Central Moments Calculator
Use this unified calculator to find the first four central moments for both ungrouped (raw) data and grouped (frequency distribution) data. Central moments measure the spread and shape characteristics of your distribution.
Quick Start
| Central Moments Calculator | |
|---|---|
| Data Type | Ungrouped (Raw Data) Grouped (Frequency Distribution) |
| Enter the X Values (Separated by comma,) | |
| Type of Frequency Distribution | DiscreteContinuous |
| Enter the Classes for X (Separated by comma,) | |
| Enter the frequencies (f) (Separated by comma,) | |
| Results | |
| Number of Observations (n): | |
| Mean of X values: | |
| First Central Moment (μ₁): | |
| Second Central Moment (μ₂): | |
| Third Central Moment (μ₃): | |
| Fourth Central Moment (μ₄): | |
Understanding Central Moments
Central Moments measure the distribution of data around the mean:
- First Central Moment (μ₁): Always equals zero (mathematical property)
- Second Central Moment (μ₂): Variance - measures spread around mean
- Third Central Moment (μ₃): Related to skewness - measures asymmetry
- Fourth Central Moment (μ₄): Related to kurtosis - measures tail heaviness
Formulas
Ungrouped Data: $$\mu_r = \frac{1}{n}\sum_{i=1}^{n}(x_i - \overline{x})^r$$
Grouped Data: $$\mu_r = \frac{1}{N}\sum_{i=1}^{k}f_i(x_i - \overline{x})^r$$
Where:
- r = moment number (1, 2, 3, 4)
- x̄ = mean
- N = total frequency
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