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Arithmetic Mean Calculator

Calculate the arithmetic mean, median, sum, count, variance, and standard deviation for any dataset with step-by-step steps.

Mean & Statistical Summary
Arithmetic Mean (x̄)
24.333333
Median
22
Sum (Σx)
146
Std Deviation (σ)
10.934146
Deduction Steps:
Total items (n) = 6
Sum of elements: Σx = 12 + 15 + 20 + 24 + 30 + 45 = 146
Arithmetic Mean: x̄ = (Σx) / n = 146 / 6 = 24.333333
Sorted dataset: [12, 15, 20, 24, 30, 45] → Median = 22
Population Variance: σ² = (Σ(x - x̄)²) / n = 119.555556
Standard Deviation: σ = √σ² = 10.934146
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What Is a Arithmetic Mean Calculator?

The arithmetic mean (commonly called the average) is the central tendency measure calculated by dividing the sum of a collection of numbers by the total count of numbers in the collection.

How to Use This Calculator

  1. Enter your dataset numbers separated by commas or spaces.
  2. View the calculated Arithmetic Mean (x̄), Median, Sum, and Standard Deviation.
  3. Review the step-by-step summation and division steps.

Arithmetic Mean Formula

\bar{x} = \frac{1}{n}\sum_{i=1}^n x_i = \frac{x_1 + x_2 + \dots + x_n}{n}

Sum all values x_i in the dataset, then divide by the sample size n.

Worked Example

Scenario: Find the arithmetic mean of 12, 18, 25, 30, 40

Count items: n = 5

Sum items: Σx = 12 + 18 + 25 + 30 + 40 = 125

Compute mean: x̄ = 125 / 5 = 25

Median: The middle value in sorted order is 25

Result: Mean = 25, Sum = 125, Count = 5

Tips & Key Notes

  • The arithmetic mean is sensitive to extreme outliers; if data is skewed, consider using the median.
  • The sum of the signed deviations of each data point from the arithmetic mean is always equal to 0.
  • For grouped data or values with unequal importance, use the Weighted Arithmetic Mean.

Frequently Asked Questions

What is the difference between mean and average?

In general speech, average refers to the arithmetic mean. In statistics, average can refer to any measure of central tendency including mean, median, or mode.

How do outliers affect the mean?

Because the mean incorporates every value equally, unusually large or small values will pull the mean toward the extreme, potentially distorting the representative center.

What is the difference between sample mean and population mean?

Sample mean (x̄) is computed from a subset of a population, while population mean (μ) is the true mean of all members in the entire population.

Can the arithmetic mean be negative?

Yes, if the sum of the numbers in the dataset is negative, the resulting arithmetic mean will be negative.

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