2.1: Stem-and-Leaf Graphs, Line Graphs, and Bar Graphs

Introduction to Descriptive Statistics

• Overview of Descriptive Statistics:

  • Descriptive statistics involves organizing and displaying data in clear, readable, and pictorial formats such as tables, charts, and graphs.

  • The primary goal of descriptive statistics is to present data in a structured manner that enables readers to observe patterns and easily draw valid conclusions.

• Common Graphical Depictions of Data:

  • Dot plot

  • Histogram

  • Frequency polygon

  • Pie chart

  • Bar graph

  • Stem-and-leaf plot (stemplot)

  • Line graph

  • Box plot

• Core Focus Topics:

  • Stem-and-leaf plots

  • Line graphs

  • Bar graphs

Stem-and-Leaf Plots (Stemplots)

• Definition and Mechanics:

  • A stem-and-leaf plot (or stemplot) is an efficient method for making quantitative data concise and readable while retaining all original data values.

  • Numerical data is filtered by dividing each data value into two distinct components:

  • Stem: Comprises all digits except the rightmost significant digit (e.g., the tens and hundreds places).

  • Leaf: Represents the single rightmost significant digit (e.g., the ones place).

  • Stemplots enable immediate visual inspection of the entire data distribution and rapid identification of data features.

• Example: Single-Set Stemplot (Redwood Tree Heights):

  • Data Set Context: Height (in feet) of $33$ Redwood trees measured in the year $2000$

  • Raw Data Values ($33$ observations):

  • $307$, $300$, $320$, $330$, $299$, $330$, $270$, $341$, $337$, $262$, $275$, $244$, $286$, $355$, $255$, $313$, $288$, $301$, $360$, $354$, $264$, $369$, $320$, $315$, $314$, $297$, $296$, $267$, $316$, $320$, $325$, $331$, $215$

  • Stem and Leaf Construction (Stems = Tens, Leaves = Ones):

  • Stem $21$: Leaf $5$ (Value: $215$\,ft — noted as a potential outlier)

  • Stem $22$: (No leaves)

  • Stem $23$: (No leaves)

  • Stem $24$: Leaf $4$ (Value: $244$\,ft)

  • Stem $25$: Leaf $5$ (Value: $255$\,ft)

  • Stem $26$: Leaves $2$, $4$, $7$ (Values: $262$, $264$, $267$\,ft)

  • Stem $27$: Leaves $0$, $5$ (Values: $270$, $275$\,ft)

  • Stem $28$: Leaves $6$, $8$ (Values: $286$, $288$\,ft)

  • Stem $29$: Leaves $6$, $7$, $9$ (Values: $296$, $297$, $299$\,ft)

  • Stem $30$: Leaves $0$, $1$, $7$ (Values: $300$, $301$, $307$\,ft)

  • Stem $31$: Leaves $3$, $4$, $5$, $6$ (Values: $313$, $314$, $315$, $316$\,ft)

  • Stem $32$: Leaves $0$, $0$, $0$, $5$ (Values: $320$, $320$, $320$, $325$\,ft)

  • Stem $33$: Leaves $0$, $0$, $1$, $7$ (Values: $330$, $330$, $331$, $337$\,ft)

  • Stem $34$: Leaf $1$ (Value: $341$\,ft)

  • Stem $35$: Leaves $4$, $5$ (Values: $354$, $355$\,ft)

  • Stem $36$: Leaves $0$, $9$ (Values: $360$, $369$\,ft)

• Outliers in Stemplots:

  • Definition: Outliers are data values that are far greater or far smaller than the remaining data points and do not fit the overall pattern established by the data set.

  • The stemplot format makes outliers immediately visible, such as the value $215$\,ft separated from the main data cluster by empty stems $22$ and $23$

• Side-by-Side Stemplots:

  • Purpose: Constructed to compare two data sets simultaneously along a common stem scale under a shared heading (e.g., comparing "before" and "after" experimental or educational intervention data).

  • Example: Exam Performance Comparison:

  • Data Context: Exam $1$ and Exam $2$ test scores for a group of $20$ students who engaged in targeted studying between exams.

  • Structure: Central column contains shared Stems (Tens), left column displays Exam 1 Leaves (Ones), and right column displays Exam 2 Leaves (Ones).

  • Stem-and-Leaf Comparison Chart:

    • Stem $0$: Exam 1 Leaves: (None) | Exam 2 Leaves: (None)

    • Stem $1$: Exam 1 Leaves: (None) | Exam 2 Leaves: (None)

    • Stem $2$: Exam 1 Leaves: (None) | Exam 2 Leaves: (None)

    • Stem $3$: Exam 1 Leaves: $8$, $7$, $7$ | Exam 2 Leaves: (None)

    • Stem $4$: Exam 1 Leaves: $9$, $6$, $5$, $3$, $1$ | Exam 2 Leaves: (None)

    • Stem $5$: Exam 1 Leaves: $9$, $5$, $3$, $3$, $1$, $1$, $0$ | Exam 2 Leaves: $9$, $9$

    • Stem $6$: Exam 1 Leaves: $9$ | Exam 2 Leaves: $2$, $3$, $8$, $9$, $9$

    • Stem $7$: Exam 1 Leaves: $3$, $2$, $1$, $1$ | Exam 2 Leaves: $0$, $0$, $1$, $1$, $4$, $4$, $5$, $6$, $6$, $7$, $8$, $9$

    • Stem $8$: Exam 1 Leaves: (None) | Exam 2 Leaves: $1$

    • Stem $9$: Exam 1 Leaves: (None) | Exam 2 Leaves: (None)

  • Analytical Finding: Generally speaking, student exam scores improved significantly from Exam $1$ (where scores clustered between the $30\text{s}$ and $50\text{s}$) to Exam $2$ (where scores clustered primarily in the $60\text{s}$ and $70\text{s}$).

Line Graphs and Time Series Graphs

• Line Graphs:

  • A line graph is a visual display connecting data points with straight line segments to illustrate trends and structural shifts in variables.

  • Time Series Graphs: Most commonly used to display how a specific variable changes over a continuous sequence of time intervals.

  • Survey Frequency Representation: Line graphs can also showcase the frequency distribution of numerical values obtained from survey responses.

  • Frequency Polygons: A refined variation of a line graph that plots class frequencies against class midpoints.

Bar Graphs

• Definition and Key Distinction:

  • A bar graph visually presents categorical or quantitative data using rectangular bars whose height or length is directly proportional to the frequency or relative frequency of each category.

  • Critical Structural Requirement: Bars in a bar graph must be separated from each other by space (unlike histograms where adjacent bars touch).

  • Bars may be oriented either vertically or horizontally.

• Rules for Drawing Hand-Drawn Bar Graphs:

  • Uniform Width: All bars must be drawn with identical widths.

  • Ordered Variable Axis: If the data variable is quantitative, the category values along the variable axis must follow continuous numerical order.   

    

  • Zero Origin: The frequency scale axis must always begin at zero ($0$) to prevent baseline distortion.

  • Consistent Interval Scale: The frequency scale axis must be marked using uniform, consistent measurement intervals.

• Worked Example: Customer Complaint Resolution:

  • Scenario: A customer support center records the total number of calls required to resolve a customer complaint.

  • Data Table:

  • Number of Calls: $1$, $2$, $3$, $4$, $5+$

  • Relative Frequency: $25\%$, $45\%$, $15\%$, $10\%$, $5\%$

  • Bar Graph Axis Configuration:

  • Horizontal Axis ($X$-axis): Number of calls ($1$, $2$, $3$, $4$, $5+$)

  • Vertical Axis ($Y$-axis): Relative Frequency marked from $0\%$ to $50\%$ in uniform $5\%$ increments

  • Bar Heights: $1\text{ call} = 25\%$, $2\text{ calls} = 45\%$, $3\text{ calls} = 15\%$, $4\text{ calls} = 10\%$, $5+\text{ calls} = 5\%$

Questions & Discussion

• Question: When a customer has a complaint, how many times should they expect to call?

  • Response: $2\text{ times}$ (representing the mode of the distribution with the highest relative frequency at $45\%$).

• Question: How many calls does it take to resolve at least $80\%$ of complaints?

  • Response: $3\text{ calls}$.

  • Cumulative Calculation:

  • $1\text{ call} = 25\%$

  • $1\text{ or } 2\text{ calls} = 25\% + 45\% = 70\%$

  • $1, 2,\text{ or } 3\text{ calls} = 25\% + 45\% + 15\% = 85\%$

  • Because $85\% \ge 80\%$, resolving at least $80\%$ of complaints requires up to $3\text{ calls}$.

• Question: Does the company need to make changes to its user complaint system? Why or why not?

  • Response: Evaluation depends on organizational benchmarks. While $70\%$ of customer issues are resolved within $2$ calls, $30\%$ require $3$ or more calls, which may necessitate intervention if performance standards demand faster resolution.

• Question: Is the data "good"?

  • Response: Determining whether data is "good" depends entirely on contextual evaluation. Key evaluative factors include:

  • Bias: Is the data collection methodology free from systematic bias?

  • Sampling Representation: Is the data set drawn from a truly representative sample of all customer complaint interactions?

  • Hidden Variables: Are there confounding or unobserved hidden variables affecting customer call volume (e.g., variation in issue complexity or agent skill level)?

• Software Applications:

  • Statistical bar graphs and charts can be computer-generated using spreadsheet software applications such as Google Sheets or Microsoft Excel.