Measures of Variability, Box Plots, Variance, and Bivariate Associations

Review of Central Tendency, Percentiles, and Measures of Variability

• Measures of Central Tendency Overview:

  • Mode: The most frequently occurring score in a dataset.

  • Median: The exact middle score dividing an ordered dataset into two equal halves ($50\text{th}$ percentile).

  • Mean: The arithmetic average calculated by summing all observations and dividing by total sample size ($n$).

  • Trade-offs: Each measure possesses distinct advantages and disadvantages based on measurement scale and distribution properties.

  • Skew Indicator: Misalignment among mode, median, and mean provides direct evidence of asymmetrical distribution skew.

• Percentiles:

  • Percentiles describe an observation's relative standing within a data distribution.

  • The median corresponds precisely to the $50\text{th}$ percentile, splitting the distribution so $50\%$ of scores fall below it and $50\%$ fall above it.

• Measures of Variability:

  • Essential for understanding how data points spread around a central value.

  • Crucial for statistical inference to evaluate whether an observed score is typical or unusual.

  • Primary metrics include range, quartiles, interquartile range (IQR), variance, and standard deviation.

Quartiles and the Interquartile Range (IQR)

• Division and Properties of Quartiles:

  • Continuous and ordered distributions can be divided into four equal subsets called quartiles.

  • Each quartile subset contains exactly $\frac{1}{4}$ or $25\%$ of the sorted data.

  • $1\text{st Quartile } (Q_1)$: The $25\text{th}$ percentile mark; $25\%$ of sorted values fall at or below this point.

  • $2\text{nd Quartile } (Q_2)$: The $50\text{th}$ percentile mark, corresponding exactly to the dataset median.

  • $3\text{rd Quartile } (Q_3)$: The $75\text{th}$ percentile mark; $75\%$ of sorted values fall at or below this point.

• Interquartile Range (IQR) Definition and Formulas:

  • Definition: The range spanned by the middle $50\%$ of scores in a distribution.

  • Mathematical formula:     $IQR = Q_3 - Q_1$

  • Percentile formula:     $IQR = 75\text{th percentile} - 25\text{th percentile}$

  • Key Property: The IQR always contains approximately $50\%$ of the data points regardless of distribution shape or skewness.

  • Robustness: The IQR is highly robust to extreme scores because it ignores the lowest $25\%$ and highest $25\%$ of observations, making it insensitive to minimum and maximum value changes.

• Discrete Dataset Example Calculation:

  • Unsorted data: [20, 10, 8, 202, 18, 40, 102, 9, 101, 100, 21, 22, 88, 7, 41, 42, 33] ($n = 17$ observations).

  • Sorted data (Min to Max): 7, 8, 9, 10, 18, 20, 21, 22, 33, 40, 41, 42, 88, 100, 101, 102, 202.

  • Five key summary positions:

  • Minimum: $7$

  • $1\text{st Quartile } (Q_1) = 18$ ($25\%$ threshold)

  • $2\text{nd Quartile } (Q_2) = 33$ ($50\%$ threshold / Median)

  • $3\text{rd Quartile } (Q_3) = 88$ ($75\%$ threshold)

  • Maximum: $202$

  • IQR calculation:     $IQR = 88 - 18 = 70$

• Cumulative Proportion Plot Example (Binocular Rivalry):

  • Investigates perceptual state duration in young adults during a binocular rivalry task.

• Empirical percentile cutoffs derived from plot:

  • $25\text{th}$ percentile duration cutoff $\approx 0.80\,\text{s}$

  • $75\text{th}$ percentile duration cutoff $\approx 1.90\,\text{s}$

• Interquartile Range calculation:     $IQR = 1.90\,\text{s} - 0.80\,\text{s} = 1.10\,\text{s}$

Visualizing Distributions: Histograms, Box Plots, and Tukey's 5-Point Summary

• Tukey's 5-Point Data Summary:

  • Provides a concise numeric snapshot of distribution location and spread.

  • Consists of five sequential markers:

  1. Minimum value

  2. $1\text{st Quartile } (Q_1 / 25\text{th percentile})$

  3. $2\text{nd Quartile } (Q_2 / 50\text{th percentile / Median})$

  4. $3\text{rd Quartile } (Q_3 / 75\text{th percentile})$

  5. Maximum value

• Anatomy and Construction of a Box Plot (Box-and-Whisker Plot):

  • Central Box: Drawn between $Q_1$ ($25\text{th}$ percentile) and $Q_3$ ($75\text{th}$ percentile), spanning the $IQR$.

  • Median Line: A prominent line drawn inside the box at $Q_2$.

  • Whiskers Boundary Rule: Whiskers extend outward from box edges to the most extreme data points that fall within $1.5 \times IQR$ from box edges.

  • Lower whisker limit: $Q_1 - (1.5 \times IQR)$

  • Upper whisker limit: $Q_3 + (1.5 \times IQR)$

  • Outlier Rule: Individual observations falling more than $1.5 \times IQR$ beyond upper or lower box edges are classified as outliers and plotted as standalone points.

• Case Study: Body Weight of 146 Cats (fatCats dataset):

  • Sample size: $n = 146$ cats.

  • Median body weight: $2700\,\text{g}$

  • $1\text{st Quartile } (Q_1) = 2300\,\text{g}$

  • $3\text{rd Quartile } (Q_3) = 3100\,\text{g}$

  • $IQR = 3100\,\text{g} - 2300\,\text{g} = 800\,\text{g}$

  • $1.5 \times IQR = 1.5 \times 800\,\text{g} = 1200\,\text{g}$

  • Lower whisker threshold: $2300\,\text{g} - 1200\,\text{g} = 1100\,\text{g}$ (whisker extends to lowest observation within threshold, $2000\,\text{g}$).

  • Upper whisker threshold: $3100\,\text{g} + 1200\,\text{g} = 4300\,\text{g}$ (whisker extends to highest non-outlier observation within threshold, $3900\,\text{g}$).

  • Identified Outliers: Data observations exceeding $4300\,\text{g}$ (plotted individually at $4500\,\text{g}$ and $4750\,\text{g}$).

  • Full dataset overall range: $[2000\,\text{g}, 4750\,\text{g}]$.

• Case Study: Annual Canadian Lynx Trappings (1821–1934):

  • Historical wildlife record over a 114-year period.

  • Summary metrics:

  • Overall range: $[39, 6991]$ trappings.

  • $1\text{st Quartile } (25\%) = 248$

  • $2\text{nd Quartile } (50\% / \text{Median}) = 771$

  • $3\text{rd Quartile } (75\%) = 2566$

  • $IQR = 2566 - 248 = 2318$

  • Graphically demonstrates a heavy right tail with extreme upper outliers.

• Comparing Visual Capabilities of Histograms and Box Plots:

  • Histograms: Excellent for displaying overall distribution shape, modality, peaks, skew, and full range, but cannot clearly identify exact values for median or IQR.

  • Box Plots: Clearly identify median, quartiles, IQR, and extreme outliers, revealing distribution spread and skew across distinct structural types:

- Symmetrical distribution with moderate IQR.
- Positively skewed distribution with moderate IQR.
- Negatively skewed distribution with moderate IQR.
- Positively skewed distribution with narrow IQR.

• Major Limitation of Box Plots:

  • Box plots are not complete summaries of data.

  • Two separate datasets can produce completely identical box plots and share the exact same Tukey 5-point summary (minimum, $Q_1$, median, $Q_3$, maximum).

  • However, the underlying distributions between those 5 summary points can differ radically (e.g., a uniform flat distribution versus a strongly bimodal distribution with heavy clusters around test scores of $25$ and $75$).

Variance and Standard Deviation

• Measuring Variability Around the Mean:

  • Quantifies how far data points spread from sample mean $\bar{Y}$.

  • Flaw in Simple Average Deviation:

  • Intuitively, one might calculate average deviation:       $\frac{\sum_{i=1}^n (Y_i - \bar{Y})}{n}$

  • Mathematical reality: The sum of simple deviations from the mean is always zero:       $\sum_{i=1}^n (Y_i - \bar{Y}) = 0$

  • Positive deviations (scores above mean) and negative deviations (scores below mean) always cancel each other out completely, regardless of dataset variability.

• Demonstration of Zero Deviation Sum (R Computation):

  • Sample dataset: my.scores = [111, 112, 96, 106, 107, 105, 110, 97, 132, 108] ($n = 10$ observations).

  • Sample mean: $\bar{Y} = 108.4$

  • Deviations from mean ($Y_i - \bar{Y}$): 2.6, 3.6, -12.4, -2.4, -1.4, -3.4, 1.6, -11.4, 23.6, -0.4

  • Sum of deviations:     $2.6 + 3.6 + (-12.4) + (-2.4) + (-1.4) + (-3.4) + 1.6 + (-11.4) + 23.6 + (-0.4) = 0$

• Sample Variance ($s^2$) Solution and Formula:

  • To eliminate cancellation, each deviation is squared prior to summation.

  • Formula for sample variance:     $s^2 = \frac{\sum_{i=1}^n (Y_i - \bar{Y})^2}{n - 1}$

  • Degrees of Freedom: The quantity $(n - 1)$ in the denominator represents degrees of freedom.

  • Relation to Average: Variance is almost equal to average squared deviation; if $n$ were used instead of $(n - 1)$, it would equal average squared deviation.

  • Worked Example using my.scores:

  • Squared deviations: 2.6^2 = 6.76, 3.6^2 = 12.96, (-12.4)^2 = 153.76, (-2.4)^2 = 5.76, (-1.4)^2 = 1.96, (-3.4)^2 = 11.56, 1.6^2 = 2.56, (-11.4)^2 = 129.96, 23.6^2 = 556.96, (-0.4)^2 = 0.16.

  • Sum of squared deviations: $882.4$

  • Variance calculation:       $s^2 = \frac{882.4}{10 - 1} = \frac{882.4}{9} = 98.04$

• Mathematical Rationale for $(n - 1)$ Denominator (Unbiased Estimation):

  • Dividing by $n$ makes $s^2$ a biased estimator that systematically underestimates true population variance \sigma^2$.\n - Dividing by (n - 1)$makes$s^2$an **unbiased estimator** of population variance$\sigma^2$.

  • Over repeated sampling, the long-run expected mean value of $s^2$ calculated with $(n - 1)$ equals \sigma^2$.\n\n- Sample Standard Deviation (s):\n - Drawbacks of Variance:\n 1. Units are squared relative to original measurements (e.g., if Y_i$is in$\text{cm}$,$s^2$is in$\text{cm}^2).\n 2. Numerical values are often disproportionately larger than original deviation magnitudes.\n - Definition and formula for standard deviation:\n    s = \sqrt{s^2}\n - Advantages: Standard deviation returns variability to original measurement units and provides a scale consistent with deviations around the mean.\n - Example: Test score distribution with quartiles (90, 100, 110):\n - Variance: s^2 = 235.81\n - Standard deviation: s = 15.36\n\n- Sensitivity to Extreme Outliers:\n - Both variance (s^2$) and standard deviation ($s) are highly sensitive to extreme outliers.\n - Example: M&M Contest Guessing Dataset:\n - Distribution with outlier: s^2 = 19833$,$s = 141\n - Distribution without outlier: s^2 = 7661$,$s = 87.5\n - Inclusion of a single extreme guess dramatically inflates both variance and standard deviation.\n\n- Limitations of Mean and Standard Deviation:\n - Distribution shapes are **not completely determined** by mean and standard deviation.\n - Example 1: Data Set 1 and Data Set 2 share identical mean (\bar{Y} = 100$) and standard deviation ($s = 10), but Data Set 2 is symmetric bell-shaped while Data Set 1 is strongly positively skewed.\n - Example 2: Class 1 and Class 2 test scores both feature mean \bar{Y} = 65$and standard deviation$s = 5, but differ fundamentally in **kurtosis**:\n - Class 1 exhibits **high kurtosis** (leptokurtic: tall central peak with heavy tails containing extreme high/low marks).\n - Class 2 exhibits **low kurtosis** (platykurtic: flat top with tightly bounded mark boundaries).\n\n# Summary of Variability Measures\n\n- Need for Variability Measures:\n - Required to comprehensively describe statistical distributions.\n - Essential for determining whether a specific observed score is statistically unusual.\n\n- Comparison of Common Spread Measures:\n - **Range**: Difference between max and min values; simplest measure, highly sensitive to extreme outliers.\n - **Interquartile Range (IQR)**: Middle 50\%$span ($Q_3 - Q_1); most robust measure against extreme outliers.\n - **Variance (s^2$) and Standard Deviation ($s)**: Square and root measures of mean deviation; sensitive to outliers, but algebraically convenient for statistical formulas and foundational to inferential hypothesis tests.\n\n# Describing Associations Between Two Variables: Correlation and Regression\n\n- Shift to Bivariate Analysis:\n - Univariate plots (box plots, histograms) show distributions of individual variables in isolation, but cannot reveal how two variables interact or co-vary.\n\n- Classification of Bivariate Trends:\n - **Linear relationship**: Best fitted by a straight line.\n - **Curvilinear relationship**: Best fitted by a curved non-linear line.\n - **Monotonic trend**: As X$increases,$Y continuously increases or continuously decreases without changing direction (can be linear or curvilinear).\n - **Non-monotonic trend**: As X$increases,$Y changes direction at least once (e.g., inverted U-shape).\n\n\n\n- Case Study 1: Major League Baseball (MLB) Team Wins & Team OPS (Data from Sept 16, 2024):\n - On Base Plus Slugging (OPS): Measures a team's combined ability to get on base and hit for power to score runs.\n - Empirical question: Is Team OPS associated with winning more games?\n - MLB Sample Data Table (30 Teams):\n - Arizona Diamondbacks (`ARI`): 83\,\text{Wins}$,$0.557\,\text{wPCT}$,$0.778\,\text{OPS}\n - Atlanta Braves (`ATL`): 81\,\text{Wins}$,$0.540\,\text{wPCT}$,$0.715\,\text{OPS}\n - Baltimore Orioles (`BAL`): 84\,\text{Wins}$,$0.560\,\text{wPCT}$,$0.751\,\text{OPS}\n - Chicago White Sox (`CHW`): 35\,\text{Wins}$,$0.233\,\text{wPCT}$,$0.619\,\text{OPS}\n - Philadelphia Phillies (`PHI`): 90\,\text{Wins}$,$0.604\,\text{wPCT}$,$0.751\,\text{OPS}\n - Univariate vs. Bivariate Representations:\n - Side-by-side box plots show spread of Wins and OPS individually, but fail to demonstrate their linkage.\n - Scatter plot with linear regression line (r = 0.8) reveals a strong, positive, linear relationship between Team OPS and Team Wins.\n\n\n\n- Case Study 2: Visual Acuity vs. Age in Human Infants:\n - Examines infant visual acuity development across studies by Salomão et al. (47) and Birch (43) (monocular, binocular, preterm-monocular).\n - Y-axis measures visual acuity in \text{logMAR}$units; X-axis measures infant age in weeks (from birth to$160\,\text{weeks}).\n - Demonstrates a strong **curvilinear monotonic** trend (rapid initial decline in \text{logMAR} representing steep visual acuity improvement, gradually leveling off).\n - Empirical measurement methods:\n - Preferential looking / Teller acuity cards.\n - Visual Evoked Potentials (VEP): EEG electrodes attached to scalp over visual cortex measure electrical brain responses to visual stimuli on a monitor screen.\n\n- Case Study 3: Brightness Discrimination Thresholds (König, 1888):\n - Investigates differential brightness threshold sensitivity \frac{\delta r}{r}$across varying log background intensities$\text{LOG } r$(range$-2$to$5).\n - Reveals a **curvilinear monotonic** trend that drops sharply at low background intensity levels before flattening; visual theories must account for this threshold-to-background relationship.\n\n- Case Study 4: Global Average Temperature (1850–2023):\n - Prepared by Berkeley Earth (land data) and UK Hadley Centre (ocean data), measuring global temperature anomalies relative to 1850–1900 average.\n - Plot includes vertical error bars indicating 95\% confidence intervals.\n - Displays a clear **curvilinear monotonic** upward trend in global temperature anomalies, accelerating significantly over recent decades to reach over +1.4\,^\circ\text{C}$$.