10.3 Histograms, Box Plots & Scatter Plots

Histograms, Box Plots & Scatter Plots

Quick Summary: While standard bar and line graphs display categorical counts and basic time series, advanced statistical displays summarize the distribution, spread, and relationships of quantitative variables. Histograms group continuous numeric data into contiguous bin intervals. Box-and-Whisker plots graphically present the five-number summary and isolate statistical outliers using the $1.5 \times \text{IQR}$ rule. Scatter plots analyze bivariate relationships between explanatory ($x$) and response ($y$) variables to evaluate correlation and make predictions using lines of best fit.

Mastering these three visual displays is essential for answering higher-level data interpretation questions on the HiSET Mathematics subtest.

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1. Dot Plots: Small Data Sets, Every Value Visible

The blueprint names three displays together — dot plots, histograms, and box plots — so start with the simplest. A dot plot (also called a line plot) places one dot above a number line for each occurrence of that value.

Quiz scores for 15 students (out of 10):

              ·
              ·     ·
        ·     ·     ·     ·
  ·     ·     ·     ·     ·     ·
  ·     ·     ·     ·     ·     ·
--+-----+-----+-----+-----+-----+--
  5     6     7     8     9    10
  2     3     4     3     2     1   <- frequency

What a dot plot shows instantly:

• Center — the median is the middle dot when you count across; here the 8th of 15 dots sits at 7.
• Mode — the tallest stack; here 7, with 4 dots.
• Spread — the range, 10 $-$ 5 $=$ 5.
• Shape — roughly symmetric here; a long tail on one side would indicate skew.
• Outliers — a lone dot far from the cluster.

Dot plot vs. histogram — when each is used: a dot plot keeps every individual data value and works only for small sets (roughly $n < 30$) with few distinct values. A histogram groups values into bins, losing individual identities but handling large or continuous data sets. If a HiSET question asks you to recover an exact data value, the display must be a dot plot; a histogram can only give you a bin.

Worked Example: Reading a Dot Plot

Using the dot plot above, find the mean. Multiply each value by its frequency and divide by 15:

$\text{Mean} = \frac{5(2) + 6(3) + 7(4) + 8(3) + 9(2) + 10(1)}{15} = \frac{10 + 18 + 28 + 24 + 18 + 10}{15} = \frac{108}{15} = 7.2$

The mean (7.2) sits slightly above the median (7), the signature of a mild right skew.

2. Histograms: Continuous Data Distributions

A histogram displays the frequency distribution of a continuous quantitative dataset grouped into equal-width, contiguous numerical intervals known as bins (or class intervals).

   Histogram of Exam Scores (n = 30 students)
   
   Frequency (Students)
   10│             ┌─────────┐
    8│             │         │
    6│   ┌─────────┤         ├─────────┐
    4│   │         │         │         │
    2│───┤         │         │         ├───┐
    0└───┴─────────┴─────────┴─────────┴───┴──► Score Intervals
        50-59     60-69     70-79     80-89   90-99
        (f=2)     (f=6)     (f=10)    (f=8)   (f=4)

Critical Differences: Bar Graphs vs. Histograms

Worked Example 1: Extracting Statistics from a Histogram

Refer to the histogram above showing exam scores for a class:

• Scores $50\text{--}59$: $2$ students
• Scores $60\text{--}69$: $6$ students
• Scores $70\text{--}79$: $10$ students
• Scores $80\text{--}89$: $8$ students
• Scores $90\text{--}99$: $4$ students

Question 1: What is the total number of students in the class? $N = 2 + 6 + 10 + 8 + 4 = 30\text{ students}$

Question 2: What percentage of students scored $80$ or higher on the exam? $\text{Students with score } \ge 80 = 8 + 4 = 12$ $\text{Percentage} = \frac{12}{30} \times 100\% = \frac{2}{5} \times 100\% = 40\%$

3. The Five-Number Summary & Box Plots (Box-and-Whisker Plots)

A Box Plot (or Box-and-Whisker plot) provides a visual summary of the dispersion, central tendency, and skewness of a quantitative dataset based on its Five-Number Summary.

                    Anatomy of a Box-and-Whisker Plot

                ┌───────────────────┬─────────────────────────┐
   ├────────────┤                   │                         ├────────────┤
   │            │                   │                         │            │
   Minimum      Q1               Median (Q2)                  Q3        Maximum
  (0th Pct)  (25th Pct)          (50th Pct)               (75th Pct)   (100th Pct)
   
   │◄── 25% ───►│◄────── 25% ──────►│◄──────── 25% ──────────►│◄── 25% ───►│
   │                                                          │
   └────────────────────── Interquartile Range (IQR) ─────────┘
                           IQR = Q3 - Q1 (Middle 50%)

The Five-Number Summary Defined

1. Minimum ($\text{Min}$): The lowest data value in the dataset (excluding outliers).
2. First Quartile ($Q_1$ / 25th Percentile): The median of the lower half of data values; $25%$ of data lies below $Q_1$.
3. Median ($Q_2$ / 50th Percentile): The physical middle value dividing the entire dataset in half.
4. Third Quartile ($Q_3$ / 75th Percentile): The median of the upper half of data values; $75%$ of data lies below $Q_3$ ($25%$ lies above).
5. Maximum ($\text{Max}$): The highest data value in the dataset (excluding outliers).

The Interquartile Range (IQR)

The Interquartile Range (IQR) measures the spread of the middle $50%$ of the data: $\mathbf{\text{IQR} = Q_3 - Q_1}$

Outlier Identification: The $1.5 \times \text{IQR}$ Rule

Statisticians use mathematical boundaries called fences to formally classify outliers:

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

$\text{Value } x \text{ is an Outlier if: } x < Q_1 - 1.5(\text{IQR}) \quad \text{or} \quad x > Q_3 + 1.5(\text{IQR})$

Worked Example 2: Constructing a 5-Number Summary and Detecting Outliers

Calculate the five-number summary, IQR, and determine whether any outliers exist for the $12$-number dataset: $\{14, 18, 20, 22, 25, 26, 28, 30, 31, 35, 38, 72\}$

1. Verify data is sorted: $14, 18, 20, 22, 25, 26 \mid 28, 30, 31, 35, 38, 72$.
2. Find Median ($Q_2$): $n = 12$ (even). Average of 6th and 7th values: $Q_2 = \frac{26 + 28}{2} = 27$
3. Find $Q_1$ (median of lower 6 values ${14, 18, 20, 22, 25, 26}$): $Q_1 = \frac{20 + 22}{2} = 21$
4. Find $Q_3$ (median of upper 6 values ${28, 30, 31, 35, 38, 72}$): $Q_3 = \frac{31 + 35}{2} = 33$
5. Calculate IQR: $\text{IQR} = Q_3 - Q_1 = 33 - 21 = 12$
6. Compute Outlier Fences ($1.5 \times \text{IQR} = 1.5 \times 12 = 18$):
   • $\text{Lower Fence} = 21 - 18 = 3$
   • $\text{Upper Fence} = 33 + 18 = 51$
7. Evaluate Outliers: The value $72$ exceeds the Upper Fence ($72 > 51$). Therefore, $72$ is a confirmed statistical outlier.

Core Principle: Every quartile section in a box plot contains exactly $25%$ of the data points, regardless of the physical length of the box segment or whisker. A longer whisker indicates greater dispersion/spread among those $25%$ of points, NOT a greater number of points!

4. Scatter Plots, Correlation & Lines of Best Fit

A scatter plot graphs paired quantitative measurements $(x, y)$ as individual points on a Cartesian coordinate plane. It investigates the relationship between an independent/explanatory variable ($x$) on the horizontal axis and a dependent/response variable ($y$) on the vertical axis.

   Scatter Plot Correlation Patterns
   
   1. Strong Positive             2. Strong Negative             3. No Correlation
   y                              y                              y
   │        ●   ●                 │ ●                            │   ●     ●
   │      ●   ●                   │   ●   ●                      │ ●    ●     ●
   │    ●   ●                     │     ●   ●                    │    ●    ●   ●
   │  ●   ●                       │       ●   ●                  │  ●    ●    ●
   │●                             │         ●   ●                │ ●   ●    ●
   └──────────────► x             └──────────────► x             └──────────────► x
    r ≈ +0.9 (Upward)              r ≈ -0.9 (Downward)            r ≈ 0 (Random Cloud)

Classifying Correlation

• Positive Correlation: As $x$ increases, $y$ tends to increase (points slope upward from left to right).
  • Example: Study hours vs. Exam score; Vehicle weight vs. Fuel consumption.
• Negative Correlation: As $x$ increases, $y$ tends to decrease (points slope downward from left to right).
  • Example: Speed traveled vs. Time to destination; Elevation vs. Temperature.
• No Correlation: Points are randomly distributed; changes in $x$ show no linear relationship with $y$.
  • Example: Shoe size vs. IQ score; Coffee consumption vs. Commute distance.
• Strong vs. Weak: Points tightly clustered along a line indicate strong correlation; points widely scattered indicate weak correlation.
• Non-Linear (Curvilinear): Points form a curved pattern (such as a parabola or exponential curve).

Trend Lines / Line of Best Fit

A line of best fit (or trend line) is a straight line drawn through a scatter plot that best approximates the linear association of the data points.

$\mathbf{y = mx + b}$

Where:

• $m = \frac{\Delta y}{\Delta x}$ is the slope, representing the rate of change in $y$ per unit increase in $x$.
• $b$ is the $y$-intercept, representing the expected baseline value of $y$ when $x = 0$.

Making Predictions: Interpolation vs. Extrapolation

• Interpolation: Predicting a value within the domain of observed data points (e.g., predicting $y$ for $x = 5$ when collected data ranges from $x = 1$ to $x = 10$). Highly reliable.
• Extrapolation: Predicting a value outside the domain of observed data points (e.g., predicting $y$ for $x = 25$ when data only extends to $x = 10$). Caution is required, as linear trends frequently break down over extended ranges.

Critical HiSET Distinction — Correlation Does Not Imply Causation: Even if two variables exhibit a near-perfect linear correlation ($r \approx 1.0$), one variable does not necessarily cause the other to change. Both may be driven by a third, unobserved confounding (or lurking) variable. (For example, ice cream sales and swimming pool drownings are strongly positively correlated, but both are caused by high summer temperatures).