Elementary statistics for foresters

Lecture 3

Socrates/Erasmus Program @ WAU

Spring semester 2005/2006

Statistical distributions

Statistical distributions

• Empirical distributions

• Why distributions?

• Variable types

• Sample theoretical distributions– Normal distribution– Binomial distribution

Empirical distributions

• Graphical representation of the data in a form of frequency distribution, histogram, polygon, etc.

Graphical description of data

Histogram for dk

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Graphical description of data

polygon

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Why distributions?

• In some cases it is necessary to formulate hypotheses about the specific distribution of the investigated variable. – For example, we can think of a wood density as

following the normal distribution, and use this information for modeling and inferential statistics purposes.

Why distributions?

• When using distributions for predictive purposes it is often desirable to understand the shape of the underlying distribution of the population.

• To determine this distribution, it is common to fit the observed distribution to a theoretical distribution by comparing the observed frequencies to the expected frequencies of the theoretical distribution.

Why distribution?

• To do this, maximum likelihood method or the method of moments are used.

• Another common application of theoretical distributions is to be able to verify the assumption of normality before using some parametric test.

Variable types

• Variables can be qualitative (which means: describing belonging to a group or category, eg. sex, hair color, tree species), and quantitative (which means: possible to measure using a numerical scale, or numeric values for which addition and averaging make sense, eg. DBH, height, crown ratio, ...).

Variable and distribution types

• If variables can take only a finite set of values, we are talking about discrete variables (eg. age, DBH class, ...), and about probability distribution.

• If variables can take any value (or any value from a given interval), we are talking about continuous variables (eg. height, DBH, ...), and probability density.

Variable and distribution types

• In many cases, due to measurement limitations or simplifications, continuous variables can be treated as discrete (eg., when DBH measured as rounded to 1mm).

Sample distributions

• Beta distribution is used to model the distribution of order statistics, and to representing processes with natural lower and upper limits.

• binomial distribution is used for describing binomial events, such as the number of M/F in a random sample, or the number of defective components in samples of n units taken from a production process.

Sample distributions

• chi-square distribution is most frequently used in modeling random variables representing frequencies.

• exponential distribution is frequently used to model the time interval between successive random events.

• logistic distribution is used to model binary responses.

Sample distributions

• normal distribution is a theoretical function commonly used in inferential statistics as an approximation to sampling distributions.

• Poisson distribution is used to model rare events.

• Weibull distribution is often used as a model of failure time or in reliability testing.

• ...

Normal distribution

• The most frequently used distribution in statistics

• The basic assumption of many statistical methods, such as estimation, hypotheses testing, regression and correlation, analysis of variance, ...

Normal distribution

• Usually variables whose values are determined by an infinite number of independent random events will be distributed following the normal distribution.

• The normal distribution is an example of the distribution of continuous variables. Its probability density function can be described as following:

Normal distribution

• where:– x is a variable of interest– µ is an arithmetic mean– σ is standard deviation

Normal distribution

Normal distribution properties:

• the probablility density function rises for x<µ, and lowers for x>µ

• the probability density function has its maximum at x = µ

• the expected value of the X variable E(X)=µ

• variance of the X variable: D2X = σ2

Normal distribution properties

• at x = µ the probability density function has a value of

• the distribution has 2 inflection points (the function changes from concavitate to convexitate or from convexitate to concavitate) for x=µ - σ and x = µ + σ

• the normal distribution is symmetric, and the symmetry axe is defined as x = µ

Normal distribution properties:

• if variance/standard deviation is low, the probability density function is narrower

• the probablity function of the normal distribution is an integral of the probability density function

Normal distribution properties:

Standarized normal distribution

• Every normal distribution can be normalized, i.e. can be written as the distribution with mean equal 0 and standard deviation equal 1: N(0,1).

• The expected value of the standarized normal distribution equals zero (EZ = 0) and its variance equals 1 (D2Z = 1).

Standarized normal distribution

• The standarization process is nothing else but changing variable x to z, where:

• The probability density function of such a distribution is:

Standarized normal distribution

Normal distribution properties:

• Between µ - σ and µ + σ about 68% of all variable values occur

• In the interval from μ - 2*σ to μ + 2*σ are about 95% of all values of the variable

• In the interval from μ - 3*σ to μ + 3*σ are about 99,7% of all observations

Cumulative distribution

cumulative histogram

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Cumulative dustribution

Cumulative normal distribution

Cumulative normal distribution

Cumulative normal distribution

Binomial distribution

• Example of the probability distribution

• Describes the probability of getting k number of successes in n independently repeated samples, where probability of a success in just one sample equals p

Binomial distribution

Binomial distribution

Binomial distribution properties

• the graph of the distribution is symmetric for p = 0.5

• for p < 0.5 the distribution is positively skewed

• for p > 0.5 is negatively skewed

Binomial distribution properties

• Expected value E(X) = n * p

• Variance D2X = n p q

• Standard deviation

• Sample exercises using the binomial distribution