The population mean is the average value of all the observations in the entire group you want to study. It is useful when you have the data for every member, item, score, or observation in your population of interest and want to summarize it using a single value.
For example, suppose you want to study the performance of students in an entire class. If you collect scores from every student and calculate the average, the result is the population mean. However, if you randomly select only a few students and calculate the average of their scores, the result is the sample mean instead.
In this guide, you will learn what the population mean is, its symbol, formula, how to calculate it using examples, and common mistakes to avoid.
What is the population mean?
The population mean is the average of all values in the entire group you want to study. In statistics, a population means the full set of people, items, scores, or observations under investigation.
A full group can be small or large. For example, it could be:
- All students in one classroom
- All employees in a company
- All test scores in one completed exam
- All products made in a factory in one day
- All monthly sales values for a business in one year
The key idea is simple: for a set of observations to be a population, every value in the group must be included.
For example, suppose a class has 8 students, and you calculate the average score using all 8 scores. That average describes the full class, and it is the population average.
In statistics, this type of mean is called a parameter because it describes a whole group, not just a selected part of it.
Population Mean Symbol and Formula
The population mean symbol is the Greek letter (μ), and is pronounced as mu. This convention is different from the sample mean symbol, which uses the x̄ symbol, often read as x-bar. The distinction makes it easier for statisticians and researchers to communicate the findings effectively.
The formula for finding the population mean is the same as the sample mean formula, with the only difference being changes in notation. While the population mean formula is μ = ∑Xi/N
Where:
- μ is the mean of the entire set of individuals, or items you want to estimate
- ΣXi represents the sum of all values in the entire dataset
- N is the total number of values in the dataset.
The formula simply means you add all the values in the entire group you want to study and divide the results by the size of your target group.
As you can see, the formula is slightly different from the sample mean formula, which uses a different symbol (x̄), lower case (xi) for values in the sample data, and divides the sum of all sample values by the sample size (n).
How to Find the Population Mean
Want to learn how to calculate the mean of population data manually? Follow these 3 steps:
- Add all values in the data to get ΣXi.
- Count the number of values (observations) in the data to get N
- Divide the sum of all values ( ΣXi) by the total number of values (N) to get the mean, μ.
Example 1
Suppose the scores of all 5 students in a small class are: 72, 80, 85, 90, 93. Find the mean.
Solution
Since the list includes every student in the class, we should use the population average formula and follow the steps.
Step 1. Add all values
The sum of all values in the entire group is: 72+80+85+90+93 = 420
Thus, ∑Xi = 420
Step 2. Count the number of values
There are 5 students in the entire class. Thus, the size of the entire group is N = 5
Step 3. Divide the sum of all values by N
Thus, the population mean is μ = 420/5
= 84.
This implies that the average score for the whole class is 84.
Example 2
Suppose a small company has 6 employees. Their ages are: 21, 24, 25, 27, 28, 30. Find the mean age of employees in the company.
Solution
Because the list includes every employee in the company, we need to find the population mean. Here’s how to find the correct mean for the scenario:
Step 1. Add all values
The sum of all ages in the company is: 21+24+25+27+28+30 =155
Thus, ∑Xi = 155
Step 2. Count the number of values
There are only 6 employees in the company. Hence, N = 6
Step 3. Divide the sum of all values by N
The required mean is μ =155/6
= 25.83
Therefore, the average age of all employees in the company is about 25.83 years.
You can also use our population mean calculator to calculate it quickly and check the steps for your own dataset.
Population Mean vs Sample Mean
These two terms are closely related, but they are not the same. A population mean includes every value in the group, whereas a sample mean uses only selected values from that group.
The table below provides a quick summary of the key differences between these two concepts.
| Feature | Population Mean | Sample Mean |
|---|---|---|
| Meaning | Average of the entire group | Average of a subset of the population |
| Symbol | μ | x̄ |
| Number of values | N | n |
| Type | Parameter | Statistic |
| The average score of every student in a class | Average score of every student in a class | The average score of selected students in a class |
For example, if you calculate the average score of all 30 students in a class, you have found the mean for the whole class. But if you calculate the average score of only 10 selected students, you have found the sample mean.
If your data represents a subset of the population, and you want to quickly find its average, you can use our sample mean calculator instead.
Is Population Mean the Same as Average?
Yes. It is a type of average that finds the average of all values in a complete group.
In everyday language, people often say “average” when they mean the arithmetic mean. However, in statistics, you need to distinguish clearly whether the calculation is based on all individuals in the target group or a sample of this group.
Sometimes, you may find yourself calculating the average value of a dataset without worrying whether it is a sample or a population. In this case, you’re finding the arithmetic mean. Thus, if you want to quickly find the arithmetic mean of any dataset, our arithmetic mean calculator might be useful.
Common Mistakes When Finding the Population Mean
This calculation is simple, but a few small mistakes can lead to the wrong answer. Here are the most common mistakes students make when finding the mean from the entire group under investigation:
- Using sample data instead of the full group. If your data does not include every value in the group you’re studying, then finding the population mean wouldn’t be appropriate.
- Forgetting to include all values. You should include every value when finding the average value. Otherwise, the final answer may be wrong.
- Confusing μ and x̄. You should use μ for the mean of the complete group and x̄ for the mean of a sample.
- Dividing by the wrong number. Always divide by the total number of values included in the dataset.
- Using N and n interchangeably. In many statistics formulas, N is used for the number of values in a population, while n is used for the number of values in a sample.
Limitations of the Population Mean
While the population mean provides the exact average of a dataset, it is not always practical to calculate in real-world studies.
Key limitations include:
- Requires complete population data. You must have values for every member of the population.
- Often impractical for large populations. Collecting data from very large populations can be costly and time-consuming.
- Researchers often rely on samples. In most studies, statisticians calculate the sample mean as an estimate of the population mean.
Frequently Asked Questions
It is simply the average of all values in a complete group you’re studying. It describes the center or typical value of the whole group under investigation.
The symbol is μ, pronounced mu.
You simply need to sum all values in the population (∑Xi) and divide by the number of values in that population (N). Thus, its formula is μ = ∑Xi/N
Use it when your dataset includes every member, item, score, or observation in the group you are studying. If you only have part of the data, you should use the sample mean instead.
