Picturing Distributions Comprehensive Study Guide

Core Statistical Definitions

• Statistics: The science dedicated to collecting, organizing, analyzing, and interpreting data.

• Individual: An object or entity described by a set of data. Examples include a person, a city, a animal, or a company.

• Variable: Any specific characteristic, attribute, or property of an individual that can take different values for different individuals.

Categorical and Quantitative Variables

• Categorical Variable:

  • Places an individual into one of several groups or distinct categories.

  • Arithmetic operations (such as addition or subtraction) do not make sense on categorical values.

  • Examples include age group (e.g., young, adult, senescent), eye color, gender, or state of residence.

• Quantitative Variable:

  • Takes numerical values for which arithmetic operations, such as adding, subtracting, or averaging, produce meaningful results.

  • Examples include height in inches, annual revenue growth rate as a percentage, temperature, or account balance.

Student Heights Example Analysis

• Study Objective: Determine the heights of all students enrolled in Math 243 at the University of Oregon (UO).

• Data Collection Context:

  • A class of $520$ students receives a survey questionnaire.

  • A total of $186$ students complete and report their height measured in inches.

• Component Identification:

  • Individuals: The students enrolled in Math 243 at UO.

  • Variable: Height measured in inches.

  • Variable Type: Quantitative (because height is expressed as a numerical value where taking differences or averages is meaningful).

Distributions and Display Selection

• Distribution: Describes what values a variable takes and how frequently each of those values occurs.

• Display Selection Matrix:

  • The choice of graphical display depends directly on the classification type of the variable:

• Categorical Display Rules:

  • Use a pie chart or a bar graph.

• Quantitative Display Rules:

  • Use a histogram or a stemplot (stem-and-leaf plot).

Displaying Categorical Data: Yellowstone Bison Study

• Dataset Overview:

  • Researchers observe and classify $26$ bison in Yellowstone National Park according to their age group.

• Data Summary:

  • Young: $9$ bison

  • Adult: $12$ bison

  • Senescent: $5$ bison

  • Total count: $9 + 12 + 5 = 26$ bison

• Pie Chart Construction:

  • Each category is represented by a slice (sector) of a circle.

  • The slice central angle and area are proportional to the relative frequency of the category:

    • Young sector angle: $\frac{9}{26} \times 360^\circ \approx 124.62^\circ$ ($\approx 34.62\%$)

    • Adult sector angle: $\frac{12}{26} \times 360^\circ \approx 166.15^\circ$ ($\approx 46.15\%$)

    • Senescent sector angle: $\frac{5}{26} \times 360^\circ \approx 69.23^\circ$ ($\approx 19.23\%$)

• Bar Graph Construction:

  • Categories are displayed on the horizontal axis with equal-width bars.

  • The height of each bar is proportional to the category frequency or count on the vertical axis:

    • Young bar height = $9$

    • Adult bar height = $12$

    • Senescent bar height = $5$

  • Bars in a categorical bar graph do not touch each other.

Displaying Quantitative Data: Histograms

• Definition: A graphical representation of the distribution of a continuous quantitative variable using adjacent, touching bars.

• Step-by-Step Construction Procedure:

  1. Divide into Classes: Divide the entire range of values into class intervals (bins) of equal width.

  2. Count Observations: Count the number of observations falling into each class (frequency).

  3. Draw Bars: Draw bars over each class interval with heights corresponding to the frequencies. Bars must touch each other (unlike bar graphs), unless a class has a frequency of zero.

Drug Poisoning Mortality Rates Example

• Dataset Context: Drug poisoning mortality rates in Oregon counties in $2016$ expressed in deaths per $100,000$ residents.

• Frequency Table Data:

  • $2 - 3.9$ deaths per $100,000$: Frequency = $1$

  • $4 - 5.9$ deaths per $100,000$: Frequency = $0$

  • $6 - 7.9$ deaths per $100,000$: Frequency = $1$

  • $8 - 9.9$ deaths per $100,000$: Frequency = $2$

  • $10 - 11.9$ deaths per $100,000$: Frequency = $5$

  • $12 - 13.9$ deaths per $100,000$: Frequency = $6$

  • $14 - 15.9$ deaths per $100,000$: Frequency = $4$

• Histogram Specifications:

  • Class Width: Equal bins of width $2$ units (e.g., $2-3.9$, $4-5.9$, etc.).

  • Horizontal Axis: Deaths per $100,000$

  • Vertical Axis: Frequency (count of counties)

  • Note: The class $4 - 5.9$ has a frequency of $0$, which creates a visible gap of height $0$ between the first and third bars.

Stemplots and Split Stemplots

• Stemplot Structure:

  • A stemplot (or stem-and-leaf display) presents quantitative data while retaining the exact individual raw data values.

  • Stem: Consists of all leading digits except the final rightmost digit.

  • Leaf: Consists of the final single rightmost digit of each observation.

  • Leaves Ordering: Leaves within each stem row must always be arranged in ascending numerical order.

  • Key: Every stemplot must include a key defining the scale of the stem and leaf (e.g., $3 \mid 7$ key represents $37$).

• Split Stemplots:

  • When data values are closely clustered together, a regular stemplot may have too few stems to show the shape of the distribution clearly.

  • To double the number of stems, write each stem digit twice:

    • First Stem Row: Receives leaves with final digits in the range $0 - 4$ ($0, 1, 2, 3, 4$).

    • Second Stem Row: Receives leaves with final digits in the range $5 - 9$ ($5, 6, 7, 8, 9$).

• Split Stemplot Demonstration:

  • Values: $31, 34, 36, 38$

  • First row (stem $3$, leaves $0-4$): $1 \quad 4$

  • Second row (stem $3$, leaves $5-9$): $6 \quad 8$

  • Key: $3 \mid 1$ represents $31$

Tech Company Revenue Growth Example

• Dataset: Annual revenue growth rates for selected technology companies during fiscal year $2021$:

• Data Values:

  • Alphabet: $41\%$

  • Amazon: $22\%$

  • Apple: $33\%$

  • eBay: $5\%$

  • Meta: $37\%$

  • Microsoft: $18\%$

  • Netflix: $19\%$

  • PayPal: $17\%$

  • Twitter: $37\%$

• Ordered Dataset: $5\%, 17\%, 18\%, 19\%, 22\%, 33\%, 37\%, 37\%, 41\%$

• Regular Stemplot Construction:

\begin{array}{c|l}\n0 & 5 \\\n1 & 7 \quad 8 \quad 9 \\\n2 & 2 \\\n3 & 3 \quad 7 \quad 7 \\\n4 & 1\n\end{array}

Key: $1 \mid 7$ represents $17\%$

• Split Stemplot Construction:

\begin{array}{c|l}\n0 & \\\n0 & 5 \\\n1 & \\\n1 & 7 \quad 8 \quad 9 \\\n2 & 2 \\\n2 & \\\n3 & 3 \\\n3 & 7 \quad 7 \\\n4 & 1 \\\n4 &\n\end{array}

Key: $1 \mid 7$ represents $17\%$

Describing Distributions: Symmetry and Skewness

• Symmetric Distribution:

  • The right and left halves of the histogram or stemplot are approximately mirror images of each other.

• Skewed Left Distribution:

  • The left tail of the distribution (lower values) extends much further out than the right tail.

• Skewed Right Distribution:

  • The right tail of the distribution (higher values) extends much further out than the left tail.

Describing Distributions: Modality and Peaks

• Unimodal:

  • A distribution that features a single distinct peak or high-frequency area.

• Bimodal:

  • A distribution that features two distinct, separated peaks.

Outliers in Data Distributions

• Definition: An individual observation that falls noticeably outside the overall pattern of the rest of the data.

• Identification Strategy: Look specifically for observations that lie unusually far above or far below the main cluster of data values.

• Evaluative Judgment: Use visual graphical displays (stemplots or histograms) together with the real-world context of the data to judge whether a extreme value is an outlier.

County Unemployment Rates Example

• Dataset: Random sample of $12$ Oregon counties and their unemployment rates in January $2015$:

• Data Values:

  • Benton: $5.1\%$

  • Deschutes: $7.4\%$

  • Douglas: $8.8\%$

  • Gilliam: $6.9\%$

  • Hood River: $5.5\%$

  • Jefferson: $9.9\%$

  • Josephine: $9.1\%$

  • Lane: $6.7\%$

  • Linn: $7.6\%$

  • Polk: $6.6\%$

  • Sherman: $8.3\%$

  • Washington: $5.3\%$

• Ordered Dataset: $5.1, 5.3, 5.5, 6.6, 6.7, 6.9, 7.4, 7.6, 8.3, 8.8, 9.1, 9.9$

• Split Stemplot Construction (Stems = Whole numbers, Leaves = Tenths digit):

\begin{array}{c|l}\n5 & 1 \quad 3 \\\n5 & 5 \\\n6 & \\\n6 & 6 \quad 7 \quad 9 \\\n7 & 4 \\\n7 & 6 \\\n8 & 3 \\\n8 & 8 \\\n9 & 1 \\\n9 & 9\n\end{array}

Key: $5 \mid 1$ represents $5.1\%$

• Outlier Evaluation:

  • None of the data values appear to be outliers. The unemployment rates range continuously from $5.1\%$ to $9.9\%$ without any extreme gaps separating values from the rest of the distribution.

• Distribution Shape Analysis:

  • Symmetry: The distribution is fairly symmetric across the range from $5\%$ to $10\%$

  • Modality: The distribution does not have a sharp single peak; it is relatively evenly spread across the $5\%$ to $9\%$ bins.

Concept Questions and Solutions

• Concept Question 1: Individuals and Variable Type

  • Prompt: A study classifies $26$ bison in Yellowstone National Park as young, adult, or senescent. Identify the individuals and the variable type.

    • a) Bison; quantitative

    • b) Bison; categorical

    • c) Parks; quantitative

    • d) Parks; categorical

  • Answer: b) Bison; categorical

  • Explanation: The $26$ bison are the objects being evaluated (individuals). The characteristic being recorded is age category (young, adult, or senescent), which places each bison into a category rather than measuring a numerical quantity.

• Concept Question 2: Histogram with Four Classes

  • Prompt: Given the quantitative dataset $\left\{0, 2, 4, 8, 12, 14, 16, 18\right\}$, which histogram accurately represents this data using four equal-width classes?

    • Class bins of width $5$ spanning $[0, 20)$: $[0, 5)$, $[5, 10)$, $[10, 15)$, $[15, 20)$

    • Bin counts:

      • Class $[0, 5)$: Contains $0, 2, 4$ (Frequency = $3$)

      • Class $[5, 10)$: Contains $8$ (Frequency = $1$)

      • Class $[10, 15)$: Contains $12, 14$ (Frequency = $2$)

      • Class $[15, 20)$: Contains $16, 18$ (Frequency = $2$)

  • Histograms Evaluated:

*   *Answer*: **b) [0,5) [5,10) [10,15) [15,20)** (Figure 13)
*   *Explanation*: Bins of width $5$ divide the dataset into frequencies $3, 1, 2, 2$ across four equal bars.

• Concept Question 3: Shape of Exam Scores

  • Prompt: What is the best description of the shape of the distribution for the dataset $\left\{0, 46, 62, 78, 78, 83, 85, 87, 91, 92, 94, 94, 98\right\}$?

    • a) Unimodal and skewed left

    • b) Unimodal and skewed right

    • c) Unimodal and symmetric

    • d) Bimodal and symmetric

    • e) Bimodal and skewed

  • Answer: a) Unimodal and skewed left

  • Explanation: The majority of score values are tightly clustered at the higher end of the scale between $78$ and $98$, forming a single main peak (unimodal) in the upper $80\text{s} - 90\text{s}$. The much lower score values ($0$ and $46$) extend the left side of the graph into a long left-hand tail, making the distribution skewed to the left.