13.2 Interpreting Bar Graphs, Histograms, Box Plots & Scatterplots
Key Takeaways
- Data displays are categorized into categorical formats (frequency tables, single/grouped bar graphs, and pie charts where angle = percent × 3.6°) and quantitative formats (dot plots, stem-and-leaf plots, and histograms).
- Histograms display continuous numerical data grouped into equal-width bins with no gaps between bars; bar charts display discrete categorical groups with distinct spaces between bars.
- Box-and-whisker plots visually display the five-number summary (Min, Q1, Median, Q3, Max); data points falling beyond the 1.5 × IQR outlier fences [Q1 - 1.5·IQR, Q3 + 1.5·IQR] are classified as statistical outliers.
- Scatterplots illustrate bivariate relationships: trends are evaluated as positive, negative, or no correlation; linear models (trend lines) allow interpolation within observed domain bounds and cautious extrapolation outside them.
- Misleading graphical representations distort visual perception through truncated y-axes (non-zero baselines), disproportionate 2D/3D area scaling, or irregular bin intervals.
Interpreting Bar Graphs, Histograms, Box Plots & Scatterplots
Quick Answer: On the WEST-B Mathematics subtest (Objective 0016), data display questions test your skill in extracting values and analyzing trends from visual representations. Key distinctions include: Bar graphs display categorical groups with separated bars; Histograms display continuous intervals with adjacent touching bars; Circle graphs convert percentages to central angles via $\text{Angle} = \text{Percentage} \times 3.6^\circ$; Box plots graph the five-number summary ($\text{Min}, Q_1, \text{Median}, Q_3, \text{Max}$) and detect outliers beyond $Q_1 - 1.5\times\text{IQR}$ and $Q_3 + 1.5\times\text{IQR}$; and Scatterplots show bivariate relationships where correlation does not imply causation.
1. Categorical Data Displays
Categorical data groups qualitative variables into discrete classes or labels.
+---------------------------------------------------------------------------------------------------+
| CATEGORICAL DATA DISPLAYS MATRIX |
| |
| +-----------------------+-----------------------------+-------------------------------------+ |
| | DISPLAY TYPE | STRUCTURE & PROPERTIES | TYPICAL USES & EXAM TRAPS | |
| +-----------------------+-----------------------------+-------------------------------------+ |
| | Frequency Table | Lists categories in rows | Sum of relative frequencies must | |
| | | with counts and percentages | equal 1.00 (100%) | |
| +-----------------------+-----------------------------+-------------------------------------+ |
| | Simple Bar Graph | Discrete bars separated by | Heights proportional to category | |
| | | spaces along category axis | counts; check vertical scale start | |
| +-----------------------+-----------------------------+-------------------------------------+ |
| | Double / Grouped Bar | Side-by-side clustered bars | Compares sub-groups across same | |
| | Graph | with distinct legends | categories (e.g., Male vs. Female) | |
| +-----------------------+-----------------------------+-------------------------------------+ |
| | Circle Graph (Pie) | Circular slices proportional| Sector central angle: | |
| | | to relative share of total | θ = (Relative Frequency) × 360° | |
+---------------------------------------------------------------------------------------------------+
Circle Graph (Pie Chart) Angle Proportions
Since a complete circle contains $360^\circ$, each $1%$ of a data total corresponds to an angle of:
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| PIE CHART ANGLE CONVERSION REFERENCE |
| |
| +-----------------------+-----------------------------+-------------------------------------+ |
| | PERCENTAGE OF TOTAL | FRACTION OF CIRCLE | CENTRAL ANGLE MEASURE (θ) | |
| +-----------------------+-----------------------------+-------------------------------------+ |
| | 10% | 1/10 | 36° | |
| | 25% (One Quarter) | 1/4 | 90° (Right angle sector) | |
| | 33.33% (One Third) | 1/3 | 120° | |
| | 50% (One Half) | 1/2 | 180° (Straight line semi-circle) | |
| | 75% (Three Quarters) | 3/4 | 270° | |
+---------------------------------------------------------------------------------------------------+
2. Quantitative Data Displays
Quantitative data displays illustrate numerical measurements, spreads, clusters, and distributions.
A. Dot Plots
A dot plot places dots above a horizontal number line to represent individual data points, making it easy to identify peaks, gaps, and clusters.
Sample Dot Plot (Quiz Scores)
•
• • •
• • • • • •
• • • • • • •
+---+---+---+---+---+---+---+
70 75 80 85 90 95 100
B. Stem-and-Leaf Plots
A stem-and-leaf plot separates each numerical value into a "stem" (leading digit(s)) and a "leaf" (final trailing unit digit), preserving exact raw data values while showing shape.
Stem | Leaf Key: 8 | 2 = 82 points
------+-------------------
6 | 4 8
7 | 1 5 5 9
8 | 2 2 6 8 8
9 | 0 3 7
C. Histograms vs. Bar Graphs
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| HISTOGRAM VS. BAR GRAPH COMPARISON |
| |
| +-----------------------+-----------------------------+-------------------------------------+ |
| | FEATURE | HISTOGRAM | BAR GRAPH | |
| +-----------------------+-----------------------------+-------------------------------------+ |
| | Data Type | Continuous quantitative data| Discrete categorical / qualitative | |
| +-----------------------+-----------------------------+-------------------------------------+ |
| | Horizontal Axis | Numerical intervals / bins | Named categories / labels | |
| +-----------------------+-----------------------------+-------------------------------------+ |
| | Spaces Between Bars | NO spaces (bars touch) | Spaces exist between distinct bars | |
| +-----------------------+-----------------------------+-------------------------------------+ |
| | Bar Width Meaning | Bin width / interval size | Arbitrary visual width | |
+---------------------------------------------------------------------------------------------------+
3. Five-Number Summary & Box-and-Whisker Plots
A box-and-whisker plot (box plot) visually divides a dataset into four quartiles (each containing exactly $25%$ of the data points) using a five-number summary:
BOX-AND-WHISKER PLOT ANATOMY
|--- 25% ---|------ 25% ------|------ 25% ------|--- 25% ---|
Min Q1 Median Q3 Max
|----------+=================+=================+-----------|
| | | | |
|----------+=================+=================+-----------|
|<------------- IQR --------------->|
----+----------+-----------------+-----------------+-----------+----
40 55 70 85 100
Determining Skewness from a Box Plot
- Symmetric: The median line is centered in the box, and the left and right whiskers are approximately equal in length.
- Right-Skewed (Positive Skew): The right whisker is substantially longer than the left whisker, and the median line is shifted toward $Q_1$ (left side of box).
- Left-Skewed (Negative Skew): The left whisker is substantially longer than the right whisker, and the median line is shifted toward $Q_3$ (right side of box).
The $1.5 \times \text{IQR}$ Outlier Fences Rule
Statistical outliers are formally detected using upper and lower fences:
- Any data point $x < \text{Lower Fence}$ or $x > \text{Upper Fence}$ is an outlier.
4. Scatterplots & Bivariate Data
A scatterplot graphs paired numerical observations $(x, y)$ on a coordinate grid to identify relationships between two quantitative variables.
SCATTERPLOT PATTERNS
Positive Correlation Negative Correlation No Correlation
y y y
| • • | • • | • • •
| • • • | • • • | • • • •
| • • • | • • • | • • •
| • • | • • | • • •
+--------------> x +--------------> x +--------------> x
As x ↑ , y ↑ As x ↑ , y ↓ No systematic trend
Key Bivariate Concepts
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| BIVARIATE ANALYSIS CONCEPTS |
| |
| +-----------------------+-----------------------------+-------------------------------------+ |
| | CONCEPT | DEFINITION | KEY PROPERTY / APPLICATION | |
| +-----------------------+-----------------------------+-------------------------------------+ |
| | Independent (x) vs. | x = Explanatory (input); | Plotted on x-axis; y is plotted | |
| | Dependent (y) | y = Response (output) | on y-axis | |
| +-----------------------+-----------------------------+-------------------------------------+ |
| | Line of Best Fit | Linear trend line | ŷ = mx + b minimizes squared | |
| | (Regression Line) | approximating scatter | vertical residuals | |
| +-----------------------+-----------------------------+-------------------------------------+ |
| | Interpolation | Predicting y within range | Highly reliable mathematical | |
| | | of observed x values | estimate | |
| +-----------------------+-----------------------------+-------------------------------------+ |
| | Extrapolation | Predicting y outside range | Unreliable; assumes linear trend | |
| | | of observed x data | continues indefinitely | |
| +-----------------------+-----------------------------+-------------------------------------+ |
| | Correlation vs. | Association does NOT prove | Confounding / lurking variables may | |
| | Causation | direct cause-and-effect | drive apparent correlation | |
| +-----------------------+-----------------------------+-------------------------------------+ |
+---------------------------------------------------------------------------------------------------+
5. Misleading Graphs & Visual Distortions
WEST-B questions frequently require candidates to identify why a particular graph is deceptive.
- Truncated Vertical Axis (Broken Axis): When the vertical y-axis begins at a non-zero value (e.g., starts at $80$ instead of $0$), small differences appear enormous.
- Disproportionate 3D & Pictograph Scaling: When an icon's height is doubled, its 2D area increases by $2^2 = 4\times$, and a 3D volume increases by $2^3 = 8\times$, misleading the viewer.
- Unequal Interval Bins: Using varying bin widths on a histogram without adjusting density skews visual area proportions.
6. Step-by-Step Worked Graphical Interpretation Exercises
Problem 1: Outlier Identification from Five-Number Summary
Problem: A survey of weekly employee commute times produces the following five-number summary (in minutes): Calculate the IQR, determine the upper and lower outlier fences, and determine whether the maximum value of $84\text{ minutes}$ is an outlier.
Step-by-Step Solution:
- Calculate Interquartile Range (IQR):
- Calculate $1.5 \times \text{IQR}$:
- Compute Lower Fence: (Since commute time cannot be negative, no lower outliers exist).
- Compute Upper Fence:
- Evaluate Maximum Value: Since $84$ exceeds the upper fence of $78$, $84\text{ minutes}$ is a statistical outlier.
Problem 2: Pie Chart Sector Angle Calculation
Problem: A school library budget of $$12,000$ is allocated across four departments: Digital Media ($$4,800$), Non-Fiction Books ($$3,600$), Fiction Books ($$2,400$), and Administrative Supplies ($$1,200$). What is the central angle measure for the Non-Fiction Books sector in a circle graph?
Step-by-Step Solution:
- Determine the proportion for Non-Fiction Books:
- Calculate central angle:
- Verify all angles sum to $360^\circ$:
- Digital Media: $(4,800 / 12,000) \times 360^\circ = 0.40 \times 360^\circ = 144^\circ$
- Non-Fiction: $108^\circ$
- Fiction: $(2,400 / 12,000) \times 360^\circ = 0.20 \times 360^\circ = 72^\circ$
- Admin Supplies: $(1,200 / 12,000) \times 360^\circ = 0.10 \times 360^\circ = 36^\circ$
- Check: $144^\circ + 108^\circ + 72^\circ + 36^\circ = 360^\circ$. Correct!
Problem 3: Scatterplot Trend Line Interpolation vs. Extrapolation
Problem: A high school track coach records athletes' weekly sprint training hours ($x$) and their $100\text{m}$ dash race times ($y$ in seconds) between $2\text{ hours}$ and $10\text{ hours}$ of training per week. The calculated linear regression line is $\hat{y} = -0.35x + 14.8$.
- Predict the sprint time for an athlete training $6\text{ hours/week}$.
- Predict the sprint time for an athlete training $20\text{ hours/week}$, and explain why this prediction is unreliable.
Step-by-Step Solution:
- Prediction for $x = 6\text{ hours}$: Since $x = 6$ falls within the observed domain of $[2, 10]$, this is an interpolation and is statistically reliable.
- Prediction for $x = 20\text{ hours}$: $x = 20$ is an extrapolation far outside the $[2, 10]$ domain. A sprint time of $7.8\text{ seconds}$ is physically impossible (beating world records), showing why extrapolating linear trends beyond observed data is invalid.
A dataset of weekly student study hours has a five-number summary: Min = 4, Q1 = 12, Median = 18, Q3 = 24, and Max = 48. Using the standard 1.5 × IQR rule, which threshold defines an upper outlier, and is the maximum value of 48 classified as an outlier?
A school district conducts a survey of 600 high school graduates regarding their post-graduation plans: 270 plan to attend a 4-year university, 150 plan to attend a community college, 90 plan to enter the workforce, 60 plan to join the military, and 30 are undecided. In a circle graph (pie chart) representing this data, what is the central angle measure for the sector representing students attending a community college?
A biologist creates a scatterplot comparing ambient temperature (x in °F) to the number of cricket chirps per minute (y). The linear regression trend line is given by ŷ = 4x - 160, based on observed data collected between 55°F and 85°F. Based on this model, what is the predicted chirp rate at 72°F, and is this estimation considered an interpolation or an extrapolation?
A histogram displays student test scores grouped into bins of width 10: [50–59: 2 students], [60–69: 3 students], [70–79: 7 students], [80–89: 14 students], [90–99: 18 students]. Which statement accurately characterizes this distribution and the relationship between its mean and median?