

In a series of tests throughout the school year, a student gets the following percentage scores: 72, 55, 83, 62, 77, 80 and 87. Assuming all tests count for the same, what is the student's average score? To get the answer you add all the scores to get 516 and you divide by the number of tests, which is 7 to get 73.7 or, rounding up, 74 percent.

Improving the Accuracy of X-Bar

You can only calculate the true mean of a population by measuring every individual in the population. Statisticians denote this true mean by the lowercase Greek letter mu (µ). Because it's an approximation, x-bar doesn't necessarily equal µ, but the approximation gets closer as you increase the sample size. Another way to increase accuracy is to measure several samples, calculate x-bar for each sample and find the mean of all the x-bars you calculated.

The clothing designer measuring the height of individuals would probably want to take more than one sample and calculate x-bar for each sample. That helps avoid anomalies. For example, a sample taken at a basketball practice isn't as likely to be indicative of the population as a whole as a series of samples taken across different sectors of the population. The more measurements you take when calculating x-bar, and the more separate calculations of x-bar you are able to average into a final number, the lower the standard deviation of the resulting mean.

References

About the Author

Chris Deziel holds a Bachelor's degree in physics and a Master's degree in Humanities, He has taught science, math and English at the university level, both in his native Canada and in Japan. He began writing online in 2010 with the goal of exploring scientific, cultural and practical topics, and at last count had reached over a hundred million readers through various sites.

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