Standard Deviation

Q.

If the variance of the terms in an increasing A.P., $b_1,b_2,b_3,...,b_{11}$ is $90$, then the common difference of this A.P. is

Q. Ques: coefficient of mean deviation about mean for the first 9 natural numbers is?

Q.

In an experiment with 15 observations on $x$, the following results were available: $\sum x^2=2830,\sum x=170$ One observation that was $20$ was found to be wrong and was replaced by the correct value $30$. The corrected variance is

1. 8.33

2. 78.00

3. 177.33

4. 188.66

Q. The S.D. of scores 1, 2, 3, 4, 5 is:

1. 
2. 
3. 
4. 

Q.

The sum of absolute deviation is least when taken from

1. Mean

2. Median

3. Mode

4. None of the above

Q. The mean and S.D. of the marks of 200 candidates were found to be 40 and 15 respectively. Later, it was discovered that a score of 40 was wrongly read as 50. The correct mean and S.D. respectively are

1. 14.98, 39.95
2. 39.95, 14.98
3. 39.95, 224.5
4. None the these

Q.

How do you compute confidence intervals?

Q.

The standard deviation of $9,16,23,30,37,44,51$ is

1. $7$

2. $9$

3. $12$

4. $14$

5. $16$

Q.

The variance of first 50 even natural numbers is

1. $\frac{437}{4}$

2. 833

3. 437

4. $\frac{833}{4}$

Q. The arithmetic mean of first five natural numbers is

1. $3$
2. $6$
3. $4$
4. $5$

Q. If the standard deviation of a variable $X$ is $\sigma$, then the standard deviation of variable $\frac{aX+b}{c}$, is

1. $a\sigma$
2. $\frac{a}{c}\sigma$
3. $\left|\frac{a}{c}\right|\sigma$
4. $\frac{a\sigma +b}{c}$

Q. Factorize: $36a^2-12+\frac{1}{a^2}$

1. $(a-\frac{1}{6a}{)}^2$
2. $(6a-\frac{1}{6a}{)}^2$
3. $(6a-\frac{1}{a}{)}^2$
4. $(6a-\frac{6}{a}{)}^2$

Q. For two data sets, each size 5, the variances are given to be 4 and 5 and the corresponding means are given to be 2 and 4, respectively. The variance of the combined data set is

1. $\frac{13}{2}$
2. $\frac{5}{2}$
3. $\frac{11}{2}$
4. $\sqrt{\frac{11}{2}}$

Q. Consider first $10$ positive integers having standard deviation $2.87$. If we multiply each number by $-1$ and then add $1$ to each number, the standard deviation of the numbers so obtained is

1. $8.25$
2. $2.87$
3. $-2.87$
4. $-8.25$

Q. For a group of $200$ candidates, the mean and $S.D.$ were found to be $40$ and $15$ respectively. Late on it was found that the score $43$ was misread as $34$. Find the correct mean and correct $S.D.$.

Q.

Let $x_1,x_2,\cdots ,x_n$ be n observations, and let $\overline{x}$ be their arithmetic mean and ${\sigma }^2$ be the variance.
Statement – 1: Variance of $2x_1,2x_2,\cdots ,2x_n\text{is}4{\sigma }^2$
Statement – 2: Arithmetic mean $2x_1,2x_2,\cdots ,2x_n\text{is}4\overline{x}$

1. Statement – 1 is false, Statement – 2 is true

2. Statement – 1 is true, statement – 2 is false.

3. Statement – 1 is true, statement – 2 is true; statement – 2 is a correct explanation for Statement – 1.

4. Statement – 1 is true, statement – 2 is true; statement – 2 is not a correct explanation for Statement – 1.

Q.

If the standard deviation of the numbers 2, 3, a and 11 is 3.5, then which of the following is true?

1. $3a^2–26a+55=0$

2. $3a^2–32a+84=0$

3. $3a^2–34a+91=0$

4. $3a^2–23a+44=0$

Q.

Let $S$ be the standard deviation of $n$ observations. Each of the $n$ observations is multiplied by a constant $C$. Then the standard deviation of the resulting number is

1. $S\sqrt{C}$

2. S

3. CS

4. none of these

Q.

The standard deviation of 17 numbers is zero. Then

1. the numbers are in geometric progression with common ratio not equal to one.

2. eight numbers are positive, eight are negative and one is zero.

3. either (a) or (b)

4. none of these

Q. Let $n\ge 3$. A list of numbers $x_1,x_2,.....x_n$ has mean $\mu$ and standard deviation $\sigma$. A new list of numbers $y_1,y_2,....y_n$ is made as follows : $y_1=\frac{x_1+x_2}{2},y_2=\frac{x_1+x_2}{2}$ and $y_j=x_j$ for $j=3,4,....n$. The mean and the standard deviation of the list are $\hat{\mu }$ and $\hat{\sigma }$. Then which of the following is necessarily true?

1. $\sigma =\hat{\sigma }$
2. $\mu \ne \hat{\mu }$
3. $\mu =\hat{\mu }$ and $\sigma \ge \hat{\sigma }$
4. $\mu =\hat{\sigma }$ and $\sigma \le \hat{\sigma }$

Q.

For two data sets, each of size 5, the variances are given to be 4 and 5 and the corresponding means are given to be 2 and 4, respectivley. The variance of the combined data set is

1. $\frac{11}{2}$

2. 6

3. $\frac{13}{2}$

4. $\frac{5}{2}$

Q. If the difference between the standard deviation and
variance of a data is 12 then find the sum of the variance and standard deviation of that data

1. 20
2. 15
3. 18
4. 22

Q. The standard deviation of 25 numbers is 40. if each of the numbers is increased by 5, then the new standard deviation will be

1. $40+\frac{21}{25}$
2. None of these
3. 45
4. 40

Q. If $X$ is a variate taking values $x_1,x_2,.....x_n$ and $Y$ is another variate taking values $y_1,y_2,.....y_n$ such that $y_i=2x_i+3$ and ${\sigma }_Y=8$, then the variance of variable $Z$, taking values $\frac{3}{2}{x}_i;i=1,2,....n$ is

1. $6$
2. $24$
3. $18$
4. $36$

Q.

All the students of a class performed poorly in Mathematics. The teacher decided to give grace marks of 10 to each of the students. Which of the following statistical measures will not change even after the grace marks were given?

1. Mean

2. Mode

3. Median

4. Variance

Q. What is the wine of all the 250 items ?

1. $44$
2. $53.3$
3. $55.6$
4. $59.3$