9.1 Statistical Questions & Data Displays

9.1 Statistical Questions & Data Displays

Quick Answer: A statistical question is an inquiry that anticipates variability in the data collected (e.g., "How many minutes do sixth-graders spend reading each night?"). In contrast, a non-statistical question yields a single deterministic value (e.g., "How tall is the school flagpole?"). Data is partitioned into categorical (qualitative labels) and numerical (quantitative measurements or counts). Numerical data is displayed via dot plots (individual discrete points), stem-and-leaf plots (place-value digits preserved with a key), or histograms (contiguous, touching bars spanning equal numerical intervals). The shape of a distribution is described by its symmetry, peaks (modality), clusters, gaps, and skewness—where the tail dictates the direction of skew.

---

Formulating Statistical Questions vs. Non-Statistical Questions

Under Florida B.E.S.T. benchmark MA.6.DP.1.1, the statistical inquiry cycle begins with formulating a question that can be answered by collecting and analyzing empirical data. The defining hallmark of a statistical question is the anticipation of variability.

The Variability Criterion

When assessing a question, ask: "Will the responses from different subjects or different trials vary, or is there only one fixed, deterministic answer?"

• Statistical Questions: Anticipate a range of responses because different individuals, objects, or trials naturally exhibit different characteristics or measurements.
• Non-Statistical Questions: Direct inquiries toward a single, unchanging factual value, a specific historical datum, or a deterministic measurement of a single object.

Transforming Non-Statistical Questions into Statistical Inquiries

On the FAST assessment, students are frequently asked to identify how a non-statistical question can be restructured into a valid statistical inquiry:

• Deterministic: "How many books are in Marcus's backpack right now?"
• Statistical Revision: "How many books do students in Marcus's class typically carry in their backpacks?"
• Deterministic: "Did it rain at Miami International Airport yesterday?"
• Statistical Revision: "What is the annual daily rainfall distribution recorded at Miami International Airport over the past ten years?"

---

Categorical vs. Numerical Data

Once a statistical question is formulated, data collection yields observations that fall into one of two fundamental domains:

1. Categorical (Qualitative) Data

Categorical data consists of non-numerical characteristics, descriptions, or attributes that place individuals into specific groups or categories. Arithmetic operations (such as computing an arithmetic average) are meaningless for categorical data.

• Examples: Eye color, favorite middle school subject, pet species, preferred transportation method to school (bus, car, walking, bicycle).
• Special Trap (Categorical Numbers): Numerical codes used purely as labels—such as telephone area codes, zip codes (e.g., 32801), or athletic jersey numbers (e.g., #23)—are categorical data, because adding them or averaging them produces no mathematically meaningful measurement.

2. Numerical (Quantitative) Data

Numerical data consists of numbers that represent measurable quantities or counts where mathematical computations (addition, subtraction, averaging) are valid and meaningful.

• Discrete Numerical Data: Data obtained by counting whole entities. Values exist only as isolated points on a number line, typically non-negative integers ${0, 1, 2, 3, \dots}$. Examples: Number of siblings, number of goals scored in a soccer match, number of library books checked out.
• Continuous Numerical Data: Data obtained by measuring along a continuum. Values can take on any real value within an interval, including fractions and decimals, bounded only by the precision of the measuring instrument. Examples: Student height in centimeters, body temperature in degrees Fahrenheit, time in seconds to run a $100\text{-meter}$ sprint, volume of water in a hydration flask in milliliters.

---

Constructing and Interpreting Data Displays

Under Florida B.E.S.T. standards, middle school students must select, construct, and interpret appropriate graphical representations for both categorical and numerical datasets.

1. Dot Plots (Line Plots)

A dot plot (historically called a line plot) displays discrete numerical data along a horizontal number line. Each recorded observation is marked with a single dot (or "x") stacked vertically above its numerical coordinate.

• Key Strengths: Shows every individual raw data point; instantly highlights the mode (tallest stack of dots), clusters (dense concentrations of points), gaps (intervals with zero dots), and outliers (isolated dots separated from the primary cluster).
• Limitations: Impractical for very large datasets ($n > 100$) or continuous data with infinite decimal variations.

Number of Siblings Reported by 15 Sixth-Graders:
        •
        •   •
    •   •   •   •
    •   •   •   •       •
<---|---|---|---|---|---|--->
    0   1   2   3   4   5

Analysis: Mode $= 1$ sibling (stack of $4$ dots); cluster spans from $0$ to $3$; gap at $4$; total sample size $n = 2 + 4 + 3 + 2 + 0 + 1 = 12$ students.

2. Frequency Tables and Relative Frequency

A frequency table organizes raw data by partitioning values into categories or intervals, paired with their corresponding counts (frequencies). A relative frequency table expresses each count as a fraction, decimal, or percentage of the total sample size $n$:

$\text{Relative Frequency} = \frac{\text{Class Frequency}}{\text{Total Number of Observations } (n)}$

3. Bar Graphs

A bar graph represents categorical data. Categorical groups are aligned along one axis (usually horizontal), and frequencies or relative frequencies are scaled along the other axis.

• Crucial Rule: The rectangular bars in a bar graph must have visible spaces/gaps between them to signify that the categories are discrete, non-contiguous qualitative groups (e.g., Pizza, Tacos, Burgers).

4. Stem-and-Leaf Plots

A stem-and-leaf plot organizes numerical data in place-value order while preserving every individual raw data value. Each data point is split into:

• The Stem: The leading digit(s) representing higher place values (e.g., tens, hundreds).
• The Leaf: The final trailing digit (always a single digit from $0$ to $9$, representing units or tenths).

[!IMPORTANT] The Mandatory Key: A stem-and-leaf plot is mathematically incomplete and uninterpretable without a Key (legend). For example, a stem of $4$ and leaf of $7$ ($4 \mid 7$) represents $47$ if the key states $4 \mid 7 = 47\text{ points}$, but represents $4.7$ if the key states $4 \mid 7 = 4.7\text{ seconds}$. Always check the key before computing statistics!

• Ordered Stems and Leaves: Leaves on each horizontal stem line must be written in ascending order from least to greatest, evenly spaced to depict the distribution's shape accurately.

Math Quiz Scores (n = 16 students):
Stem | Leaf
  5  | 4  8
  6  | 2  5  9
  7  | 1  4  4  8
  8  | 0  3  5  7  9
  9  | 2  6
Key: 7 | 1 = 71 points

Interpretation: The lowest score is $54$; the highest score is $96$; the mode is $74$ (repeated leaf $4$ on stem $7$).

5. Histograms

A histogram represents continuous or grouped numerical data. The horizontal axis is divided into contiguous, equal-width numerical intervals (called bins or class intervals), and the vertical axis represents the frequency of data points falling into each bin.

• Crucial Rule (Bars Touch): Unlike bar graphs, the bars in a histogram must touch each other without spaces between them (unless a bin has a frequency of zero). Touching bars visually communicate that the underlying numerical scale is continuous.
• Interval Boundary Convention: Standard convention dictates that class intervals include the lower boundary and exclude the upper boundary: $[a, b) = {x \mid a \le x < b}$. For example, in the interval $10-19$, a score of $10$ is included, but a score of $20$ belongs in the next bin ($20-29$).

---

Analyzing Distribution Shapes

Under Florida B.E.S.T. benchmark MA.6.DP.1.2, students must describe and interpret the overall shape of a numerical distribution displayed on a dot plot or histogram.

1. Symmetrical / Bell-Shaped Distributions

• Definition: The left half and right half of the display are mirror images across a central vertical line of symmetry.
• Characteristics: Displays a prominent single peak in the center, tapering off evenly on both sides. In a perfectly symmetrical distribution, the Mean $\approx$ Median $\approx$ Mode.

2. Skewed Right (Positively Skewed) Distributions

• Definition: The bulk of the data points are clustered at the lower (left) values, while an elongated tail stretches out toward the higher (right) values.
• Directional Rule: The tail dictates the direction of the skew. Because the tail extends to the right, it is skewed right.
• Relationship of Center Measures: The extreme high values in the right tail pull the arithmetic average upward, resulting in: $\text{Mean} > \text{Median}$
• Authentic Contexts: Household income distributions, housing sale prices, miles driven per week.

3. Skewed Left (Negatively Skewed) Distributions

• Definition: The bulk of the data points are clustered at the higher (right) values, while an elongated tail stretches out toward the lower (left) values.
• Directional Rule: The tail extends to the left, so the distribution is skewed left.
• Relationship of Center Measures: Extreme low values in the left tail drag the mean downward, resulting in: $\text{Mean} < \text{Median}$
• Authentic Contexts: Test scores on an exam where most students mastered the material, age at retirement.

4. Uniform Distributions

• Definition: Every numerical value or class interval has approximately the same frequency.
• Characteristics: The graph appears flat and rectangular, with no distinct peak or tail.
• Authentic Context: Rolling a fair six-sided die $600$ times (each face appears approximately $100$ times).

5. Bimodal Distributions

• Definition: A distribution exhibiting two distinct, prominent peaks separated by a central valley (trough).
• Significance: Often indicates that the dataset is composed of two distinct overlapping sub-populations (e.g., combining adult male and adult female shoe sizes into one graph).

$\begin{array}{|l|c|c|l|} \hline \textbf{Distribution Shape} & \textbf{Tail Location} & \textbf{Center Relationship} & \textbf{Visual Profile} \\ \hline \text{Symmetric / Bell} & \text{Balanced both sides} & \text{Mean } \approx \text{ Median} & \text{Single central peak, mirror halves} \\ \hline \text{Skewed Right (Positive)} & \text{Elongated to the right} & \text{Mean } > \text{ Median} & \text{Left cluster, long right tail} \\ \hline \text{Skewed Left (Negative)} & \text{Elongated to the left} & \text{Mean } < \text{ Median} & \text{Right cluster, long left tail} \\ \hline \text{Uniform} & \text{No tails} & \text{Mean } \approx \text{ Median} & \text{Flat rectangular profile} \\ \hline \text{Bimodal} & \text{Variable} & \text{Depends on peak symmetry} & \text{Two distinct dominant peaks} \\ \hline \end{array}$

---

Step-by-Step Worked Example: Constructing & Interpreting a Histogram

Problem: The following raw data represents the daily commute times in minutes for $20$ Florida teachers: $\{12, 18, 22, 25, 27, 28, 31, 33, 34, 35, 36, 38, 39, 41, 44, 45, 48, 52, 58, 65\}$

Step 1: Determine equal class intervals (bins). The values range from $12$ to $65$. A convenient bin width of $10\text{ minutes}$ yields six contiguous intervals:

• $10$ to $19\text{ min}$
• $20$ to $29\text{ min}$
• $30$ to $39\text{ min}$
• $40$ to $49\text{ min}$
• $50$ to $59\text{ min}$
• $60$ to $69\text{ min}$

Step 2: Construct the frequency table.

• $10-19$: $12, 18 \implies \text{Frequency} = 2$
• $20-29$: $22, 25, 27, 28 \implies \text{Frequency} = 4$
• $30-39$: $31, 33, 34, 35, 36, 38, 39 \implies \text{Frequency} = 7$ (Peak / Modal Interval)
• $40-49$: $41, 44, 45, 48 \implies \text{Frequency} = 4$
• $50-59$: $52, 58 \implies \text{Frequency} = 2$
• $60-69$: $65 \implies \text{Frequency} = 1$
• Total $n = 2 + 4 + 7 + 4 + 2 + 1 = 20$.

Step 3: Analyze distribution shape. The peak occurs in the central interval $30-39\text{ min}$. The frequencies taper smoothly and almost symmetrically on both sides ($2, 4, 7, 4, 2, 1$), with a slight extension into the $60-69$ bin. The distribution is approximately symmetrical and bell-shaped, meaning the mean commute time will be very close to the median commute time.

---

Common Exam Traps & Misconceptions

[!WARNING]

Exam Trap 1: Naming Skewness by the Peak Instead of the Tail

A widespread mistake on the FAST assessment is identifying a distribution as "skewed left" because the tall cluster (peak) is on the left side. The direction of skewness is always determined by the tail, not the peak. If the peak is on the left and the tail extends out to the right, the distribution is skewed right (positively skewed). Think of the tail as having been "dragged out" toward the skew.

[!WARNING]

Exam Trap 2: Confusing Histograms with Bar Graphs

When presented with a graph of test score intervals, students often incorrectly call it a bar graph. Remember: If the horizontal axis is a continuous numerical scale divided into equal bins and the vertical bars touch each other, it is a histogram. If the bars have spaces between them and represent discrete categories (e.g., Apple, Banana, Orange), it is a bar graph.

[!WARNING]

Exam Trap 3: Ignoring the Key in Stem-and-Leaf Plots

In stem-and-leaf questions, test items frequently set the key to decimals (e.g., $5 \mid 2 = 5.2$) or hundreds (e.g., $12 \mid 4 = 124$). Students who glance at the plot without checking the key assume $5 \mid 2 = 52$, leading to order-of-magnitude errors in calculating means, ranges, or medians. Always inspect the key before doing any math!