What is the Sample Mean? Definition, Symbol, Formula, and Examples

The sample mean is the average of observations collected from a sample. Researchers often use it to estimate the population mean, especially when it is not possible, practical, or necessary to collect data from every member of the population. This makes the sample mean one of the most important concepts in descriptive and inferential statistics.

In this guide, you’ll learn what the sample mean is, the symbol used to represent it, the formula, how to calculate it manually, and when to use it. You’ll also learn common mistakes people make when calculating the sample mean and how to avoid them.

Want to quickly find the sample mean for your own dataset? Use our free sample mean calculator to get instant results, along with a clear step-by-step explanation showing how the mean was calculated.

Definition

In statistics, the sample mean is the average of all observations in a sample. It provides a simple way to summarize a sample dataset using a single number.

The sample mean is useful because it helps estimate the population mean, especially when it is impractical, expensive, or time-consuming to collect data from every individual in the population.

For instance, suppose a company wants to know the average satisfaction score of all its customers. If the company has more than 100,000 customers, it may be impractical or very expensive to collect responses from every customer. In this case, a better option is to collect data from a sample of customers and then calculate the sample mean. If the sample is randomly selected, the sample mean can act as an unbiased estimator of the population mean.

Note: The sample mean is a statistic because it is calculated from sample data, whereas the population mean is a parameter because it describes the entire population.

Want to learn more about the difference between a parameter and a statistic? Check out this parameter vs. statistic guide.

Symbol

The sample mean symbol is x̄, which is read as x-bar. The x in the symbol represents the variable being measured, whereas the bar above it shows that you are working with the mean of the x-values.

Therefore, whenever you see x̄ in statistics, it usually denotes the average of sample data.

Formula

The sample mean formula is x̄ = Σxi/n

Where:

• x̄ is the sample mean.
• Σxi is the sum of all observations in the sample data
• n is the total number of observations in the sample.

From the formula, finding the sample average requires you to add all values in the sample data and divide the result by the number of observations in that sample data.

In some textbooks, you may also see the formula written in expanded form as follows:

The formula simply means that you add the first value, second value, third value, and continue until every value in the sample has been included and then divide the results by the number of observations (n).

For example, if the sample values are 4, 6, and 8, we can apply the expanded form formula as follows:

= 6

Therefore, the sample average of the three observations is 6.

How to Find the Sample Mean

To find the sample average of any given data by hand, follow these steps:

• Step 1: Add all the values in the sample data to get Σxi.
• Step 2: Count the number of observations in the dataset to get n.
• Step 3: Divide the sum of all values (Σxi) by the total number of observations (n)

Example 1

A lecturer wants to estimate the average time students spend studying for a statistics exam. Instead of surveying the entire class, she randomly selects 8 students and records the number of hours they studied in the week before the exam.

Hours: 6, 8, 5, 7, 9, 6, 4, 5

Calculate the sample mean study time.

Solution

To find the sample mean of the number of hours spent studying, follow these steps:

Step 1: Add all the values in the sample data

6 + 8 + 5 + 7 + 9 + 6 + 4 + 5 = 50

Thus, Σxi = 50

Step 2: Count the number of observations in the sample data

From the sample data, there are 8 observations. Thus, n = 8

Step 3: Divide the sum of all values by the total number of observations in the sample data

The sample mean, x̄ = Σxi/n

= 50/8

=6.25

Therefore, the sample mean study time is 6.25

Example 2

A researcher wants to estimate the average monthly internet cost for students living off-campus. Instead of collecting data from every student, he randomly selects 10 students and records their monthly internet costs.

Monthly internet costs: 25, 30, 28, 35, 32, 30, 27, 33, 29, 31

Calculate the sample mean monthly internet cost.

Solution

To find the sample mean of the monthly internet costs, follow these steps:

Step 1. Add all the values in the sample data

The sum of all values in the data is: 25 + 30 + 28 + 35 + 32 + 30 + 27 + 33 + 29 + 31

= 300

This means that Σxi = 300

Step 2. Count the number of observations in the data

From the sample data, there are 10 observations. Thus, n = 10

Step 3. Divide the sum of all values by the total number of observations

By definition, the sample mean, x̄ = Σxi/n

Thus, x̄ = 300/10

= 30

Therefore, the sample mean monthly internet cost is 30.

Related. How to find the sample mean in Excel

Sampling Distribution of the Sample Mean

When working with sample means, you may be required to find their sampling distribution. The sampling distribution of the sample mean describes how the sample means would vary if you repeatedly collected different samples from the same population.

Here’s a simpler explanation of this concept.

Suppose you select one random sample from a population and calculate its mean. Then you select another random sample of the same size and calculate another mean. If you repeat this process many times, the sample means will not all be exactly the same. Some will be slightly higher than the population mean, while others will be slightly lower. The resulting pattern is called the sampling distribution of the sample mean.

If the population has mean μ and standard deviation σ, then the mean of the sample mean is μ, and its standard deviation is $\frac{\sigma}{\sqrt{n}}$.

The resulting standard deviation of the sample mean ($\frac{\sigma}{\sqrt{n}})$ is also called the standard error of the mean. It is widely used in statistics to show how much the sample mean is expected to vary from sample to sample.

Tip. If the population is normally distributed, then the sample mean is also normally distributed. However, if the population is not normally distributed, the sampling distribution of the sample mean becomes approximately normal as the sample size becomes large. This idea comes from the central limit theorem.

Therefore, for large enough samples, the sample mean is normally distributed with mean (μ) and standard deviation ($\frac{\sigma}{\sqrt{n}})$. In statistics, it can be written as: $\bar{x} \sim N\left(\mu,\frac{\sigma}{\sqrt{n}}\right)$

where μ is the mean of the sampling distribution and $\frac{\sigma}{\sqrt{n}}$​ is the standard error of the sample mean.

Example1

Suppose test scores in a population have a mean of 70 and a standard deviation of 12. A researcher repeatedly selects random samples of 36 students from this population and calculates the sample mean for each sample. Find the mean and standard deviation of the sampling distribution of the sample mean.

Solution

From the question, we know that:

• Population mean, μ =70
• Population standard deviation, σ =12
• Sample size, n = 36

i) Finding the mean of the sampling distribution

By definition, the mean of the sampling distribution of the sample average is the population mean.

Thus, mean of x̄ =70

ii) Finding the standard deviation of the sampling distribution

By definition, the standard deviation of the sample mean formula is $\frac{\sigma}{\sqrt{n}}$

Substituting the values into the formula and solving, we have:

Standard deviation of the sample average = $\frac{12}{\sqrt{36}}$

= 12/6

= 2

Therefore, the sampling distribution of the sample mean has a mean of 70 and a standard deviation of 2. This means that if you select many random samples of 36 students from the same population, their sample means will tend to cluster around 70, with a standard error of 2.

Sample Mean vs Population Mean

Although the sample mean and population mean are related, they are not the same. The main difference is that the sample mean is calculated from a subset of data drawn from the population, whereas the population mean is calculated for every value in the entire population.

The table below provides a quick summary of how the sample and population means differ.

Feature	Sample Mean	Population Mean
Symbol	x̄	μ
Data used	Sample data	Entire population
Type	Statistic	Parameter
Formula	x̄ =∑xi​​/n	μ = ∑X​/N
Main use	Estimates the population mean	Describes the full population

Example Scenario. Suppose you want to know the average height of all students in a university. If you measure only 100 selected students and calculate the mean, the resulting average is the sample average. However, if you measure every student in the university and calculate the mean, the resulting average height is the population mean.

Tip. In most research settings, the population mean is unknown. As such, you’ll often be required to calculate the sample mean as an estimate of the population mean.

Is Sample Mean the Same as Average?

Yes. The sample mean is the same as the average, and the only difference is in how they are calculated. The term average is general and is often used in many everyday situations. On the other hand, the term sample mean is more specific and refers to the average calculated from sample data.

Why is the Sample Mean Important?

The sample average is widely used by researchers, businesses, and analysts because it summarizes sample data using a single value. Instead of reviewing every observation separately, the sample mean provides a quick way to understand the typical or average value in a dataset.

It is also important because it acts as an estimate of the population mean, when collecting data from every member of the population would be too expensive, time-consuming, or impractical. When a sample is selected properly, the sample mean can provide a reasonable and unbiased estimate of the average value in the larger population.

For example, a company may survey a sample of customers to estimate the average customer satisfaction score instead of asking every customer. Similarly, a researcher may calculate the average score from a sample of participants to estimate the average score in a larger group.

Applications of the Sample Mean

The sample average is one of the measures of central tendency, which is widely used in many real-world situations. It provides a quick way to summarize numerical data. Instead of looking at every single value separately, finding the average from the sample can help you understand what is typical in that group.

Here are some common areas where it is useful:

• Education. Schools and researchers may use it to estimate average test scores, study hours, attendance rates, or student satisfaction levels. For example, if a school wants to understand how students are performing in a certain grade, it can calculate the average score from a sample of students and use that information to compare classes or learning programs.
• Healthcare. Hospitals and health researchers often use it to summarize patient age, recovery time, blood pressure, treatment scores, or length of hospital stay. This helps healthcare professionals understand patient outcomes and identify areas for improvement.
• Finance. Banks and financial analysts may use it to estimate average income, credit scores, loan amounts, investment returns, or spending patterns. This can consequently help these organizations make more informed decisions about lending and investment.
• Business. Companies use it to understand customer ratings, product prices, delivery times, sales values, or support response times. For instance, an average rating from a sample of customers can inform the business whether customers are satisfied with their services.
• Manufacturing. Manufacturers may use it to monitor product weight, size, quality scores, production time, or defect rates. This helps them check whether a process is stable and whether products are meeting expected standards.
• Research. Researchers use it to summarize survey responses, compare groups, estimate population values, and prepare for statistical tests such as t-tests, ANOVA, and regression analysis. If you are working with sample data, you should first calculate the mean to get an overview of the data before moving to deeper analysis.

Common Mistakes to Avoid

Finding the sample mean is simple, but a few small errors can still affect your answer. Before you rely on your result, check for these common mistakes.

• Using the wrong symbol. Use x̄ when you are referring to the average from sample data. However, if you’re finding the average of the whole population, you should use the population mean symbol (μ).
• Confusing n and N. When working with sample data, you should divide the sum of all values by the number of observations in that sample, often denoted by n. However, N denotes the total number of observations in the entire population.
• Leaving out some values. When calculating the sample mean, you should always add all values in the dataset. Missing any of them can change the sum of all values, leading to an incorrect answer.
• Rounding too early. Avoid rounding answers to intermediate steps as this can lead to wrong results, especially when working on questions that require precision.
• Ignoring unusual values. Very large or very small values can pull the average up or down. When a value looks unusual, you should always recheck to determine whether it’s an outlier or a wrongly entered value.
• Using an average for category labels. An average only makes sense for numerical data. For categories such as gender, marital status, county, education level, or product type, you should summarize them using counts or percentages.

Frequently Asked Questions