Want to find the sample and population standard deviation instead? Use the standard deviation calculator.
How to Use the Mean Calculator
This mean calculator allows you to find the average values of sample or population data. To use the calculator:
- Select whether you want to calculate the sample mean or population mean. The sample mean option is appropriate if your values are a sample drawn from a larger group and population mean option if the values represent the entire group you want to study.
- Enter your data values. You can separate them using commas, spaces, tabs, or line breaks. Alternatively, you can paste values directly from Excel or Google Sheets.
- Click Calculate.
The calculator will instantly return the correct type of mean (sample or population) and show the correct statistics notation to help you report your answer as a sample mean or population mean. It also shows you exactly how to find the average for the selected mean, step-by-step.
Note. The sample mean and population mean are both computed by adding all the values and dividing by the number of observations in the data. However, the sample mean and population mean formulas use different symbols.
Example 1. Sample Mean
Suppose a researcher records the number of hours studied by a sample of five students as follows: 6, 8, 7, 10, 9. Find the sample mean.
Solution
Since these five students are only part of a larger group of students, we treat the data as sample data.
To find the sample mean using the calculator:
- Select Sample mean.
- Enter the values: 6, 8, 7, 10, 9.
- Click Calculate.
The calculator automatically treats your data as sample data, applies the sample mean formula and return the correct sample mean of the data. Notably, you’ll see the answer as: Sample mean, x̄ = 8. This implies that the students in the sample studied an average of 8 hours.
Example 2. Population Mean
Suppose a teacher wants to find the average quiz score for every student in a small class of 10 students. The scores are: 72, 75, 81, 84, 88, 90, 66, 74, 87, 64. Find the population mean.
Solution
Since the data include every student in the class, we should treat the values as population data.
To find the population mean using the calculator:
- Select Population mean.
- Enter the values: 72, 75, 81, 84, 88, 90, 66, 74, 87, 64.
- Click Calculate.
The calculator will treat values as population data, apply the population mean formula, and return the correct population mean using the correct notation. Specifically, the answer will be: Population mean, μ = 78.1. This means that the average quiz score for the whole class is 78.1.
What Is Mean in Math?
In mathematics, the mean refers to the average of a set of numbers. To find the mean, add all the values and divide the result by the total number of values.
For example, if the values are 4, 6, and 8, you can easily find the mean as follows:
Mean = (4 + 6 + 8)/3
= 18/3
= 6.
Thus, the mean is 6.
In statistics, the mean is usually written differently depending on whether the data represents a sample or a population. If the values come from a sample, we call it the sample mean. However, if the values represent the entire group being studied, we call it the population mean.
Note. While the calculation of the sample mean and population mean is the same, the formula symbols are onterpretation are different.
What is Sample Mean?
The sample mean is the average of a set of sample data. It is one of the most useful statistics because it helps statisticians make inferences about the population, especially when the full data of the population is unavailable.
The symbol for the sample mean is x̄ and is pronounced as “x-bar.” It is distinct from the population mean symbol, μ (read as mu). Thus, to find the sample mean, you simply need to sum all the data points and divide the results by the number of data points in that sample.
Sample Mean Formula
The sample mean formula is x̄ = Σxi/n
where:
- x̄ is the sample mean symbol.
- Σxi is the sum of all values in the sample.
- n is the total number of data points (observations) in the sample.
In essence, to calculate the sample mean, you only need to sum all the values in a dataset and divide by the number of observations. This formula provides a concise and reliable measure of the central tendency of the sample data.
How to Find Sample Mean
Want to learn how to compute the sample mean manually, without the calculator? The process is straightforward and involves only 3 simple steps:
- Sum all the values in the sample data to get Σxi
- Count the number of observations in the sample data to get n
- Divide the sum of all the values in the sample ( Σxi) by the number of observations in the sample (n) to get the sample mean (x̄)
Example
A teacher wants to estimate the average score of students in a statistics class. Instead of using all students in the class, the teacher selects a sample of 6 students. Their test scores are:
72, 80, 75, 90, 85, 78
Find the sample mean score.
Solution
We are given the sample data: 72, 80, 75, 90, 85, 78
Since the scores are from a sample of students, we can quickly find the sample mean as follows:
Step 1: Sum all the values in the sample data to get Σxi.
Σxi = 72 + 80 + 75 + 90 + 85 + 78
= 480
Thus, Σxi = 480
Step 2: Count the number of observations in the sample data to get n.
There are 6 scores in the sample. Thus, n = 6
Step 3: Divide the sum of all the values in the sample by the number of observations in the sample.
By definition, the sampe mean formula is: x̄ = Σxi/n
Substituting the values in step 1 and 2 into the formula, we get:
Sample mean, x̄ = 480 / 6
= 80
Therefore, the sample mean, x̄ = 80. This means the average test score for the sampled students is 80.
What is Population Mean?
The population mean is the average of all values in an entire population. It is used when the data includes every member, item, or observation in the group you want to study.
Unlike the sample mean, which is only accounts for only a smaller part of the population, the population mean accounts for all values in the population under study. In other words, the population mean gives the true average of the whole population, while the sample mean provides an estimate of the population mean.
The population mean symbol is μ and is pronounced as “mu.” It is distinct from the sample mean symbol, x̄, which is pronounced as “x-bar.”
Population Mean Formula
The population mean formula is: μ = Σxi / N
where:
- μ is the population mean symbol.
- Σxi is the sum of all values in the population.
- N is the total number of data points or observations in the population.
In essence, to find the population mean, you only need to sum all the values in the population data and divide by the number of observations. This formula provides the actual average of the entire group being studied.
How to Find Population Mean
Want to learn how to compute the population mean manually, without the calculator? The process is straightforward and involves only 3 simple steps:
- Sum all the values in the population data to get Σxi.
- Count the number of observations in the population data to get N.
- Divide the sum of all the values in the population, Σxi, by the number of observations in the population, N, to get the population mean, μ.
Example
A small business owner wants to find the average number of orders received over one complete working week. The number of orders received each day was:
18, 22, 20, 25, 15
Find the population mean number of orders.
Solution
Since the data includes all the working days in the week being studied, we can treat the values as population data and quickly find the population mean as follows:
Step 1: Sum all the values in the population data to get Σxi.
Σxi = 18 + 22 + 20 + 25 + 15
= 100
Thus, Σxi = 100
Step 2: Count the number of observations in the population data to get N.
There are 5 values in the population. Thus, N = 5
Step 3: Divide the sum of all the values in the population by the number of observations in the population.
By definition, the population mean formula is: μ = Σxi / N
Substituting the values in steps 1 and 2 into the formula, we get:
Population mean, μ = 100 / 5
= 20
Therefore, the population mean, μ = 20. This means the business received an average of 20 orders per day over the working week studied.
Sample Mean vs Population Mean
The sample mean and the population mean are both measures of central tendency. However, they differ in the scope of data they represent. While the sample mean measures the average of the subset of the population, the population mean represents the average of all values in the entire population.
The table below provides a quick summary of how the sample mean and population means differ.
| Feature | Sample Mean (x̄) | Population Mean (μ) |
|---|---|---|
| Definition | Average of a sample subset | Average of the entire population |
| Symbol | x̄ | μ |
| Number of Observations | n (sample size) | N (population size) |
| Usage | It provides an estimate of the population mean | It provides the true mean of the population |
| Calculation | The sample mean formula is: | The population mean formula is: |
Frequently Asked Questions
Yes. In most basic statistics problems, the mean is the same as the average. It is found by adding all values and dividing by the number of values.
This mean calculator finds the average of values in a sample or population dataset. It also shows the formula, the sum of values, the number of observations, and the step-by-step solution.
The sample mean is the average of a sample, while the population mean is the average of the whole population. The sample mean uses the symbol x̄, while the population mean uses the symbol μ.
No. The calculation is the same for the same dataset. You still add all values and divide by the number of values. What changes is the symbol and interpretation of the result.
Choose the sample mean when your data represent only part of a larger group. For example, if you survey 100 customers from a company with 10,000 customers, the 100 customers are a sample. In this case, you should use the sample mean option of the calculator.
Choose the population mean when your data includes every value in the group you want to study. For example, if you calculate the average score of every student in one class, that dataset can be treated as a population for that class. In such cases, you should use the population mean option of the calculator.
