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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A distribution where the Mean = 60, Median = 65, and Mode = 70 is:
Answer: C
We compare the values of the central tendencies.
Here, Mean (60) < Median (65) < Mode (70).
When the mean is pulled to the left of the median and mode, the distribution has a long tail on the left side and is said to be negatively skewed.
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The mean of 1, 3, 4, 5, 7, 4 is 4. The sum of deviations of each value from the mean is:
Answer: B
A fundamental property of the arithmetic mean is that the sum of the deviations of the data points from their mean is always zero.
We can verify: (1-4) + (3-4) + (4-4) + (5-4) + (7-4) + (4-4)
= (-3) + (-1) + 0 + 1 + 3 + 0 = 0.
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For which of the following is the median and mean the same?
Answer: A
The mean and median are the same for a data set that is symmetrically distributed.
The set {1, 2, 3, 4, 5} is an arithmetic progression, which is a symmetric distribution.
Mean = (1+2+3+4+5)/5 = 15/5 = 3.
Median = The middle value, which is 3.
All other sets are skewed.
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What is the mode of the data set {5, 6, 7, 7, 8, 8, 9, 9, 9, 10}?
Answer: C
The mode is the most frequently occurring value. In this set, the number 9 appears three times, which is more than any other number.
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A student calculates the mean of 5 numbers as 27. Later he realizes that he misread one number as 27 instead of 72. The correct mean is:
Answer: B
Incorrect sum = Mean × Count = 27 × 5 = 135.
The difference due to misreading = Correct value - Incorrect value = 72 - 27 = 45.
The correct sum is 45 more than the calculated sum.
Correct sum = 135 + 45 = 180.
Correct mean = \(\frac{180}{5} = 36\).
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The average monthly income of P and Q is Rs. 5050. The average monthly income of Q and R is Rs. 6250 and the average monthly income of P and R is Rs. 5200. The monthly income of P is:
Answer: B
We are given the following equations from the averages:
1) P + Q = 2 × 5050 = 10100
2) Q + R = 2 × 6250 = 12500
3) P + R = 2 × 5200 = 10400
Adding all three equations: 2(P + Q + R) = 10100 + 12500 + 10400 = 33000.
So, P + Q + R = 16500.
To find P, we subtract equation (2) from this sum: P = (P+Q+R) - (Q+R) = 16500 - 12500 = 4000.
The monthly income of P is Rs. 4000.
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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
In a perfectly symmetrical distribution (like a normal distribution or bell curve), the mean, median, and mode all coincide at the center of the distribution. Therefore, Mean = Median = Mode.
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