17saomm o Women i Women W Women m Women nsearchhsearche Women rsearchhsearch Searchforsinglesmeetsingles e Searchforsinglesmeetsingles rsearchh Szh o Szh s0se 17saomm rc 17saomm lsearchSesearchrsearchhfosearchs Searchforsinglesmeetsingles ngsearchese Searchforsinglesmeetsingles e Women r 17saomm hf Searchforsinglesmeetsingles rssearchn Searchforsinglesmeetsingles l Searchforsinglesmeetsingles saee Szh sisearchgle Women 1ssearcha Searchforsinglesmeetsingles c 17saomm igsearchesearch et Searchforsinglesmeetsingles igesearch 17saomm Searchforsinglesmeetsingles 7searchasearchm Searchforsinglesmeetsingles Szh s Searchforsinglesmeetsingles a Szh c Searchforsinglesmeetsingles t ssearch t Searchforsinglesmeetsingles esearchDescriptive Sampling Statistics JavaScript in performing some numerical experimentation for validating the above assertions for a deeper understanding.

Shape of a Distribution Function:
The Skewness-Kurtosis Chart

The pair of statistical measures, skewness and kurtosis, are measuring tools, which is used in selecting a distribution(s) to fit your data. To make an inference with respect to the population distribution, you may first compute skewness and kurtosis from your random sample from the entire population. Then, locating a point with these coordinates on the widely used skewness-kurtosis chart , guess a couple of possible distributions to fit your data. Finally, you might use the goodness-of-fit test to rigorously come up with the best candidate fitting your data. Removing outliers improves the accuracy of both skewness and kurtosis.

Skewness: Skewness is a measure of the degree to which the sample population deviates from symmetry with the mean at the center.

Skewness = S (xi - ) 3 / [ (n - 1) S 3 ],    n is at least 2.

Skewness will take on a value of zero when the distribution is a symmetrical curve. A positive value indicates the observations are clustered more to the left of the mean with most of the extreme values to the right of the mean. A negative skewness indicates clustering to the right. In this case we have: Mean £ Median £ Mode. The reverse order holds for the observations with positive skewness.

Kurtosis: Kurtosis is a measure of the relative peakedness of the curve defined by the distribution of the observations.

Kurtosis = S (xi - ) 4 / [ (n - 1) S 4 ],     n is at least 2.

Standard normal distribution has kurtosis of +3. A kurtosis larger than 3 indicates the distribution is more peaked than the standard normal distribution.

Coefficient of Excess Kurtosis = Kurtosis - 3.

A value of less than 3 for kurtosis indicates that the distribution is flatter than the standard normal distribution.

It can be shown that,

Kurtosis - Skewness 2 is greater than or equal to 1, and
    Kurtosis is less than or equal to the sample size n.

These inequalities hold for any probability distribution having finite skewness and kurtosis.

In the Skewness-Kurtosis Chart, you notice two useful families of distributions, namely the beta and gamma families.

The Beta-Type Density Function: Since the beta density has both a shape and a scale parameter, it describes many random phenomena provided the random variable is between [0, 1]. For example, when both parameters are integer with random variables the result is the binomial Probability function.

Applications: A basic distribution of statistics for variables bounded at both sides; for example x between [0, 1]. The beta density is useful for both theoretical and applied problems in many areas. Examples include distribution of proportion of population located between lowest and highest value in sample; distribution of daily per cent yield in a manufacturing process; description of elapsed times to task completion (PERT). There is also a relationship between the Beta and Normal distributions. The conventional calculation is that given a PERT Beta with highest value as b, lowest as a, and most likely as m, the equivalent normal distribution has a mean and mode of (a + 4m + b)/6 and a standard deviation of (b - a)/6.

Comments: Uniform, right triangular, and parabolic distributions are special cases. To generate beta, generate two random values from a gamma, g1, g2. The ratio g1/(g1 +g2) is distributed like a beta distribution. The beta distribution can also be thought of as the distribution of X1 given (X1+X2), when X1 and X2 are independent gamma random variables.

Gamma-Type Density Function: Some random variables are always non-negative. The density function associated with these random variables often is adequately modeled as the gamma density function. The Gamma-Type Density Function has both a shape and a scale parameter. With both the shape and scale parameters equal to 1, the result is the exponential density function. Chi-square is also a special case of gamma density function with shape parameter equal to 2.

Applications: A basic distribution of statistics for variables bounded at one side ; for example x greater than or equal to zero. The gamma density gives distribution of time required for exactly k independent events to occur, assuming events take place at a constant rate. Used frequently in queuing theory, reliability, and other industrial applications. Examples include distribution of time between re-calibrations of instrument that needs re-calibration after k uses; time between inventory restocking, time to failure for a system with standby components.

Comments: Erlangian, Exponential, and Chi-square distributions are special cases. The negative binomial is an analog to gamma distribution with discrete random variable.

What is the distribution of the product of sample observations from the uniform (0, 1) random? Like many problems with products, this becomes a familiar problem when turned into a problem about sums. If X is uniform (for simplicity of notation make it U(0,1)), Y=-log(X) is exponentially distributed, so the log of the product of X1, X2, ... Xn is the sum of Y1, Y2, ... Yn which has a gamma (scaled Chi-square) distribution. Thus, it is a gamma density with shape parameter n and scale 1.

The Log-normal Density Function: Permits representation of a random variable whose logarithm follows a normal distribution. The ratio of two log-normally random variables is also log-normal.

Applications: Model for a process arising from many small multiplicative errors. Appropriate when the value of an observed variable is a random proportion of the previously observed value.

Applications: Examples include distribution of sizes from a breakage process; distribution of income size, inheritances and bank deposits; distribution of various biological phenomena; life distribution of some transistor types.

The lognormal distribution is widely used in situations where values are positively skewed (where the distribution has a long right tail; negatively skewed distributions have a long left tail; a normal distribution has no skewness). Examples of data that"fit" a lognormal distribution include financial security valuations or real estate property valuations. Financial analysts have observed that the stock prices are usually positively skewed, rather than normally (symmetrically) distributed. Stock prices exhibit this trend because the stock price cannot fall below the lower limit of zero but may increase to any price without limit. Similarly, healthcare costs illustrate positive skewness since unit costs cannot be negative. For example, there can't be negative cost for services in a capitation contract. This distribution accurately describes most healthcare data.

In the case where the data are log-normally distributed, the Geometric Mean acts as a better data descriptor than the mean. The more closely the data follow a log-normal distribution, the closer the geometric mean is to the median, since the log re-expression produces a symmetrical distribution.

Further Reading:
Snell J., Introduction to Probability, Random House, 1987. Read section 4.2 for a link between beta and F distributions (with the advantage that tables are easy to find).
Tabachnick B., and L. Fidell, Using Multivariate Statistics, HarperCollins, 1996. Has a good discussion on applications and significance tests for skewness and kurtosis.

Numerical Example and Discussions

A Numerical Example: Given the following, small (n = 4) data set, compute the descriptive statistics: x1 = 1, x2 = 2, x3 = 3, and x4 = 6.

i
xi