What Is The Sample Mean? Symbol (X Bar), Definition, and Standard Error

By Chris Kolmar - Nov. 20, 2020
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Little is more crucial to the growth and success of a business than their statistical analysis efforts. It provides information that’s needed to formulate strategic planning and decision-making. The foundation of statistics is rooted in utilizing samples to offer predictions that can apply to larger populations.

Referring to the sample mean is a stepping stone for forming broader conclusions and development.

What Is Sample Mean?

The sample mean is defined as the average of a given sample set. The sample mean is represented mathematically as x. It’s considered a jumping-off point for initiating further analysis.

It’s common to find a sample mean in order to implement this value into a more complex and detailed formula, such as central tendency and standard deviation of a sample set.
The concept of sample mean will arise early on in learning about statistics.

Why Is Sample Mean Important?

The clearest advantage of calculating the sample mean is that it can provide information that’s accurately applicable to a larger population. This is important because it can provide statistical insight without going through the impossible task of polling every person involved.

Additionally, the sample mean is used in a variety of different industries. Any field related to science studies, like biology and chemistry, will use the sample mean in the early stages of their specific research. Data entry and IT jobs use sample mean values to accomplish daily goals. Even in business, the sample average is necessary to complete calculations of growth rates.

While you may vaguely recall sample mean as a distant memory from high school mathematics, it’s applied across many fields and can be exponentially useful for many reasons.

How to Calculate Sample Mean

Mathematics of any kind can be initially stressful to many people, but calculating a sample mean is one of the most straightforward calculations you’ll find in statistics.

Like any other standard statistical equation, you need to go through a specific formula and steps to arrive at the correct sample mean. Before attempting to solve for your organization’s sample mean, you must first consider the formula.

The formula for solving the sample mean is x̄=(Σxi) / n

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At first glance of this formula, you may already be ready to throw in the towel, but the complicated seeming language of the equation is actually fairly easy when broken down.

  • In the equation, x&#772 represents the answer you’re looking for, which is the sample mean.

  • The Σ symbol is the mathematical way of saying, “add up the following numbers.”

  • The proceeding xi within the parenthesis means “all x-values,” which would be the values for each piece of data you’re investigating.

  • Finally, the equation asks you to divide by n, which stands for the total number of values that you are comparing.

That’s a lot to take in all at once, but let’s go through it step by step. To better understand the process of finding a sample mean, consider the steps in terms of the following example.

A phone provider company is interested in learning more about the statistical trends of its customer base. To begin this process, they must determine the sample mean. They decide to compare customer base totals for the past six months to get an average of how many people are using their service.

The phone company records the following values for each of the six months:

  • January – 20,000 customers

  • February -18,000 customers

  • March- 20,400 customers

  • April- 21,050 customers

  • May – 23,000 customers

  • June – 22,300 customers

  1. Add values together. The first step of finding a sample mean asks us to add all the values in the sample together. To apply this to the example of the phone company, they begin by adding together the number of customers they had each month.

    20,000+18,000+20,400+21,050+23,000+22,300 = 124,750

    The number 124,750 represents the total number of customers that the phone provider had over the six-month sample size.

  2. Determine the value of n. The value of n in the equation for sample mean represents how many items are being compared. Since the phone company is comparing the monthly customer counts over six months, the value of n in this example would be 6.

  3. Input values and divide. The final step to figuring out a sample mean is to input the values you’ve determined into the original equation and divide to solve. For the phone company, this would mean that they divide 124,750 by 6 to arrive at their sample mean.

    124,750 / 6 = 20,792

    The sample mean of the phone company’s customer base over six months is 20,792.

What Is Variance?

Variance means how spread out the numbers in a set are. While you’ve arrived at an average with the sample mean formula, calculating the set’s variance will show you how far apart each value is from the others in the set.

While the variance is a significant value to know on its own, the primary purpose of doing this equation is to be able to complete the process for determining the standard error of the set later.

How to Calculate Variance

Once you’ve found your set’s sample mean, you can use this to find the variance.

Follow the steps below to determine the variance of a sample set.

  1. Subtract the sample mean from each value. To begin finding variance, you will need to subtract the sample mean you’ve just discovered from each value in the set. In terms of the example with the phone company, this would mean subtracting 20,792 from the value of every month.

    (20,000-20,792) (18,000-20,792) (20,400-20,792) (21,050-20,792) (23,000-29,792) (22,300-20,792) = (-792, -2,792, -392, 258, 2,028, 1,328)

  2. Square the resulting values. After subtracting the sample mean from each of the values, continue by squaring each of the new numbers you’re left with. For the phone company example, squaring each value will result in the new values being:

    (627,264) (7,795,264) (153,664) (66,564) (4,112,784) (1,763,584)

  3. Complete the with sample mean formula. The final step towards determining variance in a set is to plug in the new values you’ve found by subtracting and squaring to the original sample mean formula. Complete the equation like you normally would to arrive at the variance of the sample.

    (627,264)+(7,795,264)+(153,664)+(66,564)+(4,112,784)+(1,763,584) = 14,519,124

    14,519,124 / 6 = 2,419,854

The size of the answer describes how large the variance is in the set. The result from the phone provider example has an extremely high variance rate, which means there’s a big difference between each of the values.

If we go back to the original data points of how many customers the company had per month, we’ll see that this variance is apparent. Between January and February, the provider saw a drastic drop of 2,000 customers in only one month, which was followed by more substantial spikes. The difference in these original values is what influences the result of the variance formula.

What Is Standard Error?

Standard error is also a measure of how the numbers are spread out among the set, but it’s evaluating how far each data point is from the mean as opposed to each other. This is also known as the distribution rate.

While the sample mean will give information about the average of a sample and variance measures the difference between each value in the sample, the standard error is slightly different.

In most distributions, all the values will be within two standard deviations of the mean unless it is an outlier.

How to Calculate Standard Error

Calculating sample mean and variance requires at least a few steps to complete the process. However, you’ve already done most of the legwork to solve for standard error once you’ve run these equations. Finding the standard error of a set post-variance is only one step.

  1. Find the square root of the variance value. Determining standard error from a variance value only requires finding the square root of this number. For the example of the phone company, we would find the square root of 2,419,854 to determine the standard error.

    √2,419,854 =1,556

    The standard error for the sample set of the phone company is 1,556. This is a very large standard deviation, meaning that the values distance from the mean of the sample is spread out and more inaccurate to apply to a larger sample. In other words, the data found at the phone company demonstrates inconsistent sample values and won’t be representative of the entire phone industry’s average numbers.

Additional Example

A college professor wants to know the grade-point statistics of the last test he gave out to his class.

The test scores were for the class were: (82, 88, 83, 89, 91, 79, 85, 93, 83)

He begins the evaluation process by adding together all the test values in his class and dividing the result by 9, the number of students in the class.

82+88+83+89+91+79+85+93+83 = 773

773 / 9 = 85.89

The sample mean of the test scores in the professor’s class is 85.89. He continues by solving for the variance among the test grades. This is done by subtracting the sample mean of 85.89 from every value and squaring the resulting numbers.

(82-85.89) (88-85.89) (83-85.89) (89-85.89) (91-85.89) (79-85.89) (85-85.89) (93-85.89) (83-85.89) = (-3.89, 2.11, -2.89, 3.11, 5.11, -6.89, -0.89, 7.11, -2.89)

(-3.89, 2.11, -2.89, 3.11, 5.11, -6.89, -0.89, 7.11, -2.89)² = (15.13, 4.45, 8.35, 9.67, 26.11, 47.47, 0.79, 50.55, 8.35)

To finish solving for variance, the professor plugs in the new values he’s found and solves using the sample mean formula.

(15.13+4.45+8.35+9.67+26.11+47.47+0.79+50.55+8.35) = 170.87

170.87 / 9 = 18.99

The variance among the grades on this test is 18.99, which shows a fair amount of variation between grades, but nothing out of the ordinary.

Now that the professor has accumulated all the values he needs, he squares the variance result to find the standard error of the class sample set.

√18.99=4.36

The standard error of the grades on the test is 4.36. This means that most of the scores in the class fall within 4.36 points above or below the mean of 85.89.

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Chris Kolmar

Author

Chris Kolmar

Chris Kolmar is a co-founder of Zippia and the editor-in-chief of the Zippia career advice blog. He has hired over 50 people in his career, been hired five times, and wants to help you land your next job. His research has been featured on the New York Times, Thrillist, VOX, The Atlantic, and a host of local news. More recently, he's been quoted on USA Today, BusinessInsider, and CNBC.

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