Statistics FAQ: ANOVA, Bayes, etc

  Sample Mean = Unbiased Estimator Of Population Mean;
                Not biased, but has inherent error;
                The inherent error itself is random (usually gausian).
                Variance does not directly represent error.
                Variance of Variance somewhat represents error in sample mean.

  Sample Variance = Biased Estimator of Population Variance
                  = Multiply by N/(N-1) to get unbiased estimate
                  = Now it is not biased but has inherent error due to sample.

  Variance Of Variance = 
                 Obtained using multiple sample collections;
                 Represents error/uncertainty of variance/mean.
                 Useful when population is very large

   F = (Inter-Group Variability) / (Intra Group Variability)

   Inter-Group Variability = Variance Of Group Means wrt Overall-Groups-Mean

   Intra-Group Variability = Average Variance of all groups together.
                             (Overall Mean not involved)

   df1 = Degrees Of Freedom (Numerator) = Total Number of Groups.
   df2 = Degrees Of Freedom (Denominator) = Total Size of all Samples.

   Higher F  === means ===> Different groups.

    0 < F < Infinity

    Mostly 0 < F < 1; (Variability across group means is less);
    F > 1 when the samples are taken from different areas of population.
    (Group means vary more than intra-group)

   F probability function is ProbabilityDensity (Y Axis) vs F-Value (X Axis)
   for a given df1 and df2;

   F pdf looks like Right Skewed (long right tail) normal distribution.

   Mostly F is less than 1; It has long tail where F > 1

   F > Critical-F-Value  ==> Rejects Null Hypothesis (No Effects Assumption)
                             i.e. Groups are significantly different.

   Critical-F-Value := Use Lookup table based on Significance Level e.g. 5%
                       (and df1 and df2).

   The lookup table was constructed assuming the samples are gaussian.

   Significance level or alpha := Typically 5%

   P-value := Probability of observing the given observation;
   If P-value < Significance the null Hypothesis should be rejected.

   e.g. For a given F-Value may correspond to a P-Value of 0.03;
   So either you calculate Critical-F-Value (value for P-value 0.05),
   or find the p-value of your observation to reject/accept null hypothesis.

      Chi-Square = Sum( square(expected-observed)/ expected )

      Degrees Of Freedom = (rows-1) * (cols-1)

      Example: Category1: Male/Female; Category2: Tea/Coffee
      Rows=2; Cols=2; Degrees of Freedom = 1x1 = 1

      The table contains numeric count of peoples who prefer coffee/Tea.
      Assuming independence between variables, you compete:
           Overall_expected_Tea_drinkers = Total_tea_preference / Total_people
           Overall_expected_Coffee_drinkers = Total_coffee_preference / Total_people
      Then find the ratios for all combinations and sum it up.

      Chi-square critical value at α = 0.05 (df=1) from tables = 3.84

      Your chi-square > 3.84 means there is a relationship between categories.

      t =  (mean1 - mean2) / sqrt((variance1/n1) +(variance2/n2) )

      df = degrees-of-freedom = n - 1

      Higher t value, more likely groups are different.

      Similar to F-Statistic, there is a lookup table for T-critical-value 
      given the df and significance level (alpha - e.g. 5%);
     
      T pdf (Probability density function) looks like normal bell curve with thicker
      tails and smaller peak);