This set of 60 MCQs covers the fundamentals of correlation and covariance, including types like Pearson and Spearman, their calculations, interpretations, and applications in data analysis. Ideal for students and professionals exploring statistical relationships between variables.
1. What does covariance measure between two variables?
✅ Correct Answer: b) How they vary together from their means
📝 Explanation:
Covariance indicates the direction of linear relationship: positive if variables move together, negative if oppositely.
2. The Pearson correlation coefficient assumes which type of relationship?
✅ Correct Answer: c) Linear
📝 Explanation:
Pearson's r measures the strength and direction of the linear association between two continuous variables.
3. What is the range of values for the correlation coefficient?
✅ Correct Answer: b) -1 to 1
📝 Explanation:
Values close to 1 or -1 indicate strong positive or negative correlation; 0 indicates no linear relationship.
4. Covariance can be positive, negative, or zero, but it is not standardized. What standardizes it to create correlation?
✅ Correct Answer: a) Dividing by the product of standard deviations
📝 Explanation:
Pearson's r = Cov(X,Y) / (SD_X * SD_Y), making it unitless and comparable.
5. Which correlation measure is non-parametric and suitable for ordinal data?
✅ Correct Answer: b) Spearman
📝 Explanation:
Spearman's rank correlation assesses monotonic relationships without assuming normality.
6. A covariance matrix is a square matrix where the diagonal elements represent
✅ Correct Answer: a) Variances of variables
📝 Explanation:
Off-diagonals show pairwise covariances; diagonals are variances (Cov(X,X) = Var(X)).
7. What does a correlation of -0.85 indicate?
✅ Correct Answer: b) Strong negative relationship
📝 Explanation:
Values between -0.7 and -1 suggest a strong inverse linear association.
8. Kendall's tau measures the strength of
✅ Correct Answer: b) Monotonic association
📝 Explanation:
It counts concordant and discordant pairs, suitable for small samples or ties.
9. The formula for sample covariance is
✅ Correct Answer: b) Σ(X_i - x̄)(Y_i - ȳ) / (n-1)
📝 Explanation:
Uses n-1 for unbiased estimate; x̄ and ȳ are sample means.
10. Correlation does not imply
✅ Correct Answer: b) Causation
📝 Explanation:
Spurious correlations can occur without causal links; further analysis needed.
11. Which is affected by outliers more: covariance or correlation?
✅ Correct Answer: b) Covariance
📝 Explanation:
Covariance has units and scales with data; correlation is standardized and more robust.
12. Partial correlation measures the relationship between two variables while controlling for
✅ Correct Answer: b) One or more other variables
📝 Explanation:
It removes the effect of confounders to isolate direct association.
13. In a scatter plot, a tight cluster around the line y = x indicates
✅ Correct Answer: b) r ≈ 1
📝 Explanation:
Points aligned with positive slope show strong positive linear correlation.
14. The point-biserial correlation is used for
✅ Correct Answer: b) One continuous and one binary variable
📝 Explanation:
It's a special case of Pearson for dichotomous predictors.
15. What is the covariance of a variable with itself?
✅ Correct Answer: c) Variance
📝 Explanation:
Cov(X,X) = Var(X), the spread of the variable.
16. Spearman's rho is calculated using
✅ Correct Answer: b) Ranked data
📝 Explanation:
It applies Pearson correlation to ranks, handling non-normal data.
17. A correlation matrix diagonal is always
✅ Correct Answer: a) 1
📝 Explanation:
Correlation of a variable with itself is perfect (r=1).
18. Polychoric correlation is appropriate for
✅ Correct Answer: b) Ordinal variables assuming latent continuity
📝 Explanation:
It estimates correlation from underlying normal variables.
19. If two variables have zero covariance, their correlation is
✅ Correct Answer: b) 0
📝 Explanation:
No linear co-variation implies uncorrelated variables.
20. The sign of covariance matches the sign of
✅ Correct Answer: a) Correlation
📝 Explanation:
Both indicate direction: positive or negative association.
21. Biserial correlation assumes the binary variable is
✅ Correct Answer: b) From an underlying continuous distribution
📝 Explanation:
Used when the dichotomy is artificial, like pass/fail.
22. In hypothesis testing for correlation, the null hypothesis is usually
✅ Correct Answer: a) ρ = 0
📝 Explanation:
Tests if the true population correlation is zero (no association).
23. Tetrachoric correlation is for
✅ Correct Answer: b) Two binary variables
📝 Explanation:
Assumes underlying bivariate normal distribution.
24. Covariance is sensitive to
✅ Correct Answer: a) Units of measurement
📝 Explanation:
Changing units (e.g., cm to m) scales covariance, unlike correlation.
25. Phi coefficient measures correlation for
✅ Correct Answer: b) Two binary variables
📝 Explanation:
It's Pearson's r applied to 2x2 contingency tables.
26. A high positive covariance means variables tend to be
✅ Correct Answer: a) Both high or both low
📝 Explanation:
They deviate from means in the same direction.
27. Multiple correlation coefficient R measures
✅ Correct Answer: b) Relationship between one variable and multiple others
📝 Explanation:
R is the correlation between observed and predicted values in regression.
28. Intraclass correlation assesses
✅ Correct Answer: b) Within-group reliability
📝 Explanation:
Used for clustered data, like inter-rater agreement.
29. If X and Y are independent, their covariance is
✅ Correct Answer: c) Zero
📝 Explanation:
Independence implies uncorrelatedness (for linear relationships).
30. Cramér's V is a measure of association for
✅ Correct Answer: b) Nominal categorical variables
📝 Explanation:
Derived from chi-square, ranges 0-1 for contingency tables.
31. The sample correlation formula uses
✅ Correct Answer: a) n in denominator
📝 Explanation:
For r, it's Σ deviations product divided by (n * SD_X * SD_Y), but consistent with covariance's n-1 adjustment.
32. Canonical correlation analyzes relationships between
✅ Correct Answer: a) Two sets of variables
📝 Explanation:
Finds linear combinations maximizing correlation between groups.
33. A correlation of 0 does not mean
✅ Correct Answer: b) No relationship at all
📝 Explanation:
Non-linear relationships (e.g., quadratic) can exist with r=0.
34. Variance-covariance matrix is used in
✅ Correct Answer: a) PCA for dimensionality reduction
📝 Explanation:
Eigen decomposition reveals principal components.
35. Kendall's tau-b accounts for
✅ Correct Answer: a) Ties in data
📝 Explanation:
Adjusts for tied ranks in ordinal comparisons.
36. Negative covariance indicates
✅ Correct Answer: b) Variables move in opposite directions
📝 Explanation:
One increases as the other decreases from means.
37. Theil's U is an asymmetric measure for
✅ Correct Answer: a) Categorical prediction uncertainty
📝 Explanation:
Measures how one variable reduces uncertainty in another.
38. In time series, autocorrelation is
✅ Correct Answer: a) Covariance with lagged self
📝 Explanation:
Normalized to detect patterns like seasonality.
39. Pearson correlation requires
✅ Correct Answer: a) Linearity and homoscedasticity
📝 Explanation:
Assumes bivariate normality for significance tests.
40. Cross-correlation measures
✅ Correct Answer: a) Similarity between two series at lags
📝 Explanation:
Used in signal processing for time shifts.
41. A value of covariance close to zero suggests
✅ Correct Answer: b) Little linear co-variation
📝 Explanation:
But magnitude depends on scales; check correlation for standardization.
42. Sommer's d is a correction for
✅ Correct Answer: a) Ties in Kendall's tau
📝 Explanation:
Adjusts tau-a for tied observations.
43. In multivariate analysis, the correlation between residuals checks for
✅ Correct Answer: b) Independence assumptions
📝 Explanation:
Non-zero residual correlations violate model assumptions.
44. Covariance in portfolio theory measures
✅ Correct Answer: b) Asset co-movement
📝 Explanation:
Low or negative covariance reduces overall portfolio risk.
45. Gamma coefficient is for
✅ Correct Answer: a) Ordinal data, ignoring ties
📝 Explanation:
Proportion of concordant minus discordant pairs.
46. The determinant of a correlation matrix indicates
✅ Correct Answer: a) Multicollinearity severity
📝 Explanation:
Near zero suggests singular matrix, high redundancy.
47. Spearman correlation equals Pearson when data is
✅ Correct Answer: c) Monotonic
📝 Explanation:
Ranks preserve monotonic relationships exactly.
48. Zero correlation can occur with
✅ Correct Answer: c) Quadratic relationship
📝 Explanation:
Curvilinear patterns cancel linear signal.
49. In R, cor() function defaults to
✅ Correct Answer: c) Pearson
📝 Explanation:
Use method='spearman' for ranks.
50. Covariance stationarity in time series requires constant
✅ Correct Answer: a) Mean and variance over time
📝 Explanation:
Covariances depend only on lag, not time.
51. Yule's Q measures association in
✅ Correct Answer: a) 2x2 tables
📝 Explanation:
Based on odds ratio for dichotomous variables.
52. Bivariate correlation is
✅ Correct Answer: b) Pairwise relationship
📝 Explanation:
Focuses on two variables at a time.
53. In Excel, CORREL function computes
✅ Correct Answer: b) Pearson correlation
📝 Explanation:
For two ranges of data.
54. Heteroscedasticity affects correlation by
✅ Correct Answer: a) No impact
📝 Explanation:
Correlation is scale-invariant but assumes homoscedasticity for inference.
55. Lambda coefficient is for
✅ Correct Answer: a) Asymmetric nominal association
📝 Explanation:
Reduces prediction error using one variable to predict another.
56. The squared correlation r² represents
✅ Correct Answer: a) Variance explained
📝 Explanation:
Proportion of variance in Y predictable from X.
57. In Python pandas, df.corr() returns
✅ Correct Answer: b) Correlation matrix
📝 Explanation:
Default is Pearson; specify method for others.
58. Concordant pairs in Kendall's tau are those where
✅ Correct Answer: a) Ranks agree in order
📝 Explanation:
Both pairs increase or decrease together.
59. Covariance of standardized variables is
✅ Correct Answer: b) Correlation
📝 Explanation:
Z-scores have SD=1, so Cov(Zx, Zy) = r.
60. Spurious correlation example is
✅ Correct Answer: b) Ice cream sales and drownings
📝 Explanation:
Both correlated with summer heat, not causally.
61. In confirmatory factor analysis, correlations inform
✅ Correct Answer: b) Latent structure
📝 Explanation:
High inter-correlations suggest common factors.
62. The eta coefficient measures
✅ Correct Answer: a) Nominal predictor on continuous outcome
📝 Explanation:
Non-linear analog of point-biserial.
63. Autocovariance at lag 0 is
✅ Correct Answer: a) Variance
📝 Explanation:
Cov(Y_t, Y_t) = Var(Y).
64. Fisher's z-transformation is used to
✅ Correct Answer: a) Stabilize variance of r for inference
📝 Explanation:
z = 0.5 ln((1+r)/(1-r)) for hypothesis tests.
