Please provide numbers separated by a comma to calculate their arithmetic mean, median, mode, range, geometric mean, and step-by-step arithmetic solutions.
In mathematics, an average (most commonly referring to the arithmetic mean) represents the central value of a finite set of numbers. It is computed by summing all numeric observations and dividing by the total count of observations:
The standard average computed by adding all values and dividing by the count. Best for general symmetric datasets without extreme outliers.
The n-th root of the product of n numbers: (x₁ · x₂ · ... · xₙ)^(1/n). Ideal for calculating compound growth rates, investment returns, and exponential phenomena.
The reciprocal of the arithmetic mean of the reciprocals: n / ∑(1/xᵢ). Specifically tailored for averaging rates, speeds over fixed distances, and unit ratios.
Calculated by multiplying each value by a predetermined weight or importance percentage before summing and dividing by the total weight: ∑(wᵢ xᵢ) / ∑ wᵢ (e.g. GPA calculation).
| Measure | Definition | Best Used For | Sensitivity to Outliers |
|---|---|---|---|
| Mean (Average) | Sum divided by total count | Symmetric, bell-shaped distributions | High (heavily skewed by extreme values) |
| Median | The middle value in an ordered set | Skewed distributions (income, real estate prices) | Robust (resistant to outliers) |
| Mode | The most frequent value in the dataset | Categorical data and popularity analysis | Robust (unaffected by extreme values) |
Add all the numbers together to find the sum, then divide that sum by the count of numbers in the set. For example, for 10, 20, and 30: Sum = 60, Count = 3, Average = 60 ÷ 3 = 20.
Yes. If the sum of the numbers in the dataset is negative (e.g. -10, -20, 15 ⟹ Sum = -15), the resulting mean will be negative (-15 ÷ 3 = -5).
Population standard deviation divides the sum of squared deviations by $N$ (when you have the entire population), whereas sample standard deviation divides by $N - 1$ (Bessel's correction) to provide an unbiased estimate when sampling a subset.