Category: Level I

In comparing the formulae for the standard deviation of a population: \[\sigma\ =\ \sqrt{\frac{\sum_{i=1}^N \left(X_i\ –\ \mu_X\right)^2}{N}}\] and the standard deviation of a sample: \[s\ =\ \sqrt{\frac{\sum_{i=1}^n \left(X_i\ –\ \bar X\right)^2}{n\ –\ 1}}\] the obvious difference that strikes one immediately is the for the population standard deviation the denominator is the population size – \(N\)…

Kurtosis is generally viewed as a measure of peakedness of a probability distribution (how tall the center of the distribution is compared to, say, a normal distribution); the taller (and thinner) the center peak, the higher the kurtosis.  Another way of describing kurtosis is as a measure of how fat the tails (extreme ends, positive…

Skewness of a probability distribution is a measure of its asymmetry; the higher the (absolute value of the) skewness, the more asymmetric the distribution.  Symmetric distributions have skewness of zero.  The formula for the skewness of a sample is: \[skewness\ =\ \frac{n}{\left(n\ -\ 1\right)\left(n\ -\ 2\right)}\frac{\sum_{i=1}^n \left(X_i\ –\ \bar X\right)^3}{s^3}\ ≈\ \frac1n\frac{\sum_{i=1}^n \left(X_i\ –\ \bar…

Bayes’ Formula is frequently presented in statistics texts as important (it is), profound (it isn’t, particularly), and difficult (it isn’t, remotely).  If you understand conditional probability, then Bayes’ Formula is trivial.  Let me show you: We start with the probability of two events, A and B: \[P(AB)\ =\ P(A|B)\ ×\ P(B)\] Similarly, for the probability…