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Urban Start-up Agglomeration and Venture Capital Investment (view source)
Revision as of 19:05, 27 October 2020
, 19:05, 27 October 2020→Implementing The Elbow Method
#Variance explained = between-group variance / within-group variance (Note that this is the ANOVA F-statistic).
Using the [[https://en.wikipedia.org/wiki/Law_of_total_variance Law of total variance]], total variance = between-group variance + within-group variance.
'''From Wikipedia:'''
where <math>Y_{ij}</math> is the ''j''<sup>th</sup> observation in the ''i''<sup>th</sup> out of <math>K</math> groups and <math>N</math> is the overall sample size. This ''F''-statistic follows the ''F''-distribution with degrees of freedom <math>d_1=K-1</math> and <math>d_2=N-K</math> under the null hypothesis. The statistic will be large if the between-group variability is large relative to the within-group variability, which is unlikely to happen if the population means of the groups all have the same value.
'''Also from Wikipedia:'''
[https://en.wikipedia.org/wiki/Variance Variance] is the expectation of the squared deviation of a random variable from its mean.
:<math> \operatorname{Var}(X) = \frac{1}{n} \sum_{i=1}^n (x_i - \mu)^2 = \left( \frac{1}{n} \sum_{i=1}^n x_i^2 \right) - \mu^2, </math>
where <math>\mu</math> is the average value. That is,
:<math>\mu = \frac{1}{n}\sum_{i=1}^n x_i .</math>
====The Elbow Method Justification====
