Stat 200 Lesson 2: Describing Data - Categorical and Quantitative Variables

Review of Foundational Terminology

• Variable Types Review:     * Categorical Variables: These refer to groupings with no inherent logical order (categories) or a logical order where the differences between groups are inconsistent (rankings). For example, in a race, the difference between first and second place is not necessarily the same as the difference between tenth and eleventh place.     * Quantitative Variables: These consist of numerical values characterized by consistent intervals.

• Populations vs. Samples: Distinguishing between these is vital for notation.     * Populations: Characteristics (parameters) are often denoted by Greek symbols.         * Population Mean: $\mu$         * Population Standard Deviation: $\sigma$         * Population Proportion: $p$ or $\pi$     * Samples: Characteristics (statistics) are denoted by Roman letters.         * Sample Mean: $\bar{x}$         * Sample Standard Deviation: $s$         * Sample Proportion: $\hat{p}$

• Standard Probability Tools:     * Standard Die: A six-sided die numbered $1, 2, 3, 4, 5, 6$.     * Standard Deck of Cards: A $52$-card deck consisting of four suits: spades, hearts, diamonds, and clubs.     * Face Cards: These include the King, Queen, and Jack.

Describing One Categorical Variable: Proportion, Risk, and Odds

• Proportion, Risk, and Probability: These terms are synonymous in statistics. They must always result in a value between $0$ and $1$.     * Formula: $\text{Proportion/Risk/Probability} = \frac{x}{n}$, where $x$ is the number in the group with the trait and $n$ is the total number.     * Notation:         * Population Proportion: $p$ or $\pi$         * Sample Proportion: $\hat{p}$         * Risk: Usually written as the word "risk."         * Probability: Written as $P(\text{event})$, such as $P(\text{boy})$.

• Odds: This measures likelihood by comparing the probability of an event happening to the probability of it not happening. Unlike risk, odds can be any non-negative number.     * Formulas:         * $\text{Odds} = \frac{\text{Risk}}{1 - \text{Risk}}$         * $\text{Odds} = \frac{P}{1 - P}$         * $\text{Odds} = \frac{\text{Number with trait}}{\text{Number without trait}}$     * Example Calculation: According to the CDC, $91.9\%$ of American children (aged $19-35$ months) received the MMR vaccine.         * Proportion: $0.919$         * Odds Calculation: $\frac{0.919}{1 - 0.919} = \frac{0.919}{0.081} = 11.346$         * Interpretation: For every $11.346$ children who received the vaccine, there is $1$ child who did not ($11.346 : 1$).

Visual Representations of Categorical and Quantitative Data

• One Categorical Variable:     * Frequency Table (Tally): Lists counts and percentages for each category.     * Pie Chart: Each slice represents a proportion. In Minitab Express, the default order is alphabetical clockwise (e.g., Beer ($17.4\%$), Water, then Wine).     * Bar Chart: Displays categories on the x-axis and counts/percents on the y-axis. Bars have gaps between them to denote categorical data.

• Two Categorical Variables:     * Two-Way Table (Contingency Table or Cross Tabulation): A multi-dimensional table (e.g., a $3 \times 2$ table for preferred beverage and sex).     * Segmented Bar Chart (Stacked Bar Chart): A single bar representing the total for one category (e.g., Females) broken into segments representing a second category (e.g., Beverage choice).     * Clustered Bar Chart: Groups bars side-by-side to compare frequencies across two categorical variables.

• One Quantitative Variable:     * Dot Plot: Each dot represents an observation (or a grouping, e.g., one dot for up to two observations). Variables are on the horizontal axis.     * Histogram: Similar to a bar chart but the bars touch, signifying a continuous quantitative scale.     * Box Plot: Used to summarize quantitative data (detailed in Lesson 3).

Minitab Express Procedures for Data Visualization

• Frequency Tables: Statistics $\rightarrow$ Describe $\rightarrow$ Tally. Select the variable and check "Percents."

• Pie Charts: Graphs $\rightarrow$ Pie Chart $\rightarrow$ Counts of unique values. For raw data, select the categorical variable.

• Bar Charts: Graphs $\rightarrow$ Bar Chart $\rightarrow$ Counts of unique values. Select "Simple" for one variable or "Clustered" for two.

• Two-Way Tables: Statistics $\rightarrow$ Cross Tabulation and Chi-Square. Assign variables to Rows and Columns.

• Histograms/Dot Plots: Graphs $\rightarrow$ Histogram (or Dot Plot) $\rightarrow$ Simple. Quantitative graphs in Minitab Express often include automatic summary statistics (mean, SD).

Probability Rules: Complements, Intersections, Unions, and Conditionals

• The Complement: The probability that an event does not occur.     * Notation: $P(A^c)$ or $P(A')$.     * Rule: $P(A) + P(A^c) = 1$, therefore $P(A^c) = 1 - P(A)$.

• Intersections (AND): The probability that two events occur together.     * Notation: $P(A \text{ and } B)$ or $A \text{ upside-down U } B$.     * Calculation from Table: $\frac{\text{Count in the cell where BOTH occur}}{\text{Total sample size } (n)}$.     * General Formula: $P(A \text{ and } B) = P(A) \times P(B|A)$.     * Disjoint Events: If $P(A \text{ and } B) = 0$, the events are disjoint (they never occur together).

• Unions (OR): The probability that event A, event B, or both occur.     * Notation: $P(A \text{ or } B)$ or $A \text{ U } B$.     * General Formula: $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$.     * Calculation from Table: $\frac{\text{Count A} + \text{Count B} - \text{Overlap Count}}{\text{Total } n}$.

• Conditional Probability (GIVEN): The probability of an event given that another event is already known to have occurred.     * Notation: $P(A|B)$, read as "Probability of A given B."     * Calculation from Table: Limit the denominator to the total of the "given" group (e.g., for $P(\text{Beer}|\text{Male})$, the denominator is the total number of males).     * Formula: $P(A|B) = \frac{P(A \text{ and } B)}{P(B)}$.     * Independence: If $P(A) = P(A|B)$, the events are independent/unrelated.

Quantitative Data: Distribution Shapes and Outliers

• Outliers: Observations that fall outside the general scope of the dataset. Identifying them visually is currently subjective, using dot plots or histograms to spot isolated points.

• Distribution Shapes:     * Symmetrical: The left and right sides are roughly mirror images.         * Normal Distribution: A specific bell-shaped, symmetrical curve.     * Skewed Right: The tail of the distribution pulls to the right (higher values are more spread out).     * Skewed Left: The tail pulls to the left (lower values are more spread out).

Measures of Central Tendency and Skewness

• The Mean (Average):     * Calculation: Sum of all values divided by the number of values.     * Symbols: $\mu$ (population) or $\bar{x}$ (sample).

• The Median: The middle value of an ordered dataset. For an even number of observations, it is the average of the two middle values.

• The Mode: The most frequently occurring value.

• Impact of Skewness:     * Symmetrical Distribution: $\text{Mean} = \text{Median} = \text{Mode}$.     * Right Skewed: The mean is pulled toward the long right tail. \text{Mean} > \text{Median} > \text{Mode}.     * Left Skewed: The mean is pulled toward the long left tail. \text{Mean} < \text{Median} < \text{Mode}.

Measures of Variability: Calculating Standard Deviation

• Standard Deviation ($s$): Represents the average difference between each observation and the mean.     * Step-by-Step Manual Calculation:         1. Compute the sample mean ($\bar{x}$).         2. Compute deviations for every observation ($x - \bar{x}$). Note: the sum of all deviations must always equal $0$.         3. Square every deviation ($(x - \bar{x})^2$).         4. Sum the squared deviations (Sum of Squares or SS).         5. Divide by $n - 1$ to find the Variance ($s^2$).         6. Take the square root of the variance to find the Standard Deviation ($s$).     * Formula: $s = \text{sqrt}(\frac{\text{SS}}{n - 1})$.

Standardized Scores and Percentiles

• Percentiles: The proportion of values falling below a given value. For instance, the $85\text{th}$ percentile means the score was better than $85\%$ of peers.

• Z-Scores (Standardized Scores): Measures how many standard deviation units an observation falls above or below the mean.     * Formulas:         * Sample: $z = \frac{x - \bar{x}}{s}$         * Population: $z = \frac{x - \bar{\mu}}{\sigma}$     * Interpretation: A positive z-score is above the mean; a negative z-score is below the mean.     * The Standard Normal Distribution (Z-distribution): A normal distribution where $\mu = 0$ and $\sigma = 1$.

The Empirical Rule (68-95-99.7 Rule)

• Applies only to normal distributions. It defines where specific proportions of data reside relative to the mean ($\mu$):     * 68% of observations fall within $\pm 1\sigma$.     * 95% of observations fall within $\pm 2\sigma$.     * 99.7% of observations fall within $\pm 3\sigma$.

• Example (SAT Math): \mu = 500, ̄\sigma = 100.     * Middle $68\%$: $500 \times 1(100) = [400, 600]$.     * Middle $95\%$: $500 \times 2(100) = [300, 700]$.     * Middle $99.7\%$: $500 \times 3(100) = [200, 800]$.

Five-Number Summary and Range Measures

• The Five-Number Summary:     1. Minimum: The lowest observation.     2. Q1 (First Quartile): The $25\text{th}$ percentile.     3. Median: The middle observation ($50\text{th}$ percentile).     4. Q3 (Third Quartile): The $75\text{th}$ percentile.     5. Maximum: The highest observation.

• Range Metrics:     * Range: $\text{Maximum} - \text{Minimum}$. Note: Heavily affected by outliers.     * Interquartile Range (IQR): $Q3 - Q1$. Represents the spread of the middle $50\%$ of observations.