Statistics: SD, chi-square, t-test, ANOVA MCQs With Answer

Introduction

Understanding statistics is essential in Modern Pharmaceutics for designing robust experiments, interpreting data, and making evidence-based decisions. This MCQ set focuses on core tools used by M. Pharm students and researchers: standard deviation (SD), chi-square tests, t-tests, and ANOVA. You will reinforce concepts such as variability, assumptions underlying parametric and nonparametric tests, degrees of freedom, effect sizes, post-hoc analyses, and the interpretation of p-values. Realistic pharmaceutical contexts—like tablet weight variation, dissolution testing, stability studies, and bioequivalence—are embedded to help bridge theory and practice. Each question is crafted to deepen understanding of when and how to apply these tests correctly, and to avoid common pitfalls in pharmaceutics research.

Q1. Which statistic best quantifies the average spread of observations around the mean in the original measurement units?

• Standard deviation
• Standard error of the mean
• Variance
• Coefficient of correlation

Correct Answer: Standard deviation

Q2. For a sample of size n, which is the correct formula for the sample standard deviation (s)?

• s = √[Σ(xi − x̄)² / n]
• s = √[Σ(xi − x̄)² / (n − 1)]
• s = Σ(xi − x̄) / (n − 1)
• s = √[Σ(xi) / (n − 1)]

Correct Answer: s = √[Σ(xi − x̄)² / (n − 1)]

Q3. Which is NOT an assumption of the one-sample t-test?

• The data are approximately normally distributed (or n is sufficiently large)
• The population variance is known
• Observations are independent and randomly sampled
• The variable is measured on a continuous (interval/ratio) scale

Correct Answer: The population variance is known

Q4. In an independent-samples t-test assuming equal variances with n1 = 12 and n2 = 15, the degrees of freedom (df) are:

• 26
• 25
• 24
• 27

Correct Answer: 25

Q5. Welch’s t-test is most appropriate when:

• Group variances are unequal (heteroscedastic), regardless of equal or unequal sample sizes
• The data are severely non-normal
• More than two groups are compared
• Data are paired from the same subjects

Correct Answer: Group variances are unequal (heteroscedastic), regardless of equal or unequal sample sizes

Q6. For a chi-square test of independence to be valid in a contingency table, a common rule of thumb is:

• Observed counts must all be ≥ 5
• Expected counts should generally be ≥ 5 in each cell
• Expected counts can be zero in up to 10% of cells
• Row totals must be equal across categories

Correct Answer: Expected counts should generally be ≥ 5 in each cell

Q7. A chi-square goodness-of-fit test is appropriate when:

• Comparing means across three or more independent groups
• Testing variance equality across groups
• Comparing an observed frequency distribution of a single categorical variable to a theoretical distribution
• Assessing correlation between two continuous variables

Correct Answer: Comparing an observed frequency distribution of a single categorical variable to a theoretical distribution

Q8. What are the degrees of freedom for a chi-square test of independence in a 3 × 4 contingency table?

• 12
• 7
• 6
• 9

Correct Answer: 6

Q9. In one-way ANOVA, the F-statistic is the ratio of:

• Within-group variance to total variance
• Between-group mean square to within-group mean square
• Total variance to between-group variance
• Sum of squares within to sum of squares between

Correct Answer: Between-group mean square to within-group mean square

Q10. Which is NOT an assumption of a standard one-way ANOVA on independent groups?

• Independence of observations
• Normality of residuals within each group
• Homogeneity of variances across groups
• Sphericity of repeated measures

Correct Answer: Sphericity of repeated measures

Q11. After a significant one-way ANOVA with equal sample sizes and homogeneous variances, which post-hoc test is specifically designed for all pairwise comparisons?

• Tukey’s Honestly Significant Difference (HSD)
• Fisher’s Least Significant Difference (LSD) without correction
• Scheffé’s method
• Dunnett’s test

Correct Answer: Tukey’s Honestly Significant Difference (HSD)

Q12. Which effect size metric expresses the standardized difference between two means (e.g., treatment vs. control)?

• Eta-squared (η²)
• Partial eta-squared (ηp²)
• Cohen’s d
• Phi coefficient (φ)

Correct Answer: Cohen’s d

Q13. In one-way ANOVA, which effect size quantifies the proportion of total variance explained by the factor?

• Cohen’s d
• Eta-squared (η²)
• Hedges’ g
• Glass’s Δ

Correct Answer: Eta-squared (η²)

Q14. In bioequivalence testing for two formulations, the usual statistical framework is:

• A single two-sided t-test for mean difference equal to zero
• Two one-sided t-tests (TOST) against predefined equivalence margins
• A chi-square test of independence
• Only an ANOVA F-test without pairwise comparisons

Correct Answer: Two one-sided t-tests (TOST) against predefined equivalence margins

Q15. Which statement about the standard error of the mean (SEM) is TRUE?

• SEM equals SD for any sample
• SEM increases as sample size increases
• SEM is SD divided by the square root of sample size
• SEM is unaffected by sample size

Correct Answer: SEM is SD divided by the square root of sample size

Q16. Which scenario justifies using a paired t-test in pharmaceutics?

• Comparing dissolution rates between three independent formulations
• Comparing assay values of tablets from two different batches
• Comparing pre- and post-stability assay values of the same batches stored for 3 months
• Comparing content uniformity across four manufacturing lines

Correct Answer: Comparing pre- and post-stability assay values of the same batches stored for 3 months

Q17. In a one-way ANOVA with k = 4 groups and total sample size N = 40, the degrees of freedom are:

• Between df = 4, Within df = 36
• Between df = 3, Within df = 36
• Between df = 4, Within df = 35
• Between df = 3, Within df = 35

Correct Answer: Between df = 3, Within df = 36

Q18. If ANOVA assumptions are violated due to non-normality or outliers and you wish to compare medians across more than two independent groups, which test is most suitable?

• Kruskal–Wallis test
• Mann–Whitney U test
• Wilcoxon signed-rank test
• Fisher’s exact test

Correct Answer: Kruskal–Wallis test

Q19. Yates’ continuity correction is most commonly applied to the chi-square test when:

• The table is 3 × 3 with large expected counts
• The table is 2 × 2 with small sample sizes
• Expected counts exceed 10 in every cell
• Data are continuous and normally distributed

Correct Answer: The table is 2 × 2 with small sample sizes

Q20. In one-way ANOVA at α = 0.05, obtaining p = 0.03 implies:

• Fail to reject the null; all group means are equal
• Reject the null; at least one group mean differs from another
• All pairwise group differences are significant
• The effect size is large

Correct Answer: Reject the null; at least one group mean differs from another

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