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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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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The median of the numbers 8, 11, 13, 15, x+1, x+3, 30, 35, 40, 43 arranged in ascending order is 23. What is the value of x?
Answer: C
The data set has 10 observations (an even number). The median is the average of the 5th and 6th terms.
5th term = x + 1
6th term = x + 3
Median = \(\frac{(x + 1) + (x + 3)}{2}\)
Given that the median is 23:
\(\frac{2x + 4}{2} = 23\)
x + 2 = 23
x = 21.
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The mode of a distribution is 24 and the mean is 18. Find the median.
Answer: A
Using the empirical relationship: Mode = 3 Median - 2 Mean.
24 = 3 × Median - 2 × 18
24 = 3 × Median - 36
24 + 36 = 3 × Median
60 = 3 × Median
Median = 60 / 3 = 20.
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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 age of 5 members is 21 years. If the age of the youngest member is 5 years, then what was the average age of the family at the birth of the youngest member?
Answer: C
Current total age of 5 members = 5 × 21 = 105 years.
5 years ago, the sum of their ages would have been 105 - (5 members × 5 years) = 105 - 25 = 80 years.
At the birth of the youngest member (5 years ago), there were only 4 members in the family (the youngest was just born, age 0).
Average age at that time = Total sum of ages of the 4 members / 4. This is tricky. The sum of ages of the 4 older members 5 years ago was 80. The average age was 80/4 = 20 years. Let's re-verify. Let ages be a,b,c,d,e. (a+b+c+d+e)/5 = 21. Sum = 105. Youngest e=5. At birth of e, ages were a-5, b-5, c-5, d-5. The sum of their ages was (a+b+c+d)-20. We know a+b+c+d = 105-5=100. So sum was 100-20=80. Average = 80/4 = 20.
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A library has an average of 510 visitors on Sundays and 240 on other days. The average number of visitors per day in a month of 30 days beginning with a Sunday is:
Answer: D
If a month of 30 days begins with a Sunday, it will have 5 Sundays and 25 other days.
Total visitors on Sundays = 5 × 510 = 2550.
Total visitors on other days = 25 × 240 = 6000.
Total visitors in the month = 2550 + 6000 = 8550.
Average visitors per day = \(\frac{8550}{30} = 285\).
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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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The average of four consecutive even numbers is 27. The largest of these numbers is:
Answer: B
For a set of consecutive numbers (or consecutive even/odd numbers), the mean is the average of the first and last number. Also, the mean of four consecutive numbers lies exactly between the second and third numbers.
The mean is 27, so the second and third numbers must be 26 and 28.
The four consecutive even numbers are 24, 26, 28, 30.
The largest number is 30.
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If the mean of x, x+2, x+4, x+6, x+8 is 11, then the mean of the last three observations is:
Answer: C
Since the numbers are in an arithmetic progression, the mean is the middle term. The middle term is x+4.
So, x + 4 = 11 => x = 7.
The numbers are 7, 9, 11, 13, 15.
The last three observations are 11, 13, 15.
Their mean is the middle term, which is 13. Wait, the answer is C=15. Let me re-calculate. The mean of 11, 13, 15 is (11+13+15)/3 = 39/3 = 13. My calculation is correct. The answer key is wrong. Let me rephrase the question. 'Mean of the first three observations'. That would be (7+9+11)/3 = 9. 'Mean of the largest observation'. That is just 15. The question seems fine, but the option is wrong. I will correct the option.
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