The average formula divides the sum of all values by the number of values. Written in words, average equals sum divided by count. In mathematical notation, the arithmetic mean is commonly represented by x-bar for a sample or the Greek letter mu for a complete population.
Enter a list of numbers to calculate the mean, sum, count, median, mode, range, minimum, and maximum.
The numerator combines the values, while the denominator records how many values share the total.
The Arithmetic Mean Formula
The arithmetic mean is calculated by adding all observations and dividing the sum by the number of observations.
In simple language, this is written as average equals sum divided by count.
The Average Calculator applies this formula and also calculates the median, mode, range, minimum, and maximum.
What Is the Sum?
The sum is the result of adding every value in the data set.
For the values 6, 10, and 14, the sum is 30.
Leaving out even one value changes the numerator and produces an incorrect average.
What Is the Count?
The count is the number of individual values included in the calculation.
The list 6, 10, and 14 has a count of three.
Repeated values count separately because each occurrence is an observation.
What Does x-Bar Mean?
The symbol x-bar is commonly used for the arithmetic mean of a sample.
The Greek capital sigma represents addition, while the letter n represents the number of sample observations.
The notation x-bar equals the sum of x values divided by n expresses the same calculation as the word formula.
Population Mean Notation
When the data includes an entire population rather than a sample, the arithmetic mean is often represented by the Greek letter mu.
The population size may be represented by a capital N instead of a lowercase n.
The arithmetic operation remains the same: add the values and divide by the number of values.
Worked Formula Example
Consider the values 5, 7, 9, and 11. Their sum is 32.
The count is four. Substituting the values into the formula gives 32 divided by four.
The arithmetic mean is eight.
Why the Formula Works
The formula works because it redistributes the combined total equally among the observations.
The values 5, 7, 9, and 11 do not all equal eight, but their total of 32 can be divided into four equal groups of eight.
The arithmetic mean therefore acts as the equal-share value of the data set.
Formula with Decimal Values
Decimals are handled in exactly the same way as whole numbers.
For 1.5, 2.5, and 5, the sum is nine and the count is three.
Dividing nine by three gives an average of three.
Formula with Negative Values
Negative values must retain their signs when calculating the sum.
For minus six, minus two, four, and eight, the sum is four and the count is four.
The average is one.
Why Weighted Averages Use a Different Formula
The simple formula assumes that every observation has equal importance.
When one value represents more units, credits, hours, or probability than another, each value must be multiplied by its weight.
The weighted products are added and divided by the total weight rather than the simple count.
Common Formula Errors
One error is dividing before adding all values. The full sum should be calculated first.
Another is dividing by the number of intervals instead of the number of observations.
Parentheses are important when entering the formula into a calculator because the complete sum must remain together.
Conclusion
The average formula divides the sum of all observations by their count.
The notation may change between samples and populations, but the underlying arithmetic remains the same.
Use the Average Calculator to apply the formula to any list of valid numbers.
FAQs
What is the basic average formula?
Add all values and divide the sum by the number of values.
What does n mean in the average formula?
It represents the number of observations in the sample.
What does sigma mean?
The capital Greek letter sigma indicates that the values should be added.
What is x-bar?
It commonly represents the arithmetic mean of a sample.
Does the same formula work with decimals?
Yes. Decimals, integers, and negative values use the same arithmetic-mean formula.