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Statistics

Key Terms

StatisticsKey Terms

Key Terms

average
a number that describes the central tendency of the data; there are a number of specialized averages, including the arithmetic mean, weighted mean, median, mode, and geometric mean
central limit theorem
given a random variable (RV) with a known mean, μ, and known standard deviation, σ, and sampling with size n, we are interested in two new RVs: the sample mean, X ¯ X ¯ , and the sample sum, ΣΧ
If the size (n) of the sample is sufficiently large, then X ¯ X ¯ ~ N(μ, σ n σ n ) and ΣΧ ~ N(nμ, ( n n )(σ)). If the size (n) of the sample is sufficiently large, then the distribution of the sample means and the distribution of the sample sums will approximate a normal distribution regardless of the shape of the population. The mean of the sample means will equal the population mean, and the mean of the sample sums will equal n times the population mean. The standard deviation of the distribution of the sample means, σ n σ n , is called the standard error of the mean
exponential distribution
a continuous random variable (RV) that appears when we are interested in the intervals of time between a random events; for example, the length of time between emergency arrivals at a hospital, notation: X ~ Exp(m)
The mean is μ = 1 m 1 m and the standard deviation is σ = 1 m 1 m . The probability density function is f(x) = me–mx, x ≥ 0, and the cumulative distribution function is P(X ≤ x) = 1 – e–mx
mean
a number that measures the central tendency; a common name for mean is average; the term mean is a shortened form of arithmetic mean;.
by definition, the mean for a sample (denoted by x ¯ x ¯ ) is x ¯  =  sum of all values in the sample number of values in the sample x ¯  =  sum of all values in the sample number of values in the sample , and the mean for a population (denoted by μ) is μ =  sum of all values in the population number of values in the population μ =  sum of all values in the population number of values in the population .
normal distribution
a continuous random variable (RV) with probability density function (pdf) f(x) =  1 σ 2π   e – (x – μ) 2 2 σ 2 f(x) =  1 σ 2π   e – (x – μ) 2 2 σ 2 , where μ is the mean of the distribution and σ is the standard deviation; notation: Χ ~ N(μ, σ). If μ = 0 and σ = 1, the RV is called a standard normal distribution
sampling distribution
given simple random samples of size n from a given population with a measured characteristic such as mean, proportion, or standard deviation for each sample, the probability distribution of all the measured characteristics is called a sampling distribution.
standard error of the mean
the standard deviation of the distribution of the sample means, or σ n σ n
uniform distribution
a continuous random variable (RV) that has equally likely outcomes over the domain a < x < b; often referred as the rectangular distribution because the graph of the pdf has the form of a rectangle
Notation: X ~ U(a, b). The mean is μ =  a + b 2 μ =  a + b 2 and the standard deviation is σ =  (b–a) 2 12 σ =  (b–a) 2 12 . The probability density function is f(x) =  1 b–a f(x) =  1 b–a for a < x < b or a ≤ x ≤ b. The cumulative distribution is P(X ≤ x) = x–a b–a x–a b–a
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