A car travels from A to B at a speed of 40 km/hr and returns from B to A at a speed of 60 km/hr. What is the average speed for the entire journey?
Answer: A
When the distance is constant for two speeds x and y, the average speed is the Harmonic Mean of the speeds, not the Arithmetic Mean.
Average Speed = \(\frac{2xy}{x+y}\).
Average Speed = \(\frac{2 \times 40 \times 60}{40+60} = \frac{4800}{100} = 48\) km/hr.
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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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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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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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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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A data set can have:
Answer: D
A data set's mode has the following properties:
Therefore, all of the above statements are possible.
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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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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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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 appears most frequently in a data set.
In the set {2, 4, 6, 4, 7, 4, 8, 9}, the number 4 appears three times, which is more than any other number.
Therefore, the mode is 4.
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