Which Of The Following Is Not A Possible R Value

Understanding the Boundaries of Pearson's Correlation Coefficient (r)

The Pearson correlation coefficient, denoted as r, is one of the most fundamental and widely used statistics in research, data analysis, and decision-making. It quantifies the strength and direction of a linear relationship between two continuous variables. Its value is a single number that tells a compelling story: how closely do points on a scatterplot cluster around a straight line? However, this powerful metric operates within strict, immutable mathematical boundaries. Any reported value falling outside these boundaries is not just improbable; it is mathematically impossible. This article will definitively establish the possible range for an r-value, explain the "why" behind these limits, and equip you to identify any erroneous calculation or reporting.

The Golden Rule: The Absolute Ceiling of |r| = 1

The single most important rule to remember is that the absolute value of Pearson's r cannot exceed 1. Therefore, the complete, mathematically permissible range for r is:

-1 ≤ r ≤ 1

This means:

• r = 1 represents a perfect positive linear correlation. As one variable increases, the other increases in a perfectly predictable, linear fashion. All data points lie exactly on an upward-sloping straight line.
• r = -1 represents a perfect negative linear correlation. As one variable increases, the other decreases in a perfectly predictable, linear fashion. All data points lie exactly on a downward-sloping straight line.
• r = 0 indicates no linear correlation. There is no discernible linear trend; the points are scattered randomly. (Crucially, this does not rule out a strong non-linear relationship).

Any r value reported as greater than 1 (e.g., 1.2, 1.5, 2.0) or less than -1 (e.g., -1.3, -1.8) is not a possible r-value. It is a clear signal of a fundamental error in calculation, data entry, or conceptual application.

The Scientific Explanation: Why |r| Cannot Exceed 1

The constraint of |r| ≤ 1 is not an arbitrary statistical convention; it is a direct mathematical consequence of how r is defined. There are several ways to understand this bound.

1. The Covariance and Standard Deviations Formula

The most common formula for Pearson's r is: r = cov(X, Y) / (σ_X * σ_Y) Where:

• cov(X, Y) is the covariance between variables X and Y.
• σ_X and σ_Y are the standard deviations of X and Y, respectively.

The covariance can be positive, negative, or zero. However, its absolute magnitude is fundamentally limited by the spread (variance) of the two datasets. The denominator, the product of the standard deviations (σ_X * σ_Y), represents the maximum possible absolute covariance that could exist if the relationship were perfectly linear. This is a manifestation of the Cauchy-Schwarz inequality, a cornerstone of linear algebra, which mathematically guarantees that the absolute value of a dot product (analogous to covariance) cannot exceed the product of the vector magnitudes (analogous to standard deviations). Thus, the fraction cov(X, Y) / (σ_X * σ_Y) must always fall between -1 and 1, inclusive.

2. The Geometric Interpretation: The Cosine of an Angle

A profoundly intuitive way to grasp this limit is through geometry. Imagine plotting your two variables, X and Y, as vectors in an n-dimensional space (where n is your number of data points). Each variable is a vector from the origin to the point defined by its n values.

Pearson's r is, in fact, mathematically equivalent to the cosine of the angle (θ) between these two centered vectors. r = cos(θ)

From basic trigonometry, we know that the cosine of any angle must always be between -1 and 1.

• When θ = 0°, cos(0°) = 1. The vectors point in the exact same direction → perfect positive correlation.
• When θ = 180°, cos(180°) = -1. The vectors point in exact opposite directions → perfect negative correlation.
• When θ = 90°, cos(90°) = 0. The vectors are perpendicular → no linear correlation.

An angle cannot have a cosine of 1.5 or -2.0. Therefore, r cannot have those values either. This geometric view makes the impossibility of |r| > 1 visually and logically irrefutable.

3. The Squared Correlation (R²) and Its Bounds

The square of the Pearson correlation, R² (the coefficient of determination), represents the proportion of variance in one variable explained by the other. By definition, a proportion must be between 0 and 1 (or 0% and 100%). Therefore: 0 ≤ R² ≤ 1

Since r can be negative, R² = r². If r were 1.2, then R² would be 1.44, or 144% of the variance explained—a logical absurdity. You cannot explain more than 100% of the variance in a dataset using a linear model of another single variable. This provides a quick sanity check: if squaring your r gives a number greater than 1, your original r was impossible.

Common Pitfalls and Misinterpretations

Understanding what isn't possible is as crucial as knowing what is. Here are frequent sources of confusion:

• Confusing r with R-squared: This is the most common error. R-squared ranges from 0 to 1. If you see a value like 0.85, that could be either a strong r (0.85 or -0.85) or an R-squared value. Always check the context and notation. An r of 0.85 and an R² of 0.85 mean very different things.
• Misreading Software Output: Some statistical software packages label columns ambiguously. A column titled "R" or "R Square" must be checked. A value of 0.92 in an

"R" column is a valid r (strong positive correlation), but the same value in an "R Square" column would be impossible for r itself.

• Ignoring the Sign: The correlation coefficient can be negative. A value of -0.8 indicates a strong negative linear relationship, not an error. The magnitude (0.8) tells you the strength; the sign tells you the direction.

• Assuming Correlation Implies Causation: This is a separate but critical point. A high |r| only tells you that two variables move together linearly. It does not tell you that one causes the other. Spurious correlations are common and can be misleading.

• Over-Interpreting Near-Bound Values: An r of 0.99 is not "twice as strong" as an r of 0.5. The relationship between r and the strength of a linear association is not linear. Small differences near the bounds (like 0.98 vs. 0.99) can be statistically significant but practically negligible.

• Forgetting the Assumptions: Pearson's r measures linear correlation. If the true relationship is strongly non-linear, r can be close to zero even if the variables are perfectly related in a non-linear way. Always visualize your data.

Conclusion: The Bounds as a Built-in Sanity Check

The fact that Pearson's correlation coefficient is mathematically constrained to the interval [-1, 1] is not a limitation, but a fundamental feature. It is a consequence of the way the statistic is defined, rooted in the Cauchy-Schwarz inequality and elegantly interpreted as the cosine of an angle between vectors. This bound provides a powerful, built-in sanity check for your analysis. If your calculated value falls outside this range, it is a clear signal of a computational error, a data problem, or a misinterpretation of the output. By understanding the geometric and algebraic foundations of this constraint, you can use it as a tool to validate your work and avoid common pitfalls, ensuring that your conclusions about the relationships in your data are both accurate and meaningful.