162 practice questions covering Statistics, organised into 2 topics.
162 questions include a written explanation.
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For a set of eight observations, $\sum x_i = 40$ and $\sum x_i^2 = 272$. Find the standard deviation.
A$5.83$
B$9$
C$34$
D$3$
Show answer and explanation
Correct answer: D - $3$
The mean is $\frac{40}{8} = 5$, so the variance is $\frac{272}{8} - 5^2 = 34 - 25 = 9$ and the standard deviation is $3$. The value $9$ is the variance itself. The value $34$ is $\frac{\sum x_i^2}{n}$ with the $\bar{x}^2$ term forgotten, and $5.83$ is $\sqrt{34}$, the same slip carried into the square root.
Q2
medium
Consider these two statements. I: For any data set, the mean deviation about the median is never greater than the mean deviation about the mean. II: The mean deviation about the median is negative when most of the observations lie below the median. Which is correct?
AOnly I is true
BOnly II is true
CBoth I and II are true
DNeither I nor II is true
Show answer and explanation
Correct answer: A - Only I is true
The sum of absolute deviations is smallest when measured about the median, so the mean deviation about the median cannot exceed the one about the mean, making I true. II is false because absolute deviations are distances and can never be negative, whatever the shape of the data. So the options accepting II fail, and I cannot be rejected.
Q3
easy
Find the mean deviation about the mean of the five observations $4, 7, 8, 9, 12$.
A$2$
B$8$
C$10$
D$0$
Show answer and explanation
Correct answer: A - $2$
The mean is $\frac{40}{5} = 8$. The absolute deviations are $4, 1, 0, 1, 4$, whose sum is $10$, so the mean deviation is $\frac{10}{5} = 2$. The value $0$ comes from adding the deviations with their signs, which always cancel to zero. The value $10$ is the sum of the absolute deviations before dividing, and $8$ is the mean itself.
Q4
easy
In the mode formula $\text{Mode} = l + \left(\frac{f_1-f_0}{2f_1-f_0-f_2}\right)h$ for grouped data, what does $f_1$ stand for?
AThe size of each class
BThe total of all the frequencies
CThe frequency of the class just before the modal class
DThe frequency of the modal class
Show answer and explanation
Correct answer: D - The frequency of the modal class
In the formula $f_1$ is the frequency of the modal class itself, while $f_0$ and $f_2$ are the frequencies of the classes just before and just after it. The total frequency is written $\sum f_i$ and appears in the mean and median formulas, not here. The class size is $h$, the letter multiplying the bracket.
Q5
medium
Assertion (A): For the distribution $0-10$ ($3$), $10-20$ ($9$), $20-30$ ($7$), $30-40$ ($1$), the mode lies between $10$ and $20$. Reason (R): The mode of a grouped distribution is always equal to the class mark of the modal class.
ABoth A and R are true, and R is the correct explanation of A
BBoth A and R are true, but R is not the correct explanation of A
CA is true, but R is false
DA is false, but R is true
Show answer and explanation
Correct answer: C - A is true, but R is false
The greatest frequency is $9$, so the modal class is $10-20$ and the mode does lie between $10$ and $20$, making A true. R is false: the formula gives $10 + \frac{6}{18-3-7}\times 10 = 17.5$ here, not the class mark $15$, and in general the mode need not sit at the middle of the modal class.
Q6
medium
A factory tells a buyer that its mean daily output is $500$ units. The buyer also asks for the standard deviation of the daily output. What is the buyer trying to find out?
AHow steady the daily output is from day to day
BWhether the reported mean has been computed correctly
CThe total output of the factory for the month
DThe most frequently occurring daily output
Show answer and explanation
Correct answer: A - How steady the daily output is from day to day
The standard deviation measures how far the daily figures typically lie from their average, so a small value means a dependable, steady supply. The total needs only the mean and the number of days, not the spread. The standard deviation cannot check whether a mean was computed correctly. The most frequent value is the mode, which is a different measure altogether.
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