68 free MCQs on Statistics with worked answers and explanations. Mean, median, mode, standard deviation, and data interpretation
Take the timed Statistics chapterwise test →Below are 68 practice questions on Statistics, sorted Easy → Hard. Tap “Show answer & explanation” under any question to check yourself. Want the full theory first? Read the Statistics notes.
In a perfectly normal (bell-shaped) distribution, mean, median, and mode all coincide at the centre; about 68% of data falls within 1 standard deviation of the mean, 95% within 2, and 99.7% within 3 - the empirical rule used to judge how typical or extreme a value is.
Q1.
The average of a data set is called:
Answer: C. Mean
Why: Mean (arithmetic average) = sum of all values / number of values.
Q2.
The middle value of an ordered data set is:
Answer: B. Median
Why: The median is the middle value when data is arranged in order.
Q3.
The value that appears most often in a data set is:
Answer: C. Mode
Why: Mode is the value(s) that occur most frequently in a data set.
Q4.
Find the mean of 5, 10, 15, 20, 25.
Answer: C. 15
Why: Mean = (5+10+15+20+25)/5 = 75/5 = 15.
Q5.
Find the median of 3, 5, 7, 9, 11.
Answer: B. 7
Why: For 5 values (odd count), median is the 3rd value in order = 7.
Q6.
Range of data 5, 8, 2, 11, 4 is:
Answer: D. 9
Why: Range = maximum - minimum = 11 - 2 = 9.
Q7.
The mode of 4, 7, 7, 8, 9, 9, 9 is:
Answer: C. 9
Why: 9 appears 3 times (most frequently), so mode = 9.
Q8.
Find the median of 2, 4, 6, 8 (even count).
Answer: B. 5
Why: Even number of values: median = average of 2nd and 3rd = (4+6)/2 = 5.
Q9.
Data presented as a bar graph shows:
Answer: B. Frequencies of different categories
Why: Bar graphs display frequencies of categorical data using rectangular bars.
Q10.
What does a histogram show?
Answer: B. Frequency distribution of continuous data in intervals
Why: Histograms display frequency distributions of continuous data. Bars are joined (no gaps).
Q11.
If mean = 20 and there are 5 values, the sum is:
Answer: C. 100
Why: Sum = mean × number of values = 20 × 5 = 100.
Q12.
Standard deviation measures:
Answer: B. Spread (variability) of data around the mean
Why: Standard deviation measures how spread out values are from the mean. Higher SD = more spread.
Q13.
Which of these is a measure of central tendency?
Answer: C. Median
Why: Measures of central tendency (location): mean, median, mode. Others listed measure spread (dispersion).
Q14.
The cumulative frequency graph is also called:
Answer: C. Ogive
Why: An ogive (cumulative frequency curve) plots cumulative frequency vs. class boundaries.
Q15.
Find the mean of 6, 6, 6, 6, 6.
Answer: B. 6
Why: When all values are the same, mean = that value = 6.
Q16.
In a class of 10 students, the average score is 75. Total marks =
Answer: A. 750
Why: Total = mean × count = 75 × 10 = 750.
Q17.
Variance is the _____ of standard deviation.
Answer: B. Square
Why: Variance = (standard deviation)². Conversely, SD = square root of variance.
Q18.
Which graphical representation uses sectors of a circle?
Answer: C. Pie chart
Why: A pie chart divides a circle into sectors proportional to the frequencies of categories.
Q19.
If median > mean, the distribution is:
Answer: C. Negatively skewed
Why: Negatively skewed (left skewed): long tail to the left, mode > median > mean.
Q20.
Mean deviation from mean of 10, 20, 30 is:
Answer: A. 6.67
Why: Mean = 20. Deviations: |10-20|=10, |20-20|=0, |30-20|=10. Mean deviation = (10+0+10)/3 = 20/3 = 6.67.
Q21.
Coefficient of variation is used to:
Answer: B. Compare variability between datasets with different means
Why: CV = (SD/mean) × 100%. Allows comparison of variability between datasets with different scales.
Q22.
The median of grouped data uses the formula:
Answer: A. l + (n/2 - F)/f × h
Why: Median (grouped) = l + [(n/2 - F)/f] × h, where l = lower limit, F = cumulative freq before median class, f = freq of median class, h = class width.
Q23.
In grouped data, the mode formula is:
Answer: A. l + [(f₁-f₀)/(2f₁-f₀-f₂)] × h
Why: Mode (grouped) = l + [(f₁-f₀)/(2f₁-f₀-f₂)] × h, where f₁ = modal class freq, f₀ = prev class freq, f₂ = next class freq.
Q24.
Pearson's correlation coefficient r = 1 means:
Answer: C. Perfect positive correlation
Why: r = +1 indicates perfect positive linear correlation. As one variable increases, the other increases proportionally.
Q25.
IQR (Interquartile Range) = Q<sub>3</sub> - Q<sub>1</sub> measures:
Answer: B. Spread of middle 50% of data
Why: IQR measures the range of the middle 50% of data, making it resistant to outliers.
Q26.
Standard deviation is preferred over mean deviation because:
Answer: B. It gives more weight to extreme values and has algebraic tractability
Why: SD squares deviations (more sensitive to outliers), is algebraically tractable, and has favorable mathematical properties for further analysis.
Q27.
If all values in a dataset are multiplied by 2, the standard deviation:
Answer: B. Doubles
Why: SD scales with data. Multiplying all values by k multiplies SD by |k|. So SD doubles.
Q28.
The variance of 5, 5, 5, 5, 5 is:
Answer: C. 0
Why: All values equal the mean (5). All deviations are zero. Variance = 0.
Q29.
Ogive is used to find:
Answer: C. Median and quartiles
Why: An ogive (cumulative frequency curve) is used to graphically determine median, quartiles, and percentiles.
Q30.
For data 3, 6, 9, 12, 15, the mean deviation from mean is:
Answer: C. 4
Why: Mean = 9. Deviations: 6, 3, 0, 3, 6. Mean deviation = (6+3+0+3+6)/5 = 18/5 = 3.6. Closest to 4.
Q31.
If adding a constant k to each observation, mean:
Answer: B. Increases by k
Why: Mean shifts by the constant. New mean = original mean + k.
Q32.
The empirical relation between mean, median, and mode is:
Answer: B. Mean - Mode = 3(Mean - Median)
Why: Empirical formula: Mean - Mode = 3(Mean - Median). Or Mode = 3 Median - 2 Mean.
Q33.
For a negatively skewed distribution:
Answer: C. Mean < Median < Mode
Why: Negative (left) skew: long tail to the left. Mean pulled left by extreme low values. Mean < Median < Mode.
Q34.
Step-deviation method for mean uses:
Answer: B. Simplification by subtracting assumed mean and dividing by class width (h)
Why: Step-deviation: ui = (xi - a)/h where a is assumed mean and h is class width. Mean = a + (sum(fiui)/sum(fi)) × h.
Q35.
Spearman's rank correlation is used when:
Answer: B. Data is on ordinal scale or ranks, or non-linear relationship suspected
Why: Spearman rank correlation: non-parametric. Used for ordinal data or when linear relationship assumption fails.
Q36.
The box-plot whiskers extend to:
Answer: B. Minimum and maximum values (or 1.5×IQR from Q<sub>1</sub>/Q<sub>3</sub>)
Why: Box-plot whiskers: conventionally extend to min/max values within 1.5 × IQR from Q<sub>1</sub> and Q<sub>3</sub>. Points beyond are outliers.
Q37.
For data with high variability, standard deviation is:
Answer: C. High
Why: High variability = data spread widely from the mean = high standard deviation.
Q38.
The 50th percentile equals the:
Answer: C. Median
Why: The 50th percentile = Q2 = median. It divides the data into equal halves.
Q39.
In a frequency distribution, class mark (midpoint) of interval 30-40 is:
Answer: B. 35
Why: Class mark = (lower limit + upper limit)/2 = (30+40)/2 = 35.
Q40.
Mean of 10 numbers is 15. If one number 20 is replaced by 30, new mean is:
Answer: B. 16
Why: Sum = 150. Remove 20, add 30: new sum = 160. New mean = 160/10 = 16.
Q41.
For grouped data, Pearson's first coefficient of skewness is:
Answer: C. 3(Mean - Median)/SD
Why: Pearson's first skewness = (Mean - Mode)/SD. For grouped data the mode is ill-defined, so the second formula 3(Mean - Median)/SD is used instead via the empirical relation Mode ≈ 3Median - 2Mean. Answer: 3(Mean - Median)/SD.
Q42.
The regression line y on x passes through:
Answer: B. (x̄, ȳ) - the means
Why: The regression line y on x is ȳ = b·x̄ + a, derived by minimising Σ(yᵢ - ŷᵢ)². Setting partial derivatives to zero gives the normal equations, whose solution always passes through the point of means (x̄, ȳ). Both regression lines share this point.
Q43.
If byx × bxy = r², where byx and bxy are regression coefficients, then r =
Answer: B. sqrt(byx × bxy)
Why: By definition byx = r·(σy/σx) and bxy = r·(σx/σy). Their product: byx × bxy = r²·(σy/σx)·(σx/σy) = r². Therefore r = ±√(byx × bxy), with sign matching that of both coefficients.
Q44.
For a normal distribution N(μ, σ²), about 95% of data lies within:
Answer: B. μ ± 2σ
Why: Empirical 68-95-99.7 rule: P(μ−σ < X < μ+σ) ≈ 68%, P(μ−2σ < X < μ+2σ) ≈ 95%, P(μ−3σ < X < μ+3σ) ≈ 99.7%. Standard normal Z-table gives P(−1.96 < Z < 1.96) = 95%, confirming μ ± 2σ.
Q45.
Partial correlation measures:
Answer: A. Correlation removing effect of a third variable
Why: Partial correlation: measures linear relationship between two variables while controlling for one or more other variables.
Q46.
The variance of combined data of two groups (n₁, x̄₁, σ₁²) and (n₂, x̄₂, σ₂²) is:
Answer: B. A more complex formula involving d₁² and d₂² (distances from combined mean)
Why: Combined mean x̄ = (n₁x̄₁+n₂x̄₂)/(n₁+n₂). Let d₁ = x̄₁−x̄ and d₂ = x̄₂−x̄. Combined variance = [n₁(σ₁²+d₁²) + n₂(σ₂²+d₂²)]/(n₁+n₂). The d² terms capture the shift between group means and the combined mean.
Q47.
Moment generating function M(t) of a random variable X is defined as:
Answer: C. E[e<sup>tX</sup>]
Why: MGF: M(t) = E[e<sup>tX</sup>]. Expanding e<sup>tX</sup> = 1 + tX + t²X²/2! + … gives M(t) = 1 + tE[X] + t²E[X²]/2! + …. The n-th derivative M⁽ⁿ⁾(0) = E[Xⁿ], the n-th raw moment. Answer: E[e<sup>tX</sup>].
Q48.
Stratified random sampling ensures:
Answer: B. Proportional representation from each subgroup (stratum)
Why: Stratified sampling: population divided into strata, samples taken from each stratum proportionally or as needed. Ensures representation.
Q49.
The harmonic mean of 2, 4, 8 is approximately:
Answer: A. 3.43
Why: Harmonic mean HM = n / Σ(1/xᵢ). Here: 1/2 + 1/4 + 1/8 = 4/8 + 2/8 + 1/8 = 7/8. HM = 3 ÷ (7/8) = 3 × 8/7 = 24/7 ≈ 3.43. Useful for rates/speeds where equal time (not equal distance) is spent.
Q50.
Laspeyres price index uses base year quantities. It tends to:
Answer: B. Overestimate inflation
Why: Laspeyres (base-year weights) tends to overestimate inflation because it ignores substitution away from expensive goods. Paasche (current-year weights) tends to underestimate.
Q51.
In regression analysis, R² (coefficient of determination) measures:
Answer: B. Proportion of variance in Y explained by X
Why: SS_total = Σ(yᵢ−ȳ)², SS_residual = Σ(yᵢ−ŷᵢ)². R² = 1 − SS_residual/SS_total = SS_regression/SS_total. R² ∈ [0,1]; a value of 0.85 means 85% of variance in Y is explained by the regression model.
Q52.
Tchebychev (Chebyshev) inequality states that for any distribution, P(|X - μ| ≥ kσ) ≤:
Answer: B. 1/k²
Why: Chebyshev's inequality (no normality needed): P(|X−μ| ≥ kσ) ≤ 1/k² for any k > 1. Proof uses Markov's inequality on (X−μ)². Equivalently, at least 1−1/k² of data lies within k standard deviations of the mean.
Q53.
The standard error of mean (SEM) for sample size n from population with SD σ is:
Answer: C. σ/√n
Why: For n iid observations each with variance σ², Var(X̄) = Var(ΣXᵢ/n) = nσ²/n² = σ²/n. So SEM = SD(X̄) = σ/√n. Doubling n halves the SEM, reflecting improved precision of the sample mean estimate.
Q54.
The mean of the first five natural numbers is:
Answer: A. 3
Why: Mean = (1 + 2 + 3 + 4 + 5)/5 = 15/5 = 3.
Q55.
The median of the data 2, 4, 6, 8, 10 is:
Answer: A. 6
Why: For five ordered values, the median is the third (middle) one, which is 6.
Q56.
The mode of the data 2, 3, 3, 4, 5 is:
Answer: A. 3
Why: The mode is the most frequently occurring value; 3 appears twice, more than any other.
Q57.
The range of the data 5, 8, 12, 20 is:
Answer: A. 15
Why: Range = largest − smallest = 20 − 5 = 15.
Q58.
If the mean of five observations is 10, their total sum is:
Answer: A. 50
Why: Sum = mean × number of observations = 10 × 5 = 50.
Q59.
The standard deviation of a data set is the ___ of its variance:
Answer: A. the square root
Why: Standard deviation = √(variance).
Q60.
The variance of a set of data is always:
Answer: A. non-negative
Why: Variance is a mean of squared deviations, so it can never be negative.
Q61.
The variance of the first 10 natural numbers is:
Answer: A. 8.25
Why: Variance of first n natural numbers = (n² − 1)/12 = 99/12 = 8.25.
Q62.
If every observation in a data set is increased by 5, the variance:
Answer: C. remains unchanged
Why: Adding a constant shifts the data without changing spread, so the variance is unchanged.
Q63.
The mean of 100 observations is 50. Later one observation 100 was found to be misread as 10. The corrected mean is:
Answer: A. 50.9
Why: Corrected total increases by 90, so the mean rises by 90/100 = 0.9 to 50.9.
Q64.
The standard deviation of the data 2, 4, 6, 8, 10 is:
Answer: B. 2√2
Why: Mean = 6; squared deviations 16, 4, 0, 4, 16 sum to 40; variance 8; SD = √8 = 2√2.
Q65.
If the standard deviation of x₁, x₂, ..., xₙ is σ, the standard deviation of 3x₁, 3x₂, ..., 3xₙ is:
Answer: B. 3σ
Why: Scaling each observation by 3 scales the standard deviation by |3|, giving 3σ.
Q66.
For a data set with standard deviation 5 and mean 25, the coefficient of variation is:
Answer: B. 20%
Why: CV = (SD/mean)·100 = (5/25)·100 = 20%.
Q67.
For a moderately skewed distribution with mean 10 and median 8, the mode is:
Answer: B. 4
Why: Using mode = 3·median − 2·mean = 24 − 20 = 4.
Q68.
Two groups have sizes 20 and 30 with means 30 and 20 respectively. Their combined mean is:
Answer: B. 24
Why: Combined mean = (20·30 + 30·20)/50 = 1200/50 = 24.