8/25/25: Statistical Sampling, Errors, Data Measurement, Frequency Tables, and Experimental Design

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Statistical Sampling Errors and Sources of Bias

• Bias: A systematic sampling error that occurs when a selected sample fails to accurately represent the underlying target population.

• Sampling Error: An error caused directly by the methodology used to select the sample.

  • Example: Surveying attendees at a gerbil convention to determine the general population's favorite pet is a sampling error because the selected sample is inherently unrepresentative.

• Non-Sampling Error: An error that arises from data collection, recording, or processing glitches unrelated to the sample selection method.

  • Example: A respondent states they have $3$ siblings, but the researcher accidentally records $30$ siblings and incorporates that incorrect value into the dataset.

• Problems with Samples: Sampling bias results when an inappropriate or unrepresentative subset of the population is surveyed.

• Self-Selected Samples (Voluntary Response Bias): Occurs when individuals choose whether to participate in a sample group.

  • Example: Online polls asking questions such as "Do you think President Trump is doing a good job? Yes or no?".

  • Impact: Self-selected samples attract individuals with extreme views on either side of an issue. Moderate or undecided individuals rarely participate, producing polarized data rather than a representative sample of the full population.

• Sample Size Issues: Utilizing a sample size that is too small to draw valid statistical conclusions about a larger population.

  • Example: Surveying $3$ random Americans about their favorite cheese, receiving "cheddar" from all $3$, and concluding that all Americans prefer cheddar.

  • Principle: A sample must be large enough to minimize random variability while remaining practical for data analysis.

• Undue Influence: Occurs when data collection methods or leading question phrasing alter respondent answers.

  • Example: Asking children, "Which do you like better: gross awful broccoli or really really outstanding ice cream?" or asking subjects to choose between "stupid awful cats or really cool dogs?".

• Non-Response / Refusal to Participate: Occurs when targeted subjects refuse to participate or decline to return survey materials.

  • Distinction from Self-Selection: In non-response scenarios, the researcher selects specific subjects (e.g., sending mail-in surveys to targeted physical addresses), but selected individuals choose not to respond, introducing non-response bias.

• Causality vs. Correlation: Correlation indicates a statistical association between variables, but correlation does not imply causation ($\text{correlation} \neq \text{causation}$).

  • Historical Example: Prior to women's suffrage, $0$ world wars occurred; after women gained voting rights in the United States, $2$ world wars occurred. Claiming a causal link between suffrage and world wars is a logical fallacy.

  • Astronomical Example: Data shows that the frequency of babies named Andrea correlates with the distance between Neptune and the Moon. These variables share statistical correlation by coincidence, not direct causation.

• Self-Funded or Self-Interest Studies: Studies financed by organizations that have a commercial, financial, or political interest in the outcome.

  • Example: A study funded by Purdue Chicken Farms concluding that American consumers do not eat enough chicken and should increase purchases. Independent research requires unbiased third-party funding.

• Misleading Use of Data: The deliberate or improper presentation of data graphics to distort conclusions.

  • Case Study: A $2014$ graphic published by Reuters regarding Florida's $2005$ "Stand Your Ground" law featured an inverted y-axis (with $0$ at the top and higher values at the bottom). Visually, the line trended downward, giving the false impression that murder rates decreased after the law passed, whereas murder rates had actually increased sharply.

• Confounding: Occurs when unobserved background variables influence both the explanatory and response variables, pointing to an incorrect direct conclusion.

  • Example: Annual ice cream sales and shark attack rates show a strong positive correlation. Ice cream consumption does not cause shark attacks, nor do shark attacks drive ice cream sales. The confounding variable is summer temperature ($\text{heat}$), which increases both beach swimming and ice cream purchases.

Measurement Variation and Statistical Notation

• Variation Between Values Within a Sample: Natural individual differences observed among members of a single sample.

  • Example: A factory manufacturing $9\,\text{inch}$ nails does not produce every single nail at exactly $9.0000\,\text{inches}$. Individual nail lengths fluctuate slightly around the target average of $9\,\text{inches}$.

• Variation Between Samples: Discrepancies observed when multiple independent samples are drawn from the same population.

  • Example: Two independent researchers surveying separate samples of $1000$ people regarding ice cream flavor preferences will obtain slightly different sample statistics.

• Minimizing Variation: Sample-to-sample variation is reduced by increasing the sample size ($N$). In scientific practice, peer review and study replication validate findings across independent research groups.

• Statistical Notation:

  • Sample Mean: $\bar{x}$ ("x-bar"), which represents a calculated sample statistic.

  • Population Mean: $\mu$ ("mu"), which represents a fixed population parameter.

  • Statistical Goal: $\bar{x}$ serves as an point estimate for $\mu$. Exact equality between $\bar{x}$ and $\mu$ occurs only if the sample includes the entire population (a census).

Mathematical Rounding Rules and Precision

• Significant Digits / Figures: Rules governing precision in quantitative measurements. The term "siggy figs" is an informal phrasing that should be avoided in academic settings.

• Preserving Precision: Avoid premature or excessive rounding that forfeits numerical detail. Rounding a value like $6.487$ down to $6$ removes essential precision; retaining $6.5$ or $6.49$ is preferred.

• Rule 1: Always follow explicit rounding instructions specified in problem statements.

• Rule 2: Retain full calculator precision across all intermediate computational steps (using calculator memory keys). Perform rounding exclusively on the final output to prevent error propagation.

Levels of Data Measurement

• Nominal Scale Level: Qualitative, categorical data lacking inherent numerical rank, order, or hierarchy.

  • Examples: Eye color, animal species, yes/no survey responses, dog breeds encountered on a block.

• Ordinal Scale Level: Categorical or numerical data possessing a distinct rank or hierarchy, but differences between adjacent ranks are undefined, non-uniform, or subjective.

  • Examples: App or restaurant star ratings ($1$\text{--}$5$ stars), top $20$ college rankings, Likert scale responses (e.g., strongly agree, agree, disagree).

• Interval Scale Level: Quantitative ordered data with uniform, measurable differences between values, but lacking a true absolute zero point. Multiplication, division, and ratios are mathematically meaningless on an interval scale.

  • Examples: Fahrenheit or Celsius temperature scales ($0^\circ$ does not indicate the complete absence of heat), calendar dates of movie releases in a media franchise.

• Ratio Scale Level: Quantitative ordered data featuring uniform measurable differences and a true absolute zero point (where $0$ represents the total absence of the measured attribute). All standard mathematical operations, including ratios, multiplication, and division, are fully valid.

  • Examples: Height, length, master bedroom volume measured in cubic feet.

• Computational Rule: Mathematical calculations and statistical modeling can only be executed on data measured at the Interval or Ratio scales.

Frequency Tables and Categorical Data Analysis

• Frequency ($f$): The raw count of observations falling into a specific data category.

• Relative Frequency ($RF$): The proportion or percentage of total observations belonging to a category, calculated as:   $RF = \frac{f}{N}$   where $N$ is the total sample size.

• Cumulative Relative Frequency ($CRF$): The running total of relative frequencies accumulated sequentially across ordered categories.

• Categorical Survey Example ($N = 25$ total respondents):

  • Penguin: $f = 5$, $RF = \frac{5}{25} = 0.20 = 20\%$, $CRF = 20\%$

  • Elephant: $f = 4$, $RF = \frac{4}{25} = 0.16 = 16\%$, $CRF = 36\%$

  • Panda: $f = 4$, $RF = \frac{4}{25} = 0.16 = 16\%$, $CRF = 52\%$

  • Giraffe: $f = 2$, $RF = \frac{2}{25} = 0.08 = 8\%$, $CRF = 60\%$

  • Category 5: $f = 3$, $RF = \frac{3}{25} = 0.12 = 12\%$, $CRF = 72\%$

  • Eagle: $f = 3$, $RF = \frac{3}{25} = 0.12 = 12\%$, $CRF = 84\%$

  • Lion: $f = 4$ (or corrected count of $3$ lions yielding $12\%$), reaching a final $CRF = \frac{25}{25} = 100\%$

• Summation Property: The sum of all individual relative frequencies must equal $100\%$ ($1.00$), subject to minor rounding deviations (e.g., $99.9\%$ or $100.1\%$).

Grouped Numerical Data and Interval Analysis

• Grouped Data Structure: Continuous numerical data organized into adjacent intervals (bins) to evaluate distributions.

• NBA Player Height Dataset ($N = 167$ players, $2024$\text{--}$2025$ season):

  • Continuous bin boundaries: $61.5\,\text{inches}$, $63.5\,\text{inches}$, $65.5\,\text{inches}$, $67.5\,\text{inches}$, $69.5\,\text{inches}$, $71.5\,\text{inches}$.

  • Boundary Convention: Observations falling precisely on a bin boundary are assigned systematically to the higher adjacent interval.

• Interval Calculations:

  • Players between $65.5\,\text{inches}$ and $71.5\,\text{inches}$:

  • Frequency sum: $39 + 28 + 17 = 84$ players.

  • Percentage: $\frac{84}{167} \approx 50.3\%$.

  • Alternative method: Subtract cumulative relative frequencies ($0.952 - 0.448 = 0.504 \approx 50.4\%$ due to intermediate decimal rounding).

  • Players below $63.5\,\text{inches}$: $30$ players.

  • Players at least $63.5\,\text{inches}$: $167 - 30 = 137$ players, yielding $\frac{137}{167} \approx 82.0\%$.

  • Most Common Interval: The interval $63.5\,\text{inches}$ to $65.5\,\text{inches}$ contains the highest frequency and relative frequency.

  • Median Threshold: The cumulative relative frequency crosses the $0.5$ ($50\%$ ) mark within the $65.5\,\text{inches}$ to $67.5\,\text{inches}$ interval.

  • Evaluation of Categorical Claims: Stating that "at least $10$ players are at least $71\,\text{inches}$ tall" cannot be verified from grouped tables unless bin boundaries align directly with the threshold value.

Experimental Design and Research Ethics

• Explanatory Variable: The variable manipulated or categorized by the researcher to explain changes in an outcome (analogous to an independent variable).

• Response Variable: The outcome variable measured to observe the effect of the explanatory variable (analogous to a dependent variable). Experiments can evaluate multiple response variables simultaneously.

• Lurking Variables: Unmeasured variables that influence both the explanatory and response variables, giving rise to confounding.

• Control Groups: A baseline group in an experimental setting that receives no active treatment, used to isolate the true effect of an intervention.

• Ethical Data Collection: Human subjects research requires informed consent, adherence to safety protocols, and respect for subject autonomy (e.g., obtaining voluntary consent before drawing blood samples to test for blood type $A$).