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distribution equations

Module by: Adan Galvan. E-mail the author

Summary: A collection of equations related to distributions.

Normal Distribution

The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. Normal distributions can be transformed to standard normal distributions by the formula:

Z=X−μσ Z X μ σ
(1)
where XX is a score from the original normal distribution, μμ is the mean of the original normal distribution, and σσ is the standard deviation of original normal distribution.

Sampling Distribution

Given a population with a mean of mm and a standard deviation of σσ, the sampling distribution of the mean has a mean of mm and a standard deviation of σN σ N , where NN is the sample size. The standard deviation of the sampling distribution of the mean is called the standard error of the mean.It is designated by the symbol: σ M σ M .

Theorem 1: Central Limit Theorem

The central limit theorem states that given a distribution with a mean mm and variance s2 s 2 , the sampling distribution of the mean approaches a normal distribution with a mean (mm) and a variance s2N s 2 N as NN, the sample size, increases.

Sampling distribution, difference between independent means

This section applies only when the means are computed from independent samples. This distribution can be understood by thinking of the following sampling plan: Sample n scores from the population of people and compute the mean. This mean will be designated as M 1 M 1 . Then, sample nn scores from a second population of people not taking the drug and compute the mean. This mean will be designated as M 2 M 2 . Finally compute the difference between M 1 M 1 and M 2 M 2 . This difference will be called M d M d where the "d" stands for "difference." The mean and the variance of the sampling distribution of M d M d are:

μ M d = μ 1 − μ 2 μ M d μ 1 μ 2
(2)
and
μ M d 2= σ 1 2 n 1 + σ 2 2 n 2 μ M d 2 σ 1 2 n 1 σ 2 2 n 2
(3)
Finally the standard error of M d M d is simply the square root of the variance of the sampling distribution of M d M d .

Sampling distribution of a linear combination of means

Assuming the means from which LL is computed are independent, the mean and standard deviation of the sampling distribution of LL are: m L = a 1 ⁢ m 1 + a 2 ⁢ m 2 +…+ a k ⁢ m k m L a 1 m 1 a 2 m 2 … a k m k and σ L =∑ a i 2n⁢σ22 σ L a i 2 n σ 2 2 where m i m i is the mean for population ii, σ2 σ 2 is the variance of each population, and nn is the number of elements sampled from each population.

Sampling Distribution of Pearson's r

If NN pairs of scores were sampled over and over again the resulting Pearson rr's would form a distribution. When the absolute value of the correlation in the population is low (say less than about 0.4) then the sampling distribution of Pearson's rr is approximately normal. However, with high values of correlation, the distribution has a negative skew. Fisher's z ′ z ′ transformation converts Pearson's rr to a value that is normally distributed and with a standard error of

σ z ′ =1N−32 σ z ′ 1 N 3 2
(4)
Since NN is in the denominator of the formula, the larger the sample size, the smaller the standard error. The number of standard deviations from the mean can be calculated with the formula:
z= z ′ − μ z ′ σ z ′ z z ′ μ z ′ σ z ′
(5)
where: zz is the number of standard deviations above the z ′ z ′ associated with the population correlation, z ′ z ′ is the value of Fisher's z ′ z ′ for the sample correlation, mm is the value of z ′ z ′ for the population correlation and is the mean of the sampling distribution of z ′ z ′ . σ z ′ σ z ′ is the standard error of Fisher's z ′ z ′ .

Sampling Distribution of the Median

The standard error of the median for large samples and normal distributions is:

σ median =1.25⁢σN σ median 1.25 σ N
(6)

Sampling Distribution of the Standard Deviation

The standard error of the standard deviation is:

σ s =0.71⁢σN σ s 0.71 σ N
(7)
The distribution of the standard deviation is positively skewed for small N but is approximately normal if NN is 25 or greater. Thus, procedures for calculating the area under the normal curve work for the sampling distribution of the standard deviation as long as NN is at least 25 and the distribution is approximately normal.

Sampling Distribution of a Proportion

The sampling distribution of a proportion is equal to the binomial distribution. The mean and standard deviation of the binomial distribution are m=p m p and

σ p =π⁢(1−π)N σ p π 1 π N
(8)

Sampling Distribution of the difference between two proportions

The mean of the sampling distribution of the difference between two independent proportions p 1 − p 2 p 1 p 2 is m p 1 - p 2 = p 1 − p 2 m p 1 - p 2 p 1 p 2 . The standard error of p 1 − p 2 p 1 p 2 is

σ p 1 - p 2 = π 1 ⁢(1− π 1 ) n 1 + π 2 ⁢(1− π 2 ) n 2 σ p 1 - p 2 π 1 1 π 1 n 1 π 2 1 π 2 n 2
(9)
The sampling distribution of p 1 − p 2 p 1 p 2 is approximately normal as long as the proportions are not too close to 1 or 0 and the sample sizes are not too small. As a rule of thumb, if n 1 n 1 and n 2 n 2 are each at least 10 and neither nor are within 0.10 of 0 or 1 then the approximation is satisfactory for most purposes.

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