Business Analytics Pre-Work for QTM7800

Statistics

A mathematical framework for understanding, organizing, and drawing insights from data.

Descriptive Statistics

Techniques used to summarize and describe a dataset using a smaller set of numbers without presenting all raw data points.

Inferential Statistics

Techniques used to make judgments, predictions, or conclusions about a larger population based on sample data.

Average (Central Tendency)

General statistical term for a single representative number that locates the 'center,' 'middle,' or 'typical' value of a dataset.

Arithmetic Mean ($\bar{x}$ or $\mu$)

The sum of all observations divided by the total count of observations ($\frac{\sum x_i}{n}$). It is sensitive to extreme values.

Median

The exact middle value when data points are arranged in ascending order. It is resistant (robust) to extreme outliers.

Median of an Even Dataset

The arithmetic average of the two exact middle numbers (e.g., the median of 3 and 4 is 3.5).

Mode

The most frequently occurring value in a dataset. A dataset can have no mode or be multimodal.

Population ($N$)

The complete, specific group of elements, items, or individuals about which you want to draw conclusions or make decisions.

Sample ($n$)

A smaller subset drawn from the population that is actually measured or surveyed.

Parameter

A true numerical value summarizing an entire population (e.g., $\mu$, $\sigma$, $\sigma^2$).

Statistic

A numerical value calculated from a sample (e.g., $\bar{x}$, $s$, $s^2$) used to estimate a population parameter.

Categorical (Qualitative) Variable

Data that describes a quality, characteristic, or group label (e.g., instrument played, Zip code).

Numerical (Quantitative) Variable

Data that quantifies or measures a meaningful numeric amount (e.g., practice hours, revenue).

Rule for Numeric vs. Numerical Data

Numbers only count as numerical data if mathematical operations (like taking an average) yield meaningful insights.

Algebraic Variable ($x$)

Represents a fixed, unknown value or deterministic input that can be solved for.

Random Variable ($X$)

A function or rule that maps the numerical outcomes of a random process to numbers, governed by chance and probabilities.

Discrete Random Variable

A random variable that takes on distinct, separate, countable values (e.g., $0, 1, 2, 3$).

Continuous Random Variable

A random variable that can take on any real value within an interval or continuum (e.g., exact weight, exact time).

Measurement vs. Precision Rule

Exact physical measurements are continuous; once rounded or restricted to discrete ticks, the variable becomes discrete.

Histogram

A visual representation of data frequency where continuous numerical values are grouped into bins on the $X$-axis and their frequencies plotted on the $Y$-axis as touching bars.

Frequency

The total count of times a specific data value or range of values occurs in a dataset.

Bins / Buckets

Sub-intervals or category groupings along a numerical scale into which individual quantitative data points are sorted.

Histogram vs. Bar Chart

Histograms visualize quantitative data with continuous bins (touching bars). Bar charts visualize qualitative/categorical data with discrete, arbitrary labels (gapped bars).

Scatter Plot

A two-dimensional graph displaying individual data points for two quantitative variables to visually assess relationships or correlations.

Independent / Explanatory Variable

The variable hypothesized to drive or influence the other; plotted on the horizontal ($X$) axis.

Dependent / Response Variable

The variable being predicted or driven; plotted on the vertical ($Y$) axis.

Range

A measure of total spread or variability, calculated as Maximum minus Minimum ($\text{Max} - \text{Min}$).

Mid-Range

A rough measure of central tendency, calculated as the arithmetic mean of the maximum and minimum values ($\frac{\text{Max} + \text{Min}}{2}$).

Sample Variance ($s^2$)

A measure of the dispersion of sample data points around their sample mean.

Bessel's Correction ($n - 1$)

Dividing by $n - 1$ instead of $n$ in the sample variance formula to correct for bias and prevent underestimating the true population variance.

Unbiased Estimator

An estimator whose expected value exactly equals the true population parameter (e.g., Sample Variance $s^2$).

Biased Estimator

An estimator that systematically over- or under-estimates the true population parameter.

Sample Standard Deviation ($s$)

The square root of the sample variance ($s = \sqrt{s^2}$). It is technically a slightly biased estimator of population standard deviation due to the non-linear square root function.

Box Plot (Box-and-Whisker Plot)

A visual display summarizing data using five key boundary numbers, dividing the dataset into four $25\%$ quartiles.

Five-Number Summary

The Minimum, First Quartile ($Q_1$), Median ($Q_2$), Third Quartile ($Q_3$), and Maximum.

Interquartile Range (IQR)

The spread of the middle $50\%$ of the dataset, calculated as $Q_3 - Q_1$.

Conditional Probability ($P(A \mid B)$)

The probability of event A occurring given that event B has already occurred or is known to be true.

Independent Events

Two events where knowing one occurred does not change the probability of the other, mathematically: $P(A \mid B) = P(A)$.

Dependent Events

Two events where the occurrence of one alters the probability of the other, mathematically: $P(A \mid B) \neq P(A)$.

Joint Probability (General Multiplication Rule)

The probability of both events happening, formulated as $P(A \text{ and } B) = P(A \mid B) \cdot P(B)$.

Contingency Table (Two-Way Table)

A grid displaying frequency counts for two categorical variables simultaneously.

Marginal Distribution

The overall distribution of one variable ignoring the other, calculated using the Grand Total ($N$) as the denominator.

Conditional Distribution

The distribution of one variable restricted to a specific subgroup, calculated using the Subgroup Total as the denominator.

Chi-Square Test for Independence ($\chi^2$)

A hypothesis test used to determine if a significant association exists between two categorical variables.

Chi-Square Null Hypothesis ($H_0$)

Assumes the two categorical variables are entirely independent (no association).

Expected Frequency ($E_{ij}$)

The cell count expected if the null hypothesis is true, calculated using the formula $\frac{\text{Row Total} \cdot \text{Column Total}}{\text{Grand Total}}$.

Chi-Square Statistic Formula

$\chi^2 = \sum \frac{(O_{ij} - E_{ij})^2}{E_{ij}}$.

Degrees of Freedom ($df$) for Two-Way Tables

Calculated as $(r - 1)(c - 1)$, where $r$ is rows and $c$ is columns.

Probability Distribution

A mathematical function or table providing the probabilities of all possible outcomes for a random variable.

Rules for Discrete Probability Distribution

All individual probabilities must be between 0 and 1, and the sum of all probabilities must equal exactly 1.

Binomial Setting (BINS)

A random process requiring Binary outcomes, Independent trials, a fixed Number of trials ($n$), and the Same probability of success ($p$) for each trial.

Combinatorics (Choose Function)

Calculates the number of ways to arrange $k$ successes in $n$ trials, using the formula $\binom{n}{k} = \frac{n!}{k!(n-k)!}$.

Binomial Probability Formula

The probability of exactly $k$ successes in $n$ trials: $P(X = k) = \binom{n}{k} \cdot p^k \cdot (1-p)^{n-k}$.

Normal Distribution

A continuous, symmetrical, bell-shaped probability distribution centered around its mean.

The Empirical Rule (68-95-99.7)

In a normal distribution, roughly $68\%$ of data falls within $\pm 1$ standard deviation, $95\%$ within $\pm 2$ standard deviations, and $99.7\%$ within $\pm 3$ standard deviations.

Z-Score (Standard Score)

A measurement describing exactly how many standard deviations a raw data point is above or below the mean.

Z-Score Formula

$Z = \frac{x - \mu}{\sigma}$. (Positive means above the mean, negative means below).

Sampling Distribution of the Sample Mean

The theoretical probability distribution of all possible sample means ($\bar{x}$) calculated from repeated samples of size $n$.

Central Limit Theorem (CLT)

States that as sample size ($n$) becomes sufficiently large, the sampling distribution of the sample mean will approach a normal distribution, regardless of the original population's shape.

Standard Error of the Mean ($SE_{\bar{x}}$)

The standard deviation of the sampling distribution of the sample mean; measures how much sample means vary from the true population mean.

Standard Error Formula

$SE_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$.

Effect of Sample Size on Spread

As sample size ($n$) increases, the standard error decreases, resulting in a tighter, narrower distribution around the true mean.

Random Sample

A sampling method where members of the population are chosen by chance to avoid bias.

Bias

A systematic error causing the sample to misrepresent the true characteristics of the underlying population.

Simple Random Sample

Every individual in the population has an equal chance of being selected.

Stratified Random Sample

The population is divided into trait-based subgroups (strata), and a random selection is drawn from within every single subgroup.

Clustered Random Sample

The population is divided into mixed groups (clusters). A few entire clusters are randomly selected, and everyone in those chosen clusters is surveyed.

Voluntary Sample (Prone to Bias)

Individuals choose to participate, usually capturing only those with strong opinions.

Convenience Sample (Prone to Bias)

Sampling individuals who are easiest to reach, introducing selection bias.

Wording Bias

Phrasing questions in a leading or emotionally charged way that influences respondents.

Response Bias

When participants lie or give false answers out of fear of repercussions or a desire to look socially acceptable.