Variance 편집하기

'''Variance''' is a statistical measure that quantifies the dispersion of a dataset relative to its mean. It indicates how much the values in a dataset deviate from the average value.
==Definition==
The variance (σ²) of a dataset with n elements is calculated as:
*Population Variance (σ²):
**σ² = (1/n) * Σ (xᵢ - μ)²
*Sample Variance (s²):
**s² = (1/(n-1)) * Σ (xᵢ - x̄)²
where:
*'''xᵢ''' – Each data point.
*'''μ''' – Population mean.
*'''x̄''' – Sample mean.
*'''n''' – Number of data points.
*'''Σ''' – Summation notation.
==Interpretation==
*'''Low Variance''' – Data points are close to the mean.
*'''High Variance''' – Data points are spread out from the mean.
*'''Zero Variance''' – All values are identical.
==Example Calculation==
Given dataset: [5, 7, 9, 10, 14]
#Calculate the mean:
#*x̄ = (5 + 7 + 9 + 10 + 14) / 5 = 9
#Compute squared differences:
#*(5 - 9)² = 16
#*(7 - 9)² = 4
#*(9 - 9)² = 0
#*(10 - 9)² = 1
#*(14 - 9)² = 25
#Population variance:
#*σ² = (16 + 4 + 0 + 1 + 25) / 5 = 9.2
#Sample variance:
#*s² = (16 + 4 + 0 + 1 + 25) / (5-1) = 11.5
==Properties==
*'''Always non-negative''' – Squaring deviations ensures variance is ≥ 0.
*'''Measured in squared units''' – Different from the original data scale.
*'''Used to compute standard deviation''' – Standard deviation (σ) is the square root of variance.
==Applications==
*'''Finance''' – Measuring risk (volatility of asset returns).
*'''Machine Learning''' – Evaluating feature distribution in datasets.
*'''Quality Control''' – Assessing variability in production processes.
==Variance vs. Standard Deviation==
{| class="wikitable"
!Measure!!Formula!!Interpretation
|-
|Variance (σ²)||(1/n) * Σ (xᵢ - μ)²||Average squared deviation from mean.
|-
|Standard Deviation (σ)||√Variance||Measures dispersion in original units.
|}
==See Also==
*[[Standard Deviation]]
*[[Mean]]
*[[Covariance]]
*[[Normal Distribution]]
*[[Statistical Dispersion]]
[[Category:Statistics]]