Introductory Statistics

# 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 known mean μ and known standard deviation, σ, we are sampling with size n, and 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(, ($n n$)(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 distributions 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 some 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(Xx) = 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 , and the mean for a population (denoted by μ) is .
Normal Distribution
a continuous random variable (RV) with pdf , where μ is the mean of the distribution and σ is the standard deviation.; notation: X ~ N(μ, σ). If μ = 0 and σ = 1, the RV is called the standard normal distribution.
Normal Distribution
a continuous random variable (RV) with pdf , 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 and the standard deviation is . The probability density function is for a < x < b or axb. The cumulative distribution is P(Xx) = $x–a b–a x–a b–a$.
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