Chapter 3: Measures of Central Tendency & Dispersion, Skewness, Kurtosis (CAIIB – Paper 1)
1. The arithmetic mean is best described as:
A. The middle value of an ordered data set
B. The most frequently occurring value
C. The sum of all values divided by the number of values
D. The difference between maximum and minimum values
The arithmetic mean is calculated by adding all values and dividing by the number of observations. Median is the middle value, Mode is the most frequent value, and Range is max–min.
2. A group of 50 students has an average score of 60. Another group of 30 students has an average of 80. What is the combined mean score?
3. Which measure of central tendency is not affected by extreme values?
A. Arithmetic Mean
B. Combined Mean
C. Weighted Mean
D. Median
Median is positional and not influenced by extreme values, unlike mean and combined mean which get distorted by outliers.
4. If the distribution of data is symmetrical, then the relationship among mean, median, and mode is:
A. Mean > Median > Mode
B. Mean = Median = Mode
C. Median > Mode > Mean
D. Mode > Median > Mean
In a perfectly symmetrical distribution, mean, median, and mode coincide. In positively or negatively skewed data, they differ.
5. A distribution has mean = 50, median = 40, and mode = 30. This indicates the distribution is:
A. Symmetrical
B. Negatively skewed
C. Positively skewed
D. Normal
In a positively skewed distribution: Mean > Median > Mode. Here 50 > 40 > 30, hence positively skewed.
6. The Geometric Mean (GM) of n positive numbers is defined as:
A. The sum of all numbers divided by n
B. The middle value of the ordered data set
C. The reciprocal of the average of reciprocals
D. The nth root of the product of all n numbers
Geometric Mean (GM) = (x₁ × x₂ × ... × xₙ)^(1/n). It is used in growth rates and compound interest situations.
7. The GM of 4 and 9 is:
A. 5.5
B. 6
C. 6.5
D. 7
GM = √(4×9) = √36 = 6.
8. The Harmonic Mean (HM) of n positive numbers is given by:
A. n / (Σ (1/xᵢ))
B. (Σxᵢ) / n
C. (Πxᵢ)^(1/n)
D. The maximum value among the observations
Harmonic Mean (HM) = n / (1/x₁ + 1/x₂ + … + 1/xₙ). It is suitable for rates and ratios (e.g., speed problems).
9. If a car travels a certain distance at 60 km/h and returns at 40 km/h, what is the average speed using Harmonic Mean?
A. 50 km/h
B. 48 km/h
C. 48 km/h
D. 45 km/h
Average speed for equal distance = Harmonic Mean = 2xy / (x+y) = (2×60×40)/(60+40) = 4800/100 = 48 km/h.
10. Which of the following is the correct relationship among Mean, Geometric Mean (GM), and Harmonic Mean (HM) for positive data?
A. Arithmetic Mean < GM < HM
B. Arithmetic Mean ≥ GM ≥ HM
C. HM ≥ GM ≥ Arithmetic Mean
D. GM ≥ Arithmetic Mean ≥ HM
For any set of positive numbers: Arithmetic Mean ≥ Geometric Mean ≥ Harmonic Mean. Equality holds when all values are equal.
11. The median of a data set represents:
A. The average of all data values
B. The most frequently occurring value
C. The middle value when data is arranged in order
D. The difference between highest and lowest value
Median is the positional average. It divides the ordered data set into two equal halves. If n is odd, median is the middle value; if n is even, it is the average of two middle values.
12. For a data set with n = 20, the position of the first quartile (Q1) is:
A. (n+1)/2
B. (n+1)/4
C. (3(n+1))/4
D. (n+1)/4 = 21/4 = 5.25th item
The formula for quartiles is Qk = (k(n+1))/4. For Q1, k=1 ⇒ (n+1)/4. With n=20, position = 21/4 = 5.25th item.
13. Which measure is most suitable for categorical data such as gender, religion, or brand preference?
A. Arithmetic Mean
B. Mode
C. Median
D. Geometric Mean
Mode is most suitable for categorical or qualitative data, as it represents the most frequently occurring category.
14. In a moderately skewed distribution, the relationship between mean, median, and mode is given by:
A. Mode = 3 × Median – 2 × Mean
B. Median = 3 × Mode – 2 × Mean
C. Mean = (Median + Mode)/2
D. Mean = Median = Mode
Karl Pearson’s empirical relationship: Mode = 3 × Median – 2 × Mean. This is used in moderately skewed distributions.
15. If the quartile deviation (semi-interquartile range) is 12, what is the interquartile range (IQR)?
25. Which of the following is an advantage of Quartile Deviation?
A. It is not affected by extreme values
B. It uses all data values
C. It is the most accurate measure of dispersion
D. It is based on variance and standard deviation
Quartile Deviation considers only Q1 and Q3, so it is not influenced by extreme values, unlike range.
26. Standard Deviation is defined as:
A. Square of the variance
B. Difference between maximum and minimum values
C. Positive square root of variance
D. Average of deviations from the mean
Standard Deviation = √Variance. It measures how much the data values deviate from the mean on average.
27. Which of the following is TRUE about Standard Deviation?
A. It is always negative
B. It ignores extreme values
C. It is the same as mean deviation
D. It uses all data values and is always non-negative
SD uses all observations in its calculation and is always non-negative since it is a square root of variance.
28. If variance of a dataset is 49, the standard deviation is:
A. 49
B. 7
C. 14
D. 343
Standard Deviation = √Variance = √49 = 7.
29. Coefficient of Variation (CV) is useful because:
A. It is a relative measure of dispersion that allows comparison across datasets
B. It is always equal to the mean
C. It does not use standard deviation
D. It is independent of mean
CV = (SD / Mean) × 100. It is a relative measure, allowing comparison of variability between different datasets.
30. If Mean = 50 and Standard Deviation = 5, what is the Coefficient of Variation?
A. 2%
B. 5%
C. 10%
D. 20%
CV = (SD / Mean) × 100 = (5 / 50) × 100 = 10%.
31. Skewness measures:
A. The average of the data set
B. The spread of the middle 50% of data
C. The difference between maximum and minimum values
D. The asymmetry of the distribution around its mean
Skewness indicates whether the data distribution is symmetrical or tilted to the left (negative skew) or right (positive skew).
32. If a distribution has Mean > Median > Mode, the skewness is:
A. Symmetrical
B. Positively skewed
C. Negatively skewed
D. Uniform
Positive skewness occurs when the tail on the right side is longer, causing Mean > Median > Mode.
33. A negatively skewed distribution has:
A. Mean > Median > Mode
B. Mean = Median = Mode
C. Mean < Median < Mode
D. Mode < Median < Mean
Negative skewness occurs when the left tail is longer, causing Mean < Median < Mode.
34. Kurtosis measures:
A. The spread of the middle 50% of data
B. The peakedness or flatness of a distribution
C. The asymmetry of the distribution
D. The average deviation from mean
Kurtosis indicates whether the distribution is more peaked (leptokurtic), flatter (platykurtic), or normal (mesokurtic) compared to a normal distribution.
35. A distribution with high positive kurtosis is:
A. Leptokurtic
B. Platykurtic
C. Mesokurtic
D. Symmetrical
Leptokurtic distributions are highly peaked with heavy tails, indicating a higher probability of extreme values. Platykurtic is flat, mesokurtic is normal-like.
