Unit 1: Examining Distributions Complete Study Notes

Core Statistical Concepts and Terminology

• Statistics Defined: Statistics refers to the set of methods used for obtaining, organizing, summarizing, presenting, and analyzing data.

• Data Sources: Data is derived from characteristics measured on individuals or units. Examples of units include people, animals, places, or objects (things).

• Population: This is defined as the totality of individuals about which information is desired.

• Sample: A sample is a subset of the units in a population that are actually examined to gather information.     * Example 1: 1,000 voters are asked which candidate they support in an upcoming election. The population would be all eligible voters in that election.     * Example 2: 50 insomnia patients are given a new treatment. The population would be all patients suffering from insomnia.     * Example 3: 200 Canada geese are tagged to study migration. The population would be all Canada geese.

Variables and Data Classifications

• Variable: A characteristic or property of an individual. Examples include:     * Time until a light bulb burns out.     * Distance traveled by a taxi driver in one day.     * Number of Heads in five tosses of a quarter.     * Hair colour.     * A student's grade in a course.

• Categorical Data: Represents values of categorical variables that place individuals into one of several groups.     * Categorical and Nominal: Used when there is no natural ordering to the groups. Examples: Gender of a newborn, eye colour, favourite television show, or reason for taking a specific course.     * Categorical and Ordinal: Used when there is a logical ordering to the values. Examples: Letter grades ($A+$, $A$, $B+$, etc.), service ratings (Good, Fair, Poor), or placing in a tournament ($1^{st}$, $2^{nd}$, $3^{rd}$, etc.).

• Quantitative Data: Represents values of quantitative variables for which arithmetic operations (like adding and averaging) make sense.     * Discrete Variables: Quantitative variables that can only take certain values. Examples: The number of children in a family, number of rainy days in a month, or the highest denomination of bill in a wallet.     * Continuous Variables: Quantitative variables that can take any value within a given range. Examples: Weight, age, and distance.

• Classification Logic:     * If the data takes numerical values for which arithmetic makes sense, it is Quantitative.     * If not, it is Categorical.     * If categorical data has a sensible order, it is Categorical and Ordinal; otherwise, it is Categorical and Nominal.

Data Distribution and Categorical Visualization

• Definition of Distribution: The distribution of a data set identifies what values a variable takes and how often it takes these values. "Value" can be non-quantitative (e.g., "Blue" is a value for eye colour).

• Bar Charts:     * Displays categorical values on one axis and frequencies on the other.     * Formatting: There must be spaces between bars to indicate that the data is not continuous.

• Pie Charts: Provide a visual representation of the relative frequency (proportions) of observed values for a categorical variable.

• R Implementation for Bar Charts:     * party <- c("Bloc", "CPC", "Green", "Liberal", "NDP", "PPC", "Other")     * vote <- c(7.64, 33.74, 2.33, 32.62, 17.83, 4.94, 0.90)     * colours <- c("deepskyblue", "blue", "green3", "red", "orange", "purple4", "black")     * barplot(vote, names.arg = party, col = colours, ylab = "% Vote")

Frequency Distributions for Quantitative Data

• Frequency Distribution: A count of how many data values fall into predetermined classes or intervals.

• Interval Construction Rules:     * Intervals are chosen by the researcher; usually $5$ to $10$ intervals are used.     * The first interval must include the minimum value; the last must include the maximum.     * All intervals must be of equal length.     * Endpoint Convention: Intervals typically include the left endpoint but not the right ($[left, right)$). For example, in the interval $70 - 80$, the value $70$ is included, but $80$ is not.

• Relative Frequency Distribution: Calculated by dividing the number of data values in an interval by the total number of data values ($n$).     * Proportions: Values between $0$ and $1$. Proportions for all intervals must sum to $1$.     * Percentages: Calculated as proportion * 100.

• Continuity Concern: While intervals like $30-39$ and $40-49$ avoid overlap, continuous intervals (e.g., $30-40$, $40-50$) are preferred to ensure every possible decimal value (like $59.5$) has a place.

Histograms and Shape of Distributions

• Histogram: A graphical display of the count or proportion of data values falling into intervals.     * It is a form of bar graph with no spaces between bars to reflect data continuity.     * The height of the rectangle represents frequency or relative frequency.

• Symmetry: A distribution is symmetric if its center divides it into two approximate mirror images.

• Skewness:     * Right Skewed: The right side (tail) of the histogram (larger values) extends much further than the left.     * Left Skewed: The left side (tail) extends further than the right.

• Time Plots: Used for time series data. Time is plotted on the x-axis, and variable values on the y-axis. Points are connected to show trends.     * Seasonal Variation: A pattern that repeats at regular intervals (e.g., average monthly temperatures in Winnipeg).

Measures of Location (Center)

• Mode: The most frequently observed data value. A data set can have more than one mode.

• Median: The middle value in an ordered data set.     * Calculation: Order the data and compute the position using $\frac{n+1}{2}$.     * If $n$ is odd, the median is the value in that position.     * If $n$ is even, the median is the average of the two middle values.     * Resistance: The median is resistant to outliers. Extreme values do not change its value.

• Mean ($\bar{x}$): The arithmetic average, calculated by summing all data values and dividing by the sample size $n$.     $\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}$     * Center of Mass: The mean is the "balance point" of the data. If data points were weights on a teeter-totter, the mean is where it would balance.     * Sensitivity: The mean is not resistant to outliers; extreme values pull the mean toward them.

• Weighted Mean ($\bar{x}_w$): Used when some data values carry more weight than others.     $\bar{x}_w = \frac{\sum w_i x_i}{\sum w_i}$     * Example 1 (GPA): Credit hours serve as the weights ($w_i$) for grades ($x_i$).     * Example 2 (Combined Group Mean): To find the mean age of a class with $6$ males (mean $23.2$) and $4$ females (mean $21.7$):     $\bar{x}_c = \frac{(6 \times 23.2) + (4 \times 21.7)}{10} = 22.6$

Measures of Variability (Spread)

• Range ($R$): The difference between the maximum and minimum values ($R = \text{max} - \text{min}$). It is highly sensitive to outliers.

• Interquartile Range ($IQR$): Measures the length of the interval covering the middle $50\%$ of observations.     $IQR = Q_3 - Q_1$

• Quartiles:     * First Quartile ($Q_1$): The $25^{th}$ percentile. The median of the lower half of the data.     * Third Quartile ($Q_3$): The $75^{th}$ percentile. The median of the upper half of the data.

• Five-Number Summary: Consists of [Min, $Q_1$, Median, $Q_3$, Max]. This describes the center, shape, and spread.

Boxplots and Outlier Detection

• Quantile Boxplots: A visual representation of the five-number summary: a line at the median, a box for the IQR, and whiskers reaching the min and max.

• Outlier Boxplots (Modified Boxplots): Used to identify extreme values.     * Fences:         $\text{Lower Fence (LF)} = Q_1 - 1.5 \times IQR$         $\text{Upper Fence (UF)} = Q_3 + 1.5 \times IQR$     * Outliers: Any data point outside these fences.     * Whiskers: Extend to the lowest and highest data values that are inside the fences (the "new" min and max). Outliers are plotted as individual points.

• Comparison: Side-by-side boxplots allow comparison of different populations (e.g., heights of Blue Jays pitchers vs. fielders) with respect to center, shape, and spread using a uniform scale.

Variance and Standard Deviation

• Sample Variance ($s^2$): Loosely, the average squared deviation from the mean.     $s^2 = \frac{\sum (x_i - \bar{x})^2}{n-1}$

• Sample Standard Deviation ($s$): The positive square root of the variance.     $s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}$

• Calculation Steps:     1. Find the mean $\bar{x}$.     2. Calculate deviations $(x_i - \bar{x})$. Note: $\sum (x_i - \bar{x})$ always equals $0$.     3. Square those deviations.     4. Sum the squared deviations.     5. Divide by $n - 1$.

• Units:     * The mean and standard deviation share the same units as the individual observations (e.g., dollars).     * Variance is expressed in squared units (e.g., $\text{dollars}^2$).

Choosing Numerical Summaries

• Symmetric Distributions (no outliers): Report the mean and standard deviation because they utilize all data in the sample.

• Skewed Distributions or Outliers present: Report the five-number summary (specifically median and IQR) because they are resistant to extreme values and provide a more accurate description of the typical observation.