Measures of Statistical Data and Probability Distributions

Vocabulary flashcards covering statistics and probability topics including central tendency, dispersion, skewness, kurtosis, probability rules, random variables, and theoretical probability distributions.

Measure of Central Tendency

An average value of a statistical series of variables which represents the entire distribution, commonly measured by Arithmetic Mean, Median, Mode, Geometric Mean, and Harmonic Mean.

Arithmetic Mean

The average of a set of observations $x_1, x_2, \dots, x_n$ calculated as $\bar{x} = \frac{\sum_{i=1}^n x_i}{n}$, or for frequency distributions as $\bar{x} = \frac{\sum f_i x_i}{\sum f_i}$.

Median

A positional average that divides a distribution into two equal parts; for raw data arranged in order, it is the $\left(\frac{n+1}{2}\right)\text{th}$ term if $n$ is odd, or the average of the $\left(\frac{n}{2}\right)\text{th}$ and $\left(\frac{n}{2}+1\right)\text{th}$ terms if $n$ is even.

Continuous Data Median Formula

The median formula for continuous frequency distributions given by $\text{Median} = l + \left(\frac{\frac{N}{2} - cf}{f}\right) \times h$, where $l$ is the lower limit of the median class, $cf$ is the cumulative frequency preceding the median class, $f$ is the frequency of the median class, $N$ is total frequency, and $h$ is uniform class interval.

Mode

The value of a variable that occurs most frequently in a set of observations and around which the other items cluster most densely.

Continuous Data Mode Formula

The formula for mode in continuous frequency distributions given by $\text{Mode} = l + \left(\frac{f_1 - f_0}{2f_1 - f_0 - f_2}\right) \times h$, where $f_1$ is the modal class frequency, $f_0$ is the preceding class frequency, $f_2$ is the succeeding class frequency, $l$ is the lower limit of the modal class, and $h$ is uniform class interval.

Empirical Relation of Central Tendency

The relationship between mode, median, and mean expressed as $\text{Mode} = 3 \times \text{Median} - 2 \times \text{Mean}$.

Geometric Mean

The $n\text{th}$ root of the product of $n$ observations, calculated as $G = (x_1 \times x_2 \times \dots \times x_n)^{\frac{1}{n}}$, used to find rate of population growth and rate of interest.

Harmonic Mean

The reciprocal of the arithmetic mean of the reciprocals of the given observation values, given by $H = \frac{n}{\sum \left(\frac{1}{x_i}\right)}$.

Dispersion

The degree of scatteredness of data values around a central value, used to evaluate the homogeneity or heterogeneity of a distribution.

Range

The difference between the two extreme observations of a distribution, calculated as $\text{Range} = A - B$, where $A$ is the greatest observation and $B$ is the smallest observation.

Coefficient of Range

A relative measure of range calculated as $\text{Coefficient of Range} = \frac{A - B}{A + B}$, where $A$ is the largest observation and $B$ is the smallest observation.

Quartile Deviation

Also known as the semi-interquartile range $Q$, defined as $Q = \frac{Q_3 - Q_1}{2}$, where $Q_1$ is the first quartile and $Q_3$ is the third quartile.

Coefficient of Quartile Deviation

A relative measure of quartile deviation given by $\text{Coefficient of Q.D.} = \frac{Q_3 - Q_1}{Q_3 + Q_1}$.

Mean Deviation about Mean

The arithmetic mean of absolute deviations from the mean, given by $\text{M.D.} = \frac{\sum f_i |x_i - \bar{x}|}{N}$, with coefficient of M.D. equal to $\frac{\text{M.D.}}{\bar{x}}$.

Standard Deviation

The square root of the arithmetic mean of squared deviations from the mean, denoted by $\sigma = \sqrt{\frac{\sum f_i (x_i - \bar{x})^2}{N}}$, where $\sigma^2$ is the variance.

Coefficient of Variation

A relative measure of dispersion expressed as a percentage, given by $\text{C.V.} = \frac{\sigma}{\bar{x}} \times 100\%$.

Raw Moment

The $r\text{th}$ moment of a variable $x$ about any arbitrary point $x = A$, defined as $\mu_r' = \frac{\sum f_i (x_i - A)^r}{\sum f_i}$.

Central Moment

The $r\text{th}$ moment of a variable $x$ about its arithmetic mean $\bar{x}$, defined as $\mu_r = \frac{\sum f_i (x_i - \bar{x})^r}{\sum f_i}$, where $\mu_1 = 0$ and $\mu_2 = \sigma^2$.

Skewness

The lack of symmetry in a distribution curve where mean, median, and mode fall at different points or quartiles are not equidistant from the median.

Karl Pearson's Coefficient of Skewness

A measure of skewness defined as $S_k = \frac{\text{Mean} - \text{Mode}}{\sigma}$, or $S_k = \frac{3(\text{Mean} - \text{Median})}{\sigma}$ when mode is ill-defined.

Bowley's Coefficient of Skewness

A measure of skewness based on quartiles, defined as $S_k = \frac{Q_3 + Q_1 - 2 \times \text{Median}}{Q_3 - Q_1}$.

Kurtosis

A measure of the flatness or peakness (convexity) of a distribution curve relative to a normal curve, measured by $\beta_2 = \frac{\mu_4}{\mu_2^2}$ or $\gamma_2 = \beta_2 - 3$.

Leptokurtic

A distribution curve that is more peaked than a normal curve, characterized by $\beta_2 > 3$.

Mesokurtic

A normal distribution curve having standard peakness, characterized by $\beta_2 = 3$.

Platykurtic

A distribution curve that is flatter than a normal curve, characterized by $\beta_2 < 3$.

Random Experiment

An experiment whose set of outcomes is known in advance but specific outcome cannot be predicted prior to execution, such as tossing a fair coin or throwing an unbiased die.

Sample Space

The set of all possible outcomes of a random experiment, denoted by $S$.

Mutually Exclusive Events

Two events $A$ and $B$ that cannot occur simultaneously, meaning their intersection is empty $A \cap B = \emptyset$ and $P(A \cap B) = 0$.

Independent Events

Events $A$ and $B$ where the occurrence of one does not affect the occurrence of the other, satisfying $P(A \cap B) = P(A) \times P(B)$.

Conditional Probability

The probability of event $A$ occurring given that event $B$ has already occurred, denoted as $P(A|B) = \frac{P(A \cap B)}{P(B)}$ provided $P(B) > 0$.

Theorem of Total Probability

If $A_1, A_2, \dots, A_n$ form a partition of sample space $S$ and $B$ is any subset event of $S$, then $P(B) = \sum_{i=1}^n P(A_i) \times P(B|A_i)$.

Baye's Theorem

A theorem used to compute posterior probabilities: $P(A_i|B) = \frac{P(A_i) \times P(B|A_i)}{\sum_{j=1}^n P(A_j) \times P(B|A_j)}$.

Random Variable

A real-valued function $X$ assigned to each and every outcome of a random experiment sample space $S$.

Probability Mass Function

A function $P(X = x_i) = p(x_i)$ for a discrete random variable satisfying $0 \le p(x_i) \le 1$ and $\sum p(x_i) = 1$.

Cumulative Distribution Function

A function $F(x)$ representing the probability that a random variable $X$ takes a value less than or equal to $x$, given by $F(x) = P(X \le x)$.

Probability Density Function

A non-negative function $f(x)$ for a continuous random variable satisfying $\int_{-\infty}^{\infty} f(x)\,dx = 1$, where $P(a \le X \le b) = \int_a^b f(x)\,dx$.

Mathematical Expectation

The expected value $E(X)$ of a random variable $X$, calculated as $E(X) = \sum x_i p(x_i)$ for discrete variables or $E(X) = \int_{-\infty}^{\infty} x f(x)\,dx$ for continuous variables.

Variance of a Random Variable

The measure of dispersion of a random variable defined as $\text{Var}(X) = E(X^2) - (E(X))^2$, satisfying $\text{Var}(a X + b) = a^2 \text{Var}(X)$.

Binomial Distribution

A discrete probability distribution with PMF $P(X = x) = \binom{n}{x} p^x q^{n-x}$, mean $\mu = n p$, and variance $\sigma^2 = n p q$.

Poisson Distribution

A discrete probability distribution for rare events with PMF $P(X = x) = \frac{e^{-\lambda} \lambda^x}{x!}$ for $x = 0, 1, 2, \dots$, where mean and variance both equal $\lambda$.

Normal Distribution

A continuous bell-shaped probability distribution with PDF $f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}$ for $-\infty < x < \infty$.

Standard Normal Variable

A standardized normal variable $Z = \frac{X - \mu}{\sigma}$ having mean $E(Z) = 0$ and variance $\text{Var}(Z) = 1$.