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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The median of the numbers 8, 11, 13, 15, x+1, x+3, 30, 35, 40, 43 is 22. What is the value of x?
Answer: C
Correct answer: 20
There are ten values, already in ascending order, so the median is the mean of the 5th and 6th.
\[\frac{(x+1) + (x+3)}{2} = \frac{2x+4}{2} = x + 2\]
Setting this equal to the given median:
\[x + 2 = 22 \quad\Longrightarrow\quad x = 20\]
Substituting back, the list reads 8, 11, 13, 15, 21, 23, 30, 35, 40, 43, which is indeed in order, and the middle pair 21 and 23 averages to 22.
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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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The average age of a class of 30 students is 15 years. If the teacher's age is included, the average age becomes 16 years. What is the teacher's age?
Answer: B
Sum of ages of 30 students = 30 × 15 = 450 years.
When the teacher is included, there are 31 people and the average age is 16.
Sum of ages of 30 students + teacher = 31 × 16 = 496 years.
Teacher's age = 496 - 450 = 46 years.
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Which of the following is not a measure of central tendency?
Answer: D
Mean, Median, and Mode are all measures of central tendency, as they describe the center of a data set.
Range is a measure of dispersion or spread, as it describes how spread out the data is (Range = Maximum Value - Minimum Value).
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The arithmetic mean of 10 items is 50. If each item is increased by 5, then the new mean will be:
Answer: B
Adding a constant to every item shifts the mean by that constant.
\[ 50 + 5 = 55 \]
The number of items is irrelevant; the spread is unchanged too.
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What is the mode of the data set {5, 6, 7, 7, 8, 8, 9, 9, 9, 10}?
Answer: C
Count the repetitions: 7 twice, 8 twice, 9 three times.
So the mode is 9 - the value with the highest frequency.
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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
Correct answer: 13
The five values are equally spaced, so their mean is the middle one:
\[x + 4 = 11 \quad\Longrightarrow\quad x = 7\]
The observations are therefore 7, 9, 11, 13, 15. The last three are 11, 13 and 15, again equally spaced, so their mean is their middle value:
\[\frac{11 + 13 + 15}{3} = \frac{39}{3} = 13\]
Answering 15 mistakes the largest observation for the mean of the last three.
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