Which Of The Following Is Not A Possible R Value

Which of the following is not a possible r value? This question frequently appears in statistics textbooks, exam preparation guides, and online quizzes that test understanding of the Pearson correlation coefficient. The letter r represents the strength and direction of a linear relationship between two quantitative variables. While many learners memorize that r must lie between –1 and +1, the underlying reasons are often glossed over. This article unpacks the full meaning of r, explains why every value outside the closed interval [–1, 1] is impossible, and walks through typical multiple‑choice scenarios that ask you to spot the impossible r value. By the end, you will not only be able to answer such questions confidently but also appreciate how the mathematics of correlation safeguards against misinterpretation.

Introduction

The phrase which of the following is not a possible r value serves as both a diagnostic question and a gateway to deeper statistical insight. In educational contexts, instructors use it to assess whether students grasp the fundamental limits of the correlation coefficient. The answer hinges on three core ideas: (1) r is a standardized measure, (2) it is derived from covariance divided by the product of standard deviations, and (3) algebraic constraints force r into the range –1 ≤ r ≤ 1. Recognizing these principles enables you to evaluate any set of answer choices and identify the one that violates the rule.

What is the Correlation Coefficient?

Definition

The Pearson correlation coefficient, denoted r, quantifies the linear association between two variables X and Y. Its formal definition is

[ r = \frac{\displaystyle\sum_{i=1}^{n}(X_i-\bar X)(Y_i-\bar Y)}{\sqrt{\displaystyle\sum_{i=1}^{n}(X_i-\bar X)^2};\sqrt{\displaystyle\sum_{i=1}^{n}(Y_i-\bar Y)^2}} ]

where (\bar X) and (\bar Y) are the sample means. The numerator is the covariance of X and Y, while the denominator normalizes this covariance by the product of the individual standard deviations. Because both numerator and denominator are expressed in the same units, r is dimensionless.

Interpretation

• r = +1 indicates a perfect positive linear relationship (as X increases, Y increases proportionally).
• r = –1 indicates a perfect negative linear relationship (as X increases, Y decreases proportionally).
• r = 0 suggests no linear relationship; however, a non‑zero r does not guarantee causation.

Why the Range Matters

Since r is a ratio of two quantities that share the same scale, its value cannot exceed the absolute magnitude of 1. This constraint emerges from the Cauchy‑Schwarz inequality, a fundamental result in linear algebra that applies to any pair of vectors in Euclidean space. In practical terms, the inequality guarantees that the numerator never outgrows the denominator, preventing r from “blowing up” beyond ±1.

Possible Values of r

Theoretical Limits

• Maximum: r = +1
• Minimum: r = –1

These extremes occur only when all data points lie exactly on a straight line with a positive or negative slope, respectively. In real‑world data, perfect correlation is rare; most observed r values fall somewhere strictly between –1 and +1.

Common Misconceptions

Identifying the Impossible Value

When a question poses which of the following is not a possible r value, the answer set typically includes a mixture of plausible numbers and one outlier. The outlier often violates the –1 ≤ r ≤ 1 rule. Below is a typical example:

1. –0.85
2. 0.00
3. 1.20
4. 0.45

Step‑by‑step analysis

1. Scan each option for magnitude.
2. Discard any value whose absolute value exceeds 1.
3. The remaining numbers (–0.85, 0.00, 0.45) are all within the permissible range.
4. The value 1.20 exceeds the upper bound, making it not a possible r value.

Why 1.20 Is Impossible

If you attempted to compute r from raw data and obtained 1.20, an error must have occurred. Common sources of error include:

• Incorrect formula entry (e.g., forgetting to square the standard deviations).
• Mis‑calculation of means leading to inflated covariance.
• Using unstandardized data where variance differs dramatically between variables.

Correcting these mistakes typically forces the recomputed r back into the valid interval.

Practical Examples

Example 1: Simple Dataset

Consider two variables with the following paired observations:

Calculate r:

1. Compute means: (\bar X = 5), (\bar Y = 8).

2. Compute deviations and products:

   • (2‑5)(5‑8) = (‑3)(‑3) = 9
   • (4‑5)(7‑8) = (‑1)(‑1) = 1
   • (6‑5)(9‑8) = (1)(1) = 1
   • (8‑5)(11‑8) = (3

Continuing the calculation for the dataset:

• The product of deviations for the last pair is ((8-5)(11-8)=3 \times 3 = 9).
• Summing all four products gives (9 + 1 + 1 + 9 = 20).

Next, compute the squared deviations:

• For (X): ((-3)^2 + (-1)^2 + (1)^2 + (3)^2 = 9 + 1 + 1 + 9 = 20).
• For (Y): ((-3)^2 + (-1)^2 + (1)^2 + (3)^2 = 9 + 1 + 1 + 9 = 20).

The denominator of the Pearson formula is (\sqrt{20 \times 20} = \sqrt{400} = 20).

Thus,

[ r = \frac{\text{covariance numerator}}{\text{denominator}} = \frac{20}{20} = 1.00, ]

confirming a perfect positive linear relationship, as expected from the data lying exactly on the line (Y = X + 3).

Example 2: Imperfect Correlation

Now consider a slightly altered set:

Repeating the same steps:

1. Means: (\bar X = 5), (\bar Y = 8.75).

2. Deviations and products:

   • ((2-5)(4-8.75)=(-3)(-4.75)=14.25)
   • ((4-5)(8-8.75)=(-1)(-0.75)=0.75)
   • ((6-5)(10-8.75)=(1)(1.25)=1.25)
   • ((8-5)(13-8.75)=(3)(4.25)=12.75)

   Sum of products = (14.25+0.75+1.25+12.75 = 29.00).

3. Squared deviations:

   • (X): ((-3)^2+(-1)^2+1^2+3^2 = 20) (unchanged).
   • (Y): ((-4.75)^2+(-0.75)^2+(1.25)^2+(4.25)^2 = 22.5625+0.5625+1.5625+18.0625 = 42.75).

4. Denominator: (\sqrt{20 \times 42.75} = \sqrt{855} \approx 29.24).

5. Correlation: [ r = \frac{29.00}{29.24} \approx 0.992. ]

Although the points are still tightly clustered around a line, the slight deviation of the last observation from the perfect pattern reduces (r) just below the theoretical maximum.

Interpreting Values Near the Bounds

• Values close to +1 or –1 indicate that the data points lie near a straight line; the closer the magnitude is to 1, the less scatter there is around that line.
• Values near 0 suggest little to no linear association; however, they do not rule out a strong nonlinear relationship (e.g., a quadratic pattern). - Because the Pearson coefficient is bounded, any claim that a computed (r) exceeds 1 or falls below –1 immediately signals a computational mistake, as discussed earlier.

Conclusion

The Pearson correlation coefficient (r) is mathematically constrained to the interval ([-1,,1]). This bound arises from the Cauchy‑Schwarz inequality applied to

This bound arises from the Cauchy‑Schwarz inequality applied to the vectors of deviations from the mean. Specifically, it ensures that the covariance numerator (sum of cross-deviations) cannot exceed the geometric mean of the sum of squared deviations for (X) and (Y). Mathematically, for any dataset:
[ \left( \sum (x_i - \bar{x})(y_i - \bar{y}) \right)^2 \leq \left( \sum (x_i - \bar{x})^2 \right) \left( \sum (y_i - \bar{y})^2 \right), ]
which directly implies (|r| \leq 1). This property underscores why (r) serves as a standardized metric—its value is independent of the scales of the variables, allowing direct comparison across different datasets.

Key Implications

• Perfect Linearity: (r = \pm 1) occurs if and only if all data points lie exactly on a straight line, with the sign indicating positive or negative slope.
• No Linear Association: (r = 0) suggests no linear trend, but nonlinear relationships (e.g., parabolic or exponential) may still exist.
• Sensitivity to Outliers: Extreme values disproportionately influence (r), as they amplify both numerator and denominator terms. Robust alternatives (e.g., Spearman’s rank correlation) may be preferable in such cases.

Practical Considerations

While (r) quantifies linear strength, it does not:

• Imply causation (e.g., ice cream sales and shark attacks may correlate due to a confounding variable like temperature).
• Capture nonlinear patterns (e.g., a U-shaped relationship yields (r \approx 0)).
• Describe the slope or magnitude of the relationship (e.g., (r = 0.9) for steep vs. shallow lines is indistinguishable without regression analysis).

Final Note

The Pearson correlation coefficient’s bounded nature ((-1 \leq r \leq 1)) is not merely a mathematical curiosity—it is a safeguard against overinterpretation of weak or spurious associations. When used alongside complementary tools like scatter plots and regression diagnostics, (r) remains an indispensable tool for quantifying linear relationships in data. Always contextualize its value within the broader analytical framework to avoid drawing erroneous conclusions.