Ünite 5: Variability Measures — Önemli Kavramlar

Statistical metrics used to quantify the degree of spread, dispersion, or scatter among observations in a dataset. They complement measures of central tendency by showing how far data points lie from the center.

The difference between an individual observation and the arithmetic mean of the dataset, calculated as $(x_i - \bar{X})$. The sum of all deviations in any dataset is always mathematically equal to zero.

A measure of the average distance of data points from the arithmetic mean, calculated by dividing the sum of squared deviations by $(n-1)$ and taking the square root. It is denoted by the symbol $s$.

The square of the sample standard deviation, representing the average of the squared deviations from the mean divided by $(n-1)$. It is denoted by $s^2$ and is expressed in squared units of the original data.

The measure of variability for an entire population, denoted by $\sigma$ (Sigma). It is calculated by dividing the sum of squared deviations from the population mean $\mu$ by the total population size $N$ and taking the square root.

The square of the population standard deviation, denoted by $\sigma^2$. It represents the average of the squared deviations of all population members from the population mean $\mu$.

A sample statistic whose expected value is equal to the corresponding population parameter. Dividing the sum of squared deviations by $(n-1)$ instead of $n$ makes the sample variance an unbiased estimator of the population variance.

A data organization format where raw data is sorted into class intervals with corresponding frequencies. Calculations of spread for this format utilize class midpoints ($M_i$) as representative values.

A measure of position, denoted as $P(m)$, indicating the value below which a given percentage $m$ of observations fall. For example, the 75th percentile is the value below which 75% of the observations lie.

Specific percentiles that divide a sorted dataset into four equal parts. They consist of the first quartile ($Q_1$ or 25th percentile), second quartile ($Q_2$, median, or 50th percentile), and third quartile ($Q_3$ or 75th percentile).

The difference between the third and first quartiles ($Q_3 - Q_1$), representing the spread of the middle 50% of the data. It is highly resistant to the influence of outliers.

A graphical display of the five-number summary (minimum, $Q_1$, median, $Q_3$, and maximum) that visualizes the spread, skewness, and outliers of a dataset using a box and whiskers.

An observation that is unusually far from the rest of the data. In a box plot, it is defined as any value that lies beyond $1.5 \times IQR$ from either the first or third quartile.

An observation that lies extremely far from the main body of data, defined in box plots as any value located beyond $3 \times IQR$ from either the first or third quartile.

A measure of the asymmetry of a probability distribution or dataset. It indicates whether the observations are concentrated on one side of the mean and have a long tail on the other.

A standardized measure of skewness ranging from -3 to 3, calculated as $\frac{3(\bar{X} - \text{Median})}{s}$. Values close to zero indicate symmetry, while positive or negative values indicate right or left skewness, respectively.

A skewness measure based on the sum of the cubed standardized deviations of each observation from the mean, adjusted for sample size using the factor $\frac{n}{(n-1)(n-2)}$.

The difference between an individual observation and the arithmetic mean of the dataset, calculated as (x_i - Mean). It measures how far and in what direction a single data point lies from the average.

A measure of variation calculated by taking the average of the absolute deviations of the observations from their arithmetic mean. It is rarely used in advanced statistics due to mathematical difficulties with absolute values.

A widely used measure of variability that represents the average distance of data points from the arithmetic mean, calculated by dividing the sum of squared deviations by (n - 1) and taking the square root.

The square of the sample standard deviation, representing the average of the squared deviations from the mean. It is calculated with a divisor of (n - 1) to serve as an unbiased estimator of the population variance.

The measure of dispersion for an entire population, calculated by dividing the sum of squared deviations from the population mean by the total population size (N) and taking the square root.

The average of the squared deviations of all observations in a population from the population mean, calculated using the total population size (N) as the divisor.

A distribution in which the left and right sides are mirror images of each other. In a unimodal symmetric distribution, the mean, median, and mode are all equal.

A distribution with an asymmetric shape that has a long tail extending toward higher positive values, where the mean is pulled to the right of the median.

A numerical measure of skewness ranging between -3 and 3, calculated as 3 times the difference between the mean and median, divided by the standard deviation.