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, $\overline{X}$, and the sample sum, ΣΧ. If the size (n) of the sample is sufficiently large, then $\overline{X}$ ~ N(μ, $\frac{\sigma }{\sqrt{n}}$) and ΣΧ ~ N(nμ, ($\sqrt{n}$)(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, $\frac{\sigma }{\sqrt{n}}$, 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 = $\frac{1}{m}$ and the standard deviation is σ = $\frac{1}{m}$. 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 $\overline{x}$) is $\overline{x}\text{ = }\frac{\text{Sum of all values in the sample}}{\text{Number of values in the sample}}$, and the mean for a population (denoted by μ) is $\mu \text{ = }\frac{\text{Sum of all values in the population}}{\text{Number of values in the population}}$.

Normal Distribution

a continuous random variable (RV) with pdf $f(x)\text{ = }\frac{1}{\sigma \sqrt{2\pi }} e^{\frac{–{\text{(}x–\mu )}^2}{2{\sigma }^2}}$, 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.

Normal Distribution

a continuous random variable (RV) with pdf $f(x)\text{ = }\frac{1}{\sigma \sqrt{2\pi }} e^{\frac{–{\text{(}x–\mu )}^2}{2{\sigma }^2}}$, 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.

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 $\frac{\sigma }{\sqrt{n}}$.

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 $\mu \text{ = }\frac{a\text{ + }b}{2}$ and the standard deviation is $\sigma \text{ = }\sqrt{\frac{{(b–\text{a)}}^2}{12}}$. The probability density function is $f(x)\text{ = }\frac{1}{b–a}$ for a < x < b or a ≤ x ≤ b. The cumulative distribution is P(X ≤ x) = $\frac{x–\text{a}}{b–\text{a}}$.