Statistics - (Variance|Dispersion|Mean Square) (MS)

The variance shows how widespread the individuals are from the average.

The variance is how much that the estimate varies around its average.

It's a measure of consistency. A very large variance means that the data were all over the place, while a small variance (relatively close to the average) means that the majority of the data are closed.

See:

• Statistics - (Residual|Error Term|Prediction error|Deviation) (e|<math>\epsilon</math>)
• Statistics - Bias-variance trade-off (between overfitting and underfitting)

<MATH> \begin{array}{rrl} Variance & = & \frac{\displaystyle \sum_{i=1}^{\href{sample_size}{N}}{(\href{raw_score}{X}_i- \href{mean}{\bar{X}})^2}}{\displaystyle \href{degree_of_freedom}{\text{Degree of Freedom}}} \\ & = & \frac{\displaystyle \sum_{i=1}^{\href{sample_size}{N}}{(\href{Deviation Score}{\text{Deviation Score}}_i)^2}}{\displaystyle \href{degree_of_freedom}{\text{Degree of Freedom}}} \\ & = & (\href{Standard_Deviation}{\text{Standard Deviation}})^2 \end{array} </MATH>

where:

• <math>X</math> is a data point (raw score).
• <math>\bar{X}</math> is the mean of the distribution (all data points)
• <math>\text{Degree of Freedom}</math> is the degree of freedom which is:
  • <math>N</math> (Sample Size) for descriptive statistics
  • <math>N - 1</math> for inference statistics
• Statistics - Deviation Score (for one observation)
• Statistics - Standard Deviation (SD|s|<math>\sigma</math>|RMS width)

<MATH> Var(X + Y) = Var(X) + Var(Y) + 2 Cov(X, Y) </MATH> where:

• cov = Statistics - Covariance

• For each data point, calculate the deviation score (difference from the average)
• Square this difference (because the original sum of all deviation score is zero) (to get rid of negative differences)
• Calculate a sum of the squared differences
• The final variance is the sum of squared differences divided by the degree of freedom
• The degree of freedom is:
  • <math>N</math> (Sample Size) for descriptive statistics
  • <math>N - 1</math> for inference statistics

units = [7, 10, 9, 4, 5, 6, 5, 6, 8, 4, 1, 6, 6]
  
def units_average(units):
    average = sum(units) / len(units)
    return average

def units_variance(units,average):
    diff = 0
    for unit in units:
        diff += (unit - average) ** 2
    return diff / len(units)

print units_variance(units, units_average(units))

5