13.2 Interpreting Bar Graphs, Histograms, Box Plots & Scatterplots

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.

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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:

$$1\% = \frac{1}{100} \times 360^\circ = 3.6^\circ$$ $$\text{Central Angle } (\theta) = \frac{\text{Category Frequency}}{\text{Total Frequency}} \times 360^\circ = (\text{Percentage}) \times 3.6^\circ$$

+---------------------------------------------------------------------------------------------------+
|                             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°                                |   |
+---------------------------------------------------------------------------------------------------+

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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

+---------------------------------------------------------------------------------------------------+
|                                 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              |   |
+---------------------------------------------------------------------------------------------------+

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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:

$$\{\text{Minimum}, Q_1, \text{Median } (Q_2), Q_3, \text{Maximum}\}$$

                              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:

$$\text{IQR} = Q_3 - Q_1$$ $$\text{Lower Outlier Fence} = Q_1 - (1.5 \times \text{IQR})$$ $$\text{Upper Outlier Fence} = Q_3 + (1.5 \times \text{IQR})$$

• Any data point $x < \text{Lower Fence}$ or $x > \text{Upper Fence}$ is an outlier.

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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

+---------------------------------------------------------------------------------------------------+
|                                 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          |   |
|   +-----------------------+-----------------------------+-------------------------------------+   |
+---------------------------------------------------------------------------------------------------+

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5. Misleading Graphs & Visual Distortions

WEST-B questions frequently require candidates to identify why a particular graph is deceptive.

1. 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.
2. 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.
3. Unequal Interval Bins: Using varying bin widths on a histogram without adjusting density skews visual area proportions.

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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):

$$\text{Min} = 8, \quad Q_1 = 18, \quad \text{Median} = 28, \quad Q_3 = 42, \quad \text{Max} = 84$$

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:

1. Calculate Interquartile Range (IQR): $$\text{IQR} = Q_3 - Q_1 = 42 - 18 = 24\text{ minutes}$$
2. Calculate $1.5 \times \text{IQR}$: $$1.5 \times 24 = 36\text{ minutes}$$
3. Compute Lower Fence: $$\text{Lower Fence} = Q_1 - 36 = 18 - 36 = -18\text{ minutes}$$ (Since commute time cannot be negative, no lower outliers exist).
4. Compute Upper Fence: $$\text{Upper Fence} = Q_3 + 36 = 42 + 36 = 78\text{ minutes}$$
5. Evaluate Maximum Value: $$\text{Max} = 84 > 78\text{ minutes}$$ Since $84$ exceeds the upper fence of $78$, $84\text{ minutes}$ is a statistical outlier.

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Problem 2: Pie Chart Sector Angle Calculation

Problem: A school library budget of $\text{\textdollar}12,000$ is allocated across four departments: Digital Media ($\text{\textdollar}4,800$), Non-Fiction Books ($\text{\textdollar}3,600$), Fiction Books ($\text{\textdollar}2,400$), and Administrative Supplies ($\text{\textdollar}1,200$). What is the central angle measure for the Non-Fiction Books sector in a circle graph?

Step-by-Step Solution:

1. Determine the proportion for Non-Fiction Books: $$\text{Proportion} = \frac{\$3,600}{\$12,000} = \frac{36}{120} = \frac{3}{10} = 0.30 \quad (30\%)$$
2. Calculate central angle: $$\theta = 0.30 \times 360^\circ = 108^\circ$$
3. 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!

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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 $\hat{y} = -0.35x + 14.8$.

1. Predict the sprint time for an athlete training $6\text{ hours/week}$.
2. Predict the sprint time for an athlete training $20\text{ hours/week}$, and explain why this prediction is unreliable.

Step-by-Step Solution:

1. Prediction for $x = 6\text{ hours}$: $$\hat{y} = -0.35(6) + 14.8 = -2.10 + 14.8 = 12.7\text{ seconds}$$ Since $x = 6$ falls within the observed domain of $[2, 10]$, this is an interpolation and is statistically reliable.
2. Prediction for $x = 20\text{ hours}$: $$\hat{y} = -0.35(20) + 14.8 = -7.0 + 14.8 = 7.8\text{ seconds}$$ $x = 20$ 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.