What is the mean of the first five natural numbers?
Answer: C
The first five natural numbers are 1, 2, 3, 4, and 5.
The mean (average) is the sum of the numbers divided by the count of the numbers.
Sum = 1 + 2 + 3 + 4 + 5 = 15.
Count = 5.
Mean = \(\frac{15}{5} = 3\).
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Find the median of the following set of numbers: 7, 3, 5, 8, 2.
Answer: B
To find the median, we first need to arrange the numbers in ascending order.
Arranged set: 2, 3, 5, 7, 8.
The median is the middle value. In this set of 5 numbers, the middle value is the 3rd term, which is 5.
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What is the mode of the following data set: 2, 4, 6, 4, 7, 4, 8, 9?
Answer: B
The mode is the value that occurs most often.
\[ 4 \text{ appears three times; every other value appears once} \]
So the mode is 4.
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The mean of 5 numbers is 20. If one number is excluded, the mean of the remaining numbers becomes 23. What is the excluded number?
Answer: A
The sum of the original 5 numbers = Mean × Count = 20 × 5 = 100.
After one number is excluded, there are 4 numbers left, and their mean is 23.
The sum of the remaining 4 numbers = 23 × 4 = 92.
The excluded number is the difference between the original sum and the new sum.
Excluded number = 100 - 92 = 8.
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Find the median of the data: 12, 15, 16, 10, 8, 19.
Answer: B
First, arrange the data in ascending order: 8, 10, 12, 15, 16, 19.
Since there is an even number of observations (6), the median is the average of the two middle values (the 3rd and 4th terms).
Middle values are 12 and 15.
Median = \(\frac{12 + 15}{2} = \frac{27}{2} = 13.5\).
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A batsman scores 35, 45, 38, 42, and x runs in five innings. If his average score for the five innings is 40, what is the value of x?
Answer: B
The mean (average) score is the total score divided by the number of innings.
Total score = 35 + 45 + 38 + 42 + x = 160 + x.
Average = \(\frac{160 + x}{5} = 40\).
160 + x = 40 × 5 = 200.
x = 200 - 160 = 40.
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What is the relationship between mean, median, and mode for a moderately skewed distribution?
Answer: A
For a moderately skewed distribution the three averages satisfy the empirical relation
\[ \text{Mode} = 3\,\text{Median} - 2\,\text{Mean} \]
It says the mode lies about twice as far from the mean as the median does, on the same side.
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The mean weight of 10 students is 55 kg. If a student with weight 60 kg joins the group, what is the new mean weight?
Answer: B
Original sum of weights = 10 × 55 = 550 kg.
Weight of the new student = 60 kg.
New total weight = 550 + 60 = 610 kg.
New number of students = 10 + 1 = 11.
New mean weight = \(\frac{610}{11} \approx 55.45\) kg.
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Which measure of central tendency is most affected by extreme values (outliers)?
Answer: A
The mean is the most sensitive to extreme values because its calculation involves the sum of all values. A very large or very small value can significantly pull the mean in its direction.
The median is resistant to outliers because it only depends on the middle value(s).
The mode is also resistant as it depends only on the frequency of values.
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For a symmetrical distribution, which of the following is correct?
Answer: C
A symmetrical distribution has no skew, so the three measures of central tendency coincide at the centre.
\[ \text{Mean} = \text{Median} = \text{Mode} \]
Skew is precisely what pulls them apart: a right skew leaves mean > median > mode.
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