9.4 Five-Number Summaries, Box Plots, & Bivariate Scatter Plots

9.4 Five-Number Summaries, Box Plots, & Bivariate Scatter Plots

Advanced data analysis on TABE 13&14 Mathematics Levels D and A expands beyond single-variable summaries into multi-quartile distributions and two-variable relationships. This section covers the Five-Number Summary, Box-and-Whisker Plots, Interquartile Range (IQR) outlier calculations, and Bivariate Scatter Plots with lines of best fit.

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The Five-Number Summary

A Five-Number Summary provides a comprehensive snapshot of a dataset's center, spread, and quartile distribution:

1. Minimum ($\text{Min}$): The lowest data value.
2. First Quartile ($Q_1$): The $25\text{th}$ percentile; the median of the lower half of the data (marks the lower $25\%$ boundary).
3. Second Quartile / Median ($Q_2$): The $50\text{th}$ percentile; the physical center of the entire dataset.
4. Third Quartile ($Q_3$): The $75\text{th}$ percentile; the median of the upper half of the data (marks the upper $25\%$ boundary).
5. Maximum ($\text{Max}$): The highest data value.

Step-by-Step Algorithm for Finding Quartiles

1. Sort data in ascending order.
2. Find the overall Median ($Q_2$).
3. Split the data into a Lower Half (all values below $Q_2$) and an Upper Half (all values above $Q_2$).
   • Rule for Odd $n$: Exclude the median value itself when forming lower and upper halves.
   • Rule for Even $n$: The dataset splits evenly between the two middle numbers.
4. $Q_1$ is the median of the lower half; $Q_3$ is the median of the upper half.

Worked Example: Finding the Five-Number Summary

Consider test scores for $11$ students: $\{62, 74, 78, 80, 82, 85, 88, 91, 94, 96, 100\}$.

• $\text{Min} = \mathbf{62}$
• Median ($Q_2$) at position $\frac{11+1}{2} = 6\text{th}$ score $\implies Q_2 = \mathbf{85}$
• Lower Half ($5$ values below $85$): $\{62, 74, \mathbf{78}, 80, 82\} \implies Q_1 = \mathbf{78}$
• Upper Half ($5$ values above $85$): $\{88, 91, \mathbf{94}, 96, 100\} \implies Q_3 = \mathbf{94}$
• $\text{Max} = \mathbf{100}$
• Five-Number Summary: $[62, 78, 85, 94, 100]$.

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Interquartile Range (IQR) & The $1.5 \times \text{IQR}$ Outlier Rule

The Interquartile Range (IQR) measures the spread of the middle $50\%$ of the data, spanning from the first quartile to the third quartile:

$$\mathbf{\text{IQR} = Q_3 - Q_1}$$

Identifying Outliers Using the $1.5 \times \text{IQR}$ Rule

To formally test whether an extreme value is a statistical outlier, compute the lower and upper "fences":

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

• Any observation less than the Lower Fence is an outlier.
• Any observation greater than the Upper Fence is an outlier.

Worked Example: Using the previous test score dataset where $Q_1 = 78$ and $Q_3 = 94$:

1. $\text{IQR} = 94 - 78 = 16$.
2. $1.5 \times \text{IQR} = 1.5 \times 16 = 24$.
3. $\text{Lower Fence} = 78 - 24 = \mathbf{54}$. (Since $\text{Min} = 62 > 54$, there are no lower outliers).
4. $\text{Upper Fence} = 94 + 24 = \mathbf{118}$. (Since $\text{Max} = 100 < 118$, there are no upper outliers).

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Box-and-Whisker Plots

A box-and-whisker plot graphically renders the five-number summary above a numerical number line.

Box-and-Whisker Plot Anatomy:

        |-------[        |        ]-------|
       Min      Q1     Median     Q3     Max
       (0%)    (25%)   (50%)    (75%)   (100%)
        |<-25%->|<-25%-->|<-25%-->|<-25%->|
        |             |<- - - IQR - - ->| |
        |< - - - - - - - Range - - - - - >|

The 25% Rule per Quartile Section

Every box-and-whisker plot is partitioned into four distinct quarters, each containing exactly $25\%$ of the total data points:

• Lower Whisker (Min to $Q_1$): Contains the bottom $25\%$ of data.
• Left Box Half ($Q_1$ to Median): Contains $25\%$ of data.
• Right Box Half (Median to $Q_3$): Contains $25\%$ of data.
• Upper Whisker ($Q_3$ to Max): Contains the top $25\%$ of data.
• The central rectangular Box ($Q_1$ to $Q_3$) represents the IQR (middle $50\%$).

Comparing Parallel Box Plots

When comparing two parallel box plots (e.g., Class A vs. Class B):

1. Compare Medians: Higher median indicates higher overall central achievement.
2. Compare Box Lengths (IQRs): A wider box indicates greater variability among typical performers.
3. Compare Overall Ranges (Whiskers): Shows total data spread from minimum to maximum.

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Bivariate Data & Scatter Plots

Bivariate data involves observing two different quantitative variables on the same subject or event, plotted as coordinate pairs $(x, y)$ on a Cartesian grid:

• Explanatory (Independent) Variable ($x$): Plotted along the horizontal axis.
• Response (Dependent) Variable ($y$): Plotted along the vertical axis.

Types of Correlation

Scatter Plot Patterns:

Positive Linear:        Negative Linear:          No Correlation:         Non-Linear (Curved):
  y |       * *           y | *                     y |   *   *  *          y |       * * *
    |     * *               |   * *                   | *   *   *             |     *       *
    |   * *                 |     * *                 |   *   *               |   *           *
    | * *                   |       * *               | *   *   * *           | *               *
  0 +--------- x          0 +--------- x            0 +--------- x          0 +------------- x

Correlation Strength

• Strong Correlation: Points lie very close to an imaginary straight line ($|r| \ge 0.8$).
• Moderate / Weak Correlation: Points show a visible trend but are widely dispersed ($|r| < 0.8$).

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Line of Best Fit (Trend Line) & Linear Modeling

A line of best fit (trend line) is a straight line drawn through bivariate data that best approximates the linear trend, minimizing the vertical distances (residuals) from the points to the line.

Trend Line Equation ($y = mx + b$)

• Slope ($m$): The predicted rate of change in $y$ for every $1\text{-unit}$ increase in $x$.
• $y$-Intercept ($b$): The predicted baseline value of $y$ when $x = 0$.

Worked Example: A fitness study models body weight loss with the trend line $\hat{y} = -0.75x + 195$, where $x$ is weeks of exercise and $y$ is weight in pounds.

• Interpretation of Slope ($-0.75$): On average, participants lose $0.75\text{ pounds}$ per week of exercise.
• Interpretation of Intercept ($195$): The average starting weight at Week $0$ was $195\text{ pounds}$.
• Predicting Weight at Week $8$ (Interpolation): $\hat{y} = -0.75(8) + 195 = -6 + 195 = \mathbf{189\text{ lbs}}$.

Interpolation vs. Extrapolation

• Interpolation: Estimating a value within the range of measured $x$-data (statistically valid and reliable).
• Extrapolation: Predicting a value far beyond the measured $x$-data range (highly risky and unreliable, as trends rarely remain linear indefinitely).

The Fundamental Statistical Rule: Correlation $\neq$ Causation

Just because two variables show a strong positive or negative correlation does not prove that changes in $x$ cause changes in $y$. A hidden third variable (confounding / lurking variable) is often responsible for the mutual movement (e.g., higher summer temperatures drive both increased ice cream sales and increased swimming pool accidents).