Descriptive Statistics: Methods for Displaying and Summarizing Quantitative Data

Course Announcements & Administrative Details

• Activity 1 Grading:

  • Activity 1 grades are posted.

  • A rubric is available to review specific questions where points were lost.

  • Solutions have been posted to allow comparison of calculations and numerical results.

• Quiz 1:

  • Quiz 1 will be unlocked online for student access.

• Chapter 2 Overview:

  • Chapter 2 covers a wide variety of graphical data representations and spans multiple class sessions over two weeks (approximately three lecture days).

  • Initial focus sections cover six primary graph types:

    1. Stem-and-leaf graphs (stem plots)

    2. Line graphs

    3. Bar graphs

    4. Histograms

    5. Frequency polygons

    6. Time series graphs

Stem-and-Leaf Graphs (Stem Plots)

• Definition & Structure:

  • A stem-and-leaf graph (or stem plot) splits each data value into two components:

    • Leaf: Always contains the final significant digit.

    • Stem: Contains all leading digits preceding the final significant digit.

  • Example: For the number $432$: this is the total(end/final number)

• Construction Rules:

  • Stems are listed vertically from top to bottom in ascending order (smallest to largest).

  • Leaves are listed horizontally to the right of their corresponding stem in ascending order.

  • Decimal points are omitted in the plot layout unless explicitly specified by a key.

  • If a stem value has no corresponding data, the stem number must still be listed, and the leaf space must be left blank. Placing a $0$ as a leaf indicates a data value ending in $0$ (such as $7.0$), rather than an empty stem.

• Outliers:

  • An outlier is an observation of data that does not fit the rest of the data set (a value far removed from the remaining cluster).

  • Stem-and-leaf plots allow for immediate visual identification of outliers and concentration of values.

• Worked Example 1: Susan Bean's Spring Pre-Calculus Exam Scores:

  • Data Set: $33, 42, 49, 49, 53, 55, 55, 61, 63, 67, 68, 68, 69, 69, 72, 73, 74, 78, 80, 83, 88, 88, 88, 90, 92, 94, 94, 94, 94, 96, 100$

  • Plot Construction:

    • Stem $3$: leaf $3$ ($33$)

    • Stem $4$: leaves $2, 9, 9$ ($42, 49, 49$)

    • Stem $5$: leaves $3, 5, 5$ ($53, 55, 55$)

    • Stem $6$: leaves $1, 3, 7, 8, 8, 9, 9$ ($61, 63, 67, 68, 68, 69, 69$)

    • Stem $7$: leaves $2, 3, 4, 8$ ($72, 73, 74, 78$)

    • Stem $8$: leaves $0, 3, 8, 8, 8$ ($80, 83, 88, 88, 88$)

    • Stem $9$: leaves $0, 2, 4, 4, 4, 4, 6$ ($90, 92, 94, 94, 94, 94, 96$)

    • Stem $10$: leaf $0$ ($100$)

• Worked Example 2: Distances from Home to Local Supermarkets:

  • Data Set: $1.1, 1.5, 2.3, 2.5, 2.7, 3.2, 3.3, 3.3, 3.5, 3.8, 4.0, 4.2, 4.5, 4.5, 4.7, 4.8, 5.5, 5.6, 6.0, 6.5, 6.7, 12.3$

  • Plot Construction:

    • Stem $1$: leaves $1, 5$

    • Stem $2$: leaves $3, 5, 7$

    • Stem $3$: leaves $2, 3, 3, 5, 8$

    • Stem $4$: leaves $0, 2, 5, 5, 7, 8$

    • Stem $5$: leaves $5, 6$

    • Stem $6$: leaves $0, 5, 7$

    • Stems $7, 8, 9, 10, 11$: [blank]

    • Stem $12$: leaf $3$

  • Analysis: The data shows a concentration between $1.1$ and $6.7$. The distance $12.3$ is located far away from the rest of the data, making $12.3$ a potential outlier.

• Worked Example 3: Miles Per Gallon (MPG) Ratings for 30 Cars:

  • Data Set: MPG ratings recorded for $30$ vehicles.

  • Analysis: Stem plot values show a continuous grouping without large gaps between stems. Because all data points lie close together, there are no outliers present.

Line Graphs

• Structure & Elements:

  • Horizontal Axis ($x$-axis): Represents data values or measured categories.

  • Vertical Axis ($y$-axis): Represents frequency (or relative frequency).

  • Plotting Points: Each data point is plotted at $(x, y) = (\text{data value}, \text{frequency})$.

  • Line Segments: Sequential points are connected with straight line segments.

  • Labeling Requirement: Both axes must be explicitly labeled with titles and numbers. Omitting axis labels results in lost points.

• Worked Example 1: Teenager Chore Reminders:

  • Data pair $(0, 2)$: $0$ reminders has a frequency of $2$

  • Data pair $(1, 5)$: $1$ reminder has a frequency of $5$

  • Points are plotted at $(0, 2)$ and $(1, 5)$ and connected via a line segment.

• Worked Example 2: Car Repairs Per Year Survey:

  • Data Set ($40$ people surveyed on annual car repairs):

    • $0$ times in shop: frequency $7$

    • $1$ time in shop: frequency $10$

    • $2$ times in shop: frequency $14$

    • $3$ times in shop: frequency $9$

  • Setup Procedure:

    1. Draw $x$ and $y$ axes.

    2. Label horizontal axis: "Number of times in shop".

    3. Label vertical axis: "Frequency".

    4. Scale horizontal axis with values: $0, 1, 2, 3$.

    5. Scale vertical axis in increments of $2$ up to $14$ ($0, 2, 4, 6, 8, 10, 12, 14$).

    6. Plot points: $(0, 7)$, $(1, 10)$, $(2, 14)$, $(3, 9)$.

    7. Connect consecutive points with straight line segments.

• Worked Example 3: Store Visits Prior to Major Purchase:

  • Survey of $40$ people measuring store visits prior to a purchase.

  • Line graph constructed identically using store visit categories on the $x$-axis and frequency on the $y$-axis.

Bar Graphs

• Characteristics:

  • Used for categorical or discrete grouped data.

  • Crucial Property: The bars in a bar graph are separated from each other and do not touch.

  • Bars can be rendered vertically (up and down) or horizontally (sideways).

• Worked Example 1: Facebook User Age Distribution:

  • Age group $13$–$25$: $45\%$

  • Age group $26$–$44$: $38\%$

  • Separated bars display percentages relative to age brackets.

• Worked Example 2: US Public Schools Advanced Placement (AP) Examinees (Class of 2011):

  • Variables: Race/Ethnicity categories ($x$-axis, coded $1$ through $6$) vs. AP Examinee Population percentages ($y$-axis).

  • Category Percentages:

    • Category $1$: $10.3\%$

    • Category $2$: $9.0\%$

    • Category $3$: $17.0\%$

    • Category $4$: $0.6\%$

    • Category $5$: $57.1\%$

    • Category $6$: $6.0\%$

  • Setup: Vertical scale formatted by $10$s ($0, 10, 20, 30, 40, 50, 60$). Six distinct, non-touching bars plotted to their respective height percentages.

• Worked Example 3: Student Birthdays by Season (Ms. Ramirez's Math Class):

  • Axes: Horizontal axis labeled "Season" (Winter, Spring, Summer, Fall); Vertical axis labeled "Proportion of Population".

  • Vertical Scale: Increments of $4\%$

  • Results: Fall represents the tallest bar; Winter represents the shortest bar. All four seasonal bars remain non-touching.

Two-Way Tables, Marginal Distributions, and Conditional Distributions

• Two-Way Table Structure:

  • Categorizes data simultaneously across two qualitative or quantitative variables.

  • Sample Data: Pet Ownership by Gender

    • Dogs: Men = $4$, Women = $4$, Total = $8$

    • Cats: Men = $2$, Women = $6$, Total = $8$

    • Fish: Men = $2$, Women = $2$, Total = $4$

    • Column Totals: Men = $8$, Women = $12$

    • Grand Total: $20$

• Marginal Distribution Calculation:

  • Measures the total frequency or percentage of a single variable category relative to the overall total sample size ($N = 20$).

  • $\text{Marginal Proportion of Dogs} = \frac{4 + 4}{20} = \frac{8}{20} = 0.4$

  • $\text{Marginal Proportion of Cats} = \frac{2 + 6}{20} = \frac{8}{20} = 0.4$

  • $\text{Marginal Proportion of Fish} = \frac{2 + 2}{20} = \frac{4}{20} = 0.2$

  • Sum Check: $0.4 + 0.4 + 0.2 = 1.0$ ($100\%$ of the total population).

• Conditional Distribution Calculation:

  • Measures proportions within a specific subpopulation (restricting focus to one row or column, ignoring all other data).

  • Subpopulation Example: Subpopulation of Men ($N_{\text{men}} = 8$).

    • $\text{Conditional Proportion of Men owning Dogs} = \frac{4}{8} = 0.5$

    • $\text{Conditional Proportion of Men owning Cats} = \frac{2}{8} = 0.25$

    • $\text{Conditional Proportion of Men owning Fish} = \frac{2}{8} = 0.25$

  • Sum Check: $0.5 + 0.25 + 0.25 = 1.0$ ($100\%$ of the male subpopulation).

Histograms

• Overview & Key Properties:

  • The most important and frequently utilized graph type in statistics.

  • Unlike bar graphs, the bars in a histogram must touch.

  • Vertical Axis ($y$-axis): Frequency or Relative Frequency.

  • Horizontal Axis ($x$-axis): Quantitative data values (continuous or discrete scales).

  • Advantage over bar graphs: Captures order, intervals, continuous scale progression, and shape of numeric data distributions.

• Method 1: Continuous Data using Class Boundaries (Male Semi-Professional Soccer Players Heights):

  • Data Range: Minimum height = $60$, Maximum height = $74$

  • Step 1: Choose Starting and Ending Points:

    • Avoid starting on an exact data value so values do not land on bar boundaries.

    • $\text{Starting point} = 60 - 0.05 = 59.95$

    • $\text{Ending point} = 74 + 0.05 = 74.05$

  • Step 2: Calculate Interval Width and Bar Boundaries:

    • $\text{Total Range} = 74.05 - 59.95 = 14.1$

    • Select desired number of bars (e.g., $8$ bars).

    • $\text{Raw Bar Width} = \frac{14.1}{8} = 1.7625$

    • Round up to a convenient integer width: $2$

  • Step 3: Establish Boundaries by Repeatedly Adding Width $2$:

    • Bar 1: $59.95$ to $61.95$

    • Bar 2: $61.95$ to $63.95$

    • Bar 3: $63.95$ to $65.95$

    • Bar 4: $65.95$ to $67.95$

    • Bar 5: $67.95$ to $69.95$

    • Bar 6: $69.95$ to $71.95$

    • Bar 7: $71.95$ to $73.95$

    • Bar 8: $73.95$ to $75.95$

  • Step 4: Relative Frequency Heights:

    • $59.95 - 61.95$: relative frequency = $0.05$

    • $61.95 - 63.95$: relative frequency = $0.03$

    • $63.95 - 65.95$: relative frequency = $0.15$

    • $65.95 - 67.95$: relative frequency = $0.40$

    • $67.95 - 69.95$: relative frequency = $0.17$

    • $69.95 - 71.95$: relative frequency = $0.12$

    • $71.95 - 73.95$: relative frequency = $0.07$

    • $73.95 - 75.95$: relative frequency = $0.01$

• Method 2: Discrete Data (Books Bought by 50 Part-Time College Students at ABC College):

  • Data is discrete (countable whole numbers $1$ through $6$).

  • $x$-axis starts slightly before $1$ at $0.5$ and proceeds by integer steps ($0.5, 1.5, 2.5, 3.5, 4.5, 5.5, 6.5$).

  • Frequencies:

    • $1$ book ($0.5$ to $1.5$): frequency = $11$

    • $2$ books ($1.5$ to $2.5$): frequency = $10$

    • $3$ books ($2.5$ to $3.5$): frequency = $16$

    • $4$ books ($3.5$ to $4.5$): frequency = $6$

    • $5$ books ($4.5$ to $5.5$): frequency = $5$

    • $6$ books ($5.5$ to $6.5$): frequency = $2$

• Method 3: Continuous Data without Decimals using Left-Boundary Rule (Weekend Video Game Hours):

  • Data Range: Minimum = $0$, Maximum = $23.75$

  • Bin Setup: Intervals set at whole number steps of $5$ ($0, 5, 10, 15, 20, 25$).

  • Left-Boundary Inclusion Rule: If a data value falls exactly on an interval boundary, it is included in the bar where it serves as the left boundary (i.e., interval contains $[\text{left}, \text{right})$).

    • Bar $[0, 5)$: Includes $0$, excludes $5$. Data points: $0, 2.25$ (Frequency = $2$).

    • Bar $[5, 10)$: Includes $5$, excludes $10$. Data points: $5.5, 7.5, 9.95$ (Frequency = $3$).

    • Bar $[10, 15)$: Includes $10$, excludes $15$. Data points: $10, 10, 11, 12, 12.75$ (Frequency = $5$).

    • Bar $[15, 20)$: Includes $15$, excludes $20$. Data points: $15, 15, 16.75, 17.5, 18, 18, 18.8, 19.5$ (Frequency = $7$).

    • Bar $[20, 25)$: Includes $20$, excludes $25$. Remaining data points (Frequency = $9$).

Frequency Polygons

• Definition & Characteristics:

  • A graph constructed similarly to a line graph, with the specific modification that both ends of the line are anchored directly down to the horizontal axis ($x$-axis), forming a closed polygon.

  • Highly effective for visually overlaying and comparing multiple data distributions simultaneously (e.g., comparing calculus final exam test scores against overall calculus final course grades).

• Construction Procedure:

  1. Calculate the midpoint for each class interval:         $\text{Midpoint} = \frac{\text{Lower Bound} + \text{Upper Bound}}{2}$

  2. Plot points where $(x, y) = (\text{Midpoint}, \text{Frequency})$.

  3. Anchor the polygon: Add one extra midpoint step before the lowest class interval and one extra midpoint step after the highest class interval, setting their frequencies to $0$. Connect the line down to the axis at these anchor points.

• Worked Example: US Presidents' Ages at Inauguration:

  • Midpoint Calculations and Frequencies:

    • Interval $41.5$–$46.5$: Midpoint = $\frac{41.5 + 46.5}{2} = 44$, Frequency = $4$

    • Interval $46.5$–$51.5$: Midpoint = $49$, Frequency = $11$

    • Interval $51.5$–$56.5$: Midpoint = $54$, Frequency = $14$

    • Interval $56.5$–$61.5$: Midpoint = $59$, Frequency = $9$

    • Interval $61.5$–$66.5$: Midpoint = $64$, Frequency = $4$

    • Interval $66.5$–$71.5$: Midpoint = $69$, Frequency = $2$

  • Pattern: Midpoints increase by increments of $5$

  • Anchoring:

    • Lower anchor: Subtract $5$ from first midpoint $44 \rightarrow 39$ (Frequency = $0$).

    • Upper anchor: Add $5$ to last midpoint $69 \rightarrow 74$ (Frequency = $0$).

  • Plotting Sequence: Connect points sequentially across $x$-axis midpoints $39, 44, 49, 54, 59, 64, 69, 74$.

Time Series Graphs

• Definition & Purpose:

  • A graph generated from a paired data set where one variable explicitly represents time (such as hours, days, months, or years) and the other variable represents a measured attribute.

  • Used to observe trends, fluctuations, and progression over time.

• Worked Example: Annual Consumer Price Index (CPI) Over 10 Years:

  • Paired Variables: Year ($x$-axis) paired with Annual Consumer Price Index ($y$-axis).

  • Data Point Example: In year $2003$, the CPI was $184$. Plotted at $(2003, 184)$.

  • Structure: Years are spaced sequentially along the horizontal axis, CPI values along the vertical axis, and consecutive years are linked with line segments.

Questions & Discussion

• Question: If there is a gap in stem numbers with no data values, do you still write the stem number?

  • Response: Yes, always list all intermediate stem numbers vertically. Leave the leaf side completely blank. Do not write $0$ as a leaf because $0$ represents an actual data value ending in zero.

• Question: Should decimal points be written in the leaf section of a stem plot?

  • Response: No, decimal points do not need to be written in the leaf column.

• Question: If a graph has a stem with no leaves, but a higher stem contains values, is that higher value considered an outlier?

  • Response: Yes, if a data value is separated from the main concentration of data by empty stems/gaps, it is considered an outlier.

• Question: Why use a histogram instead of a bar graph?

  • Response: A histogram is superior for numeric data because the horizontal axis follows an ordered quantitative scale. In a bar graph, categories are often unordered qualitative groups without sequential numeric progression.