1. The empirical relationship between the three measures of central tendency is:
A. 3 Median = Mode + 2 Mean
B. Mode = 3 Mean - 2 Median
C. Mean = 3 Median - 2 Mode
D. 2 Median = Mode + 3 Mean
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Answer: A. 3 Median = Mode + 2 Mean
Explanation: Standard empirical formula connecting measures of central tendency is 3 Median = Mode + 2 Mean.
2. Class mark of a class interval is given by:
A. Upper limit - Lower limit
B. (Upper limit + Lower limit) / 2
C. Upper limit / 2
D. Lower limit / 2
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Answer: B. (Upper limit + Lower limit) / 2
Explanation: Class mark represents the mid-point of a class interval: (Upper class limit + Lower class limit) / 2.
3. For a frequency distribution, the mean is calculated using formula Σf_i x_i / Σf_i. This method is called:
A. Assumed Mean Method
B. Step-Deviation Method
C. Direct Method
D. Grouped Method
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Answer: C. Direct Method
Explanation: Finding mean directly by taking the ratio of sum of product of frequencies and class marks to total frequency is the Direct Method.
4. The mode of a grouped frequency distribution can be determined graphically from a:
A. Histogram
B. Frequency Polygon
C. Ogive
D. Pie Chart
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Answer: A. Histogram
Explanation: The highest rectangle in a histogram corresponds to the modal class, from which mode can be located graphically.
5. The median of a grouped data can be found graphically by drawing:
A. Histograms
B. Ogives
C. Bar graphs
D. Pie charts
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Answer: B. Ogives
Explanation: Median corresponds to the x-coordinate of the point of intersection of 'less than' and 'more than' cumulative frequency curves (ogives).
6. If mean = 20 and median = 22, then mode is equal to:
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Answer: B. 26
Explanation: Mode = 3 Median - 2 Mean = 3(22) - 2(20) = 66 - 40 = 26.
7. Sum of deviations of observations from their mean is always equal to:
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Answer: A. 0
Explanation: Mathematical property of arithmetic mean: Σ(x_i - Mean) = 0.
8. The class size of interval 10 - 25 is:
A. 10
B. 15
C. 25
D. 17.5
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Answer: B. 15
Explanation: Class size = Upper limit - Lower limit = 25 - 10 = 15.
9. Which measure of central tendency is affected most by extreme values?
A. Mean
B. Median
C. Mode
D. Range
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Answer: A. Mean
Explanation: Mean accounts for the numerical value of every single observation, making it highly sensitive to extreme values (outliers).
10. The mean of first 10 natural numbers is:
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Answer: B. 5.5
Explanation: Sum of first 10 natural numbers = 10 × 11 / 2 = 55. Mean = 55 / 10 = 5.5.
11. In the formula Mean = a + (Σf_i d_i / Σf_i), 'a' represents:
A. Class size
B. Assumed mean
C. Frequency
D. Cumulative frequency
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Answer: B. Assumed mean
Explanation: In Assumed Mean Method, 'a' denotes the assumed mean chosen usually from mid-values of x_i.
12. Cumulative frequency table is useful in determining the:
A. Mean
B. Median
C. Mode
D. Range
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Answer: B. Median
Explanation: Cumulative frequencies are required to locate the position N/2 to identify the median class.
13. The mode of the data 2, 3, 4, 3, 5, 3, 1, 2, 3, 4 is:
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Answer: B. 3
Explanation: 3 occurs most frequently (4 times) in the given data set.
14. If the mode of a distribution is 8 and mean is 8, then median is:
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Answer: A. 8
Explanation: 3 Median = Mode + 2 Mean = 8 + 2(8) = 24 ⇒ Median = 8.
15. The median of numbers 4, 1, 8, 3, 6, 9, 2 is:
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Answer: B. 4
Explanation: Arranging in ascending order: 1, 2, 3, 4, 6, 8, 9. Middle value (4th position) is 4.
16. Constructing a cumulative frequency table is useful in determining:
A. Mean
B. Median
C. Mode
D. All of these
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Answer: B. Median
Explanation: Cumulative frequency distribution is constructed specifically to compute the median.
17. While computing mean of grouped data, we assume that the frequencies are:
A. Evenly distributed over all classes
B. Centred at class marks of classes
C. Concentrated at lower limits
D. Concentrated at upper limits
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Answer: B. Centred at class marks of classes
Explanation: In grouped data calculation, each class interval is represented by its mid-point (class mark).
18. If mean of x, x+3, x+6, x+9, and x+12 is 10, then x is:
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Answer: D. 4
Explanation: Sum = 5x + 30. Mean = (5x + 30)/5 = x + 6 = 10 ⇒ x = 4.
19. In Step-Deviation Method, u_i is calculated as:
A. (x_i - a) / h
B. (a - x_i) / h
C. h × (x_i - a)
D. x_i - a
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Answer: A. (x_i - a) / h
Explanation: Standard definition: u_i = d_i / h = (x_i - a) / h, where h is class height.
20. In modal class formula, f_1 represents:
A. Frequency of modal class
B. Frequency of class preceding modal class
C. Frequency of class succeeding modal class
D. Total frequency
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Answer: A. Frequency of modal class
Explanation: In Mode = l + [(f_1 - f_0) / (2f_1 - f_0 - f_2)] × h, f_1 is frequency of modal class.
21. The upper limit of modal class for data: [0-10: 5, 10-20: 8, 20-30: 12, 30-40: 6] is:
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Answer: B. 30
Explanation: Maximum frequency is 12, corresponding to class interval 20 - 30. Its upper limit is 30.
22. If the mean of first n natural numbers is 15, then n is equal to:
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Answer: B. 29
Explanation: Mean = (n + 1)/2 = 15 ⇒ n + 1 = 30 ⇒ n = 29.
23. Which of the following is NOT a measure of central tendency?
A. Mean
B. Median
C. Mode
D. Standard Deviation
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Answer: D. Standard Deviation
Explanation: Standard Deviation is a measure of dispersion, whereas Mean, Median, and Mode measure central tendency.
24. The mean of 5 observations is 12. If one observation is excluded, mean becomes 10. Excluded number is:
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Answer: A. 20
Explanation: Sum of 5 observations = 5 × 12 = 60. Sum of 4 observations = 4 × 10 = 40. Excluded value = 60 - 40 = 20.
25. The algebraic sum of deviations of a set of n values from their mean is:
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Answer: B. 0
Explanation: The sum of differences between each observation and arithmetic mean is always zero.
26. Median class is the class interval whose cumulative frequency is just greater than or equal to:
A. N
B. N / 2
C. N / 4
D. 2N
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Answer: B. N / 2
Explanation: By definition, median class is identified by looking for cumulative frequency that is just ≥ N/2.
27. If each observation of a data set is multiplied by 3, its mean will:
A. Remain same
B. Become 3 times
C. Increase by 3
D. Become 1/3rd
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Answer: B. Become 3 times
Explanation: Multiplying every observation in a set by a constant k multiplies the mean by k.
28. For symmetrical distribution, relation between mean, median, and mode is:
A. Mean = Median = Mode
B. Mean > Median > Mode
C. Mean < Median < Mode
D. Mean + Mode = Median
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Answer: A. Mean = Median = Mode
Explanation: In a perfectly symmetrical distribution, Mean, Median, and Mode coincide at the same central value.
29. What is the class mark of interval 40 - 50?
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Answer: B. 45
Explanation: Class mark = (40 + 50) / 2 = 90 / 2 = 45.
30. Point of intersection of 'less than' and 'more than' ogives corresponds to x-axis coordinate value of:
A. Mean
B. Median
C. Mode
D. Class size
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Answer: B. Median
Explanation: The x-value corresponding to intersection point of both cumulative frequency curves gives the median.