Standard Deviation about Mean

Q.

Find the mean and variance for the data

6, 7, 10, 12, 13, 4, 8, 12

Q.

Find the mean and variance for the first n natural numbers.

Q.

For a frequency distribution consisting of 18 observations, the mean and the standard deviation were found to be 7 and 4 respectively. But on comparison with the original data, it was found that a figure 12 was miscopied as 21 in calculations. Find the correct mean and standard deviation.___

Q. The mean and the variance of five observations are $4$ and $5.20$, respectively. If three of the observations are $3,4$ and $4$ ; then the absolute value of the difference of the other two observations, is :

1. $1$
2. $3$
3. $5$
4. $7$

Q. 5 students of a class have an average height $150\text{cm}$ and variance $18{\text{cm}}^2$. A new student, whose height is $156\text{cm}$ joined them. The variance $\left(\text{in}{\text{cm}}^2\right)$ of the height of these six students is:

1. $20$
2. $16$
3. $18$
4. $22$

Q.

Prove that the standard deviation of the first $n$ natural numbers is $\sqrt{\frac{n^2-1}{12}}$.

Q.

What do you understand by standard deviation?

Q.

If each of the observations $x_1,x_2,x_3,\cdots ,x_n$ is increased by an amount a, where a is a negative or positive number, then show that the variance remains unchanged.

Q.

Which of the following are true statements.

1. Variance gives a better picture of dispersion or scatter compared to mean deviation

2. Variance can be zero even if all observations are not equal.

3. For given n observation x₁, x₂, ... x_n, with mean variance will be

4. Variance can be a negative quantity if all the observations are negative

Q.

Find the mean and variance for each of the data in Exercise 1 to 5:

$\begin{array}{cccccccc}{x}_i& 6& 10& 14& 18& 24& 28& 30\\ f_i& 2& 4& 7& 12& 8& 4& 3\end{array}$

Q.

Calculate the mean and the variance of first n natural numbers.

Q.

Find the value of n, so that $\frac{a^{n+1}+b^{n+1}}{a^n+b^n}$ is the geometric mean between a and b

Or

If f is a function satisfying $f(x+y)=f\left(x\right)f\left(y\right)$ for all $x,y\in N$ such that f(1)=3 and ${\sum }_{x=1}^{n}f\left(x\right)=120$ find the value of n.

Q. Prove that the standard deviation of the first n natural numbers is $\sigma =\sqrt{\frac{n^2-1}{12}}$.

Q. The mean of five observations is $4.4$ and the variance is $8.24$.Three of the five observations are $1,2$ and $6$. The remaining two are

1. $9,4$
2. $7,6$
3. $6,5$
4. $10,3$

Q.

The standard deviation of the first n natural numbers is

1. $\frac{n+1}{2}$
   

2. $\sqrt{\frac{n(n+1)}{2}}$
   

3. $\sqrt{\frac{n^2-1}{12}}$
   

4. none of these

Q.

Standard deviation for $x_1,x_2,.....x_n$ about mean can be expressed as $\sqrt{\frac{1}{n}{\sum }_{i=1}^n(x_i-\overline{x}{)}^2}$.

1. False

2. True

Q. The mean and variance of 'n' observations $x_1,x_2,x_3,......x_n$ are 5 and 0, respectively. If ${\sum }_{i=1}^n{x}_i^2=400$ then the value of n is

1. 80
2. 25
3. 20
4. 16

Q.

$x_i$	$10$	$15$	$18$	$20$	$25$
$f_i$	$3$	$2$	$5$	$8$	$2$

Find the Variance of the given data.

1. $14$
2. $15$
3. $16$
4. $17$

Q.

A scientist is weighing each of 30 fishes. Their mean weight worked out is 30 gm and a standard deviaiton of 2 gm. Later, it was found that the measuring scale was misaligned and always under-reported every fish weight by 2gm. The correct mean and standard deviation (in gm) of fishes are respectively:

1. $32,4$

2. $28,4$

3. $32,2$

4. $28,2$

Q.

Standard deviation for $x_1,x_2,.....x_n$ about mean can be expressed as $\sqrt{\frac{1}{n}{\sum }_{i=1}^n(x_i-\overline{x}{)}^2}$.

1. True

2. False

Q. Let $x_1,x_2,.....x_n$ be $n$ observations and $\overline{X}$ be their arithmetic mean. The formula for the standard deviation is given by

1. ${\sum }_{i=1}^n{{(x}_i-\overline{X})}^2$
2. $\frac{1}{n}{\sum }_{i=1}^n{{(x}_i-\overline{X})}^2$
3. $\sqrt{\frac{1}{n}{\sum }_{i=1}^n{{(x}_i-\overline{X})}^2}$
4. $\sqrt{\frac{1}{n}{\sum }_{i=1}^n{x_i}^2+{\overline{X}}^2}$

Q. For observations $x_1,x_2,x_3,..........,x_n,$ if ${\sum }_{i=1}^n(x_i+1{)}^2=9n$ and ${\sum }_{i=1}^n(x_i-1{)}^2=5n.$, then standard deviation of the data is

1. $\sqrt{10}$
2. $\sqrt{3}$
   
3. $\sqrt{2}$
   
4. $\sqrt{5}$
   

Q.

For observations $x_1,x_2,x_3,..........,x_n,$ if ${\sum }_{i=1}^n(x_i+1{)}^2=9n$ and ${\sum }_{i=1}^n(x_i-1{)}^2=5n.$, then standard deviation of the data is

1. $\sqrt{3}$
   

2. $\sqrt{5}$
   

3. $\sqrt{2}$
   

4. $\sqrt{10}$

Q. If the variance of the data $2,4,5,6,8,17$ is $23.33$, then the variance of $4,8,10,12,16,34$ is

1. $23.23$
2. $25.33$
3. $46.66$
4. $93.32$

Q. Find the mean and variance for the first 10 multiples of 3.

Q.

Given that ($\overline{x}$) is the mean and ${\sigma }^2$ is the variance of n observations $x_1,x_2,....x_n$. Prove that the mean and variance of the observations $a{x}_1,a{x}_2,a{x}_3...a{x}_n$are a$\overline{x}$ and $a^2{\sigma }^2$ respectively $(a\ne 0)$

Q. The mean and standard deviation of 20 observations are found to be 10 and 2, respectively. On rechecking, it was found that an observation 8 was incorrect. Calculate the correct mean and standard deviation in each of the following cases: (i) If wrong item is omitted. (ii) If it is replaced by 12.

Q.

The standard deviation of 4 consecutive numbers which are in A.P is $\sqrt{5}$. The common difference (d) of this A.P is

1. $\pm \sqrt{5}$

2. $\pm 2\sqrt{5}$

3. $\pm 2$
   

4. $\pm \sqrt{2}$

Q. The mean and standard deviation of a group of $100$ observations were found to be $20$ and $3$ respectively. Later on it was found that three observations were incorrect which were recorded as $21,21$ and $18$. Find the mean and standard deviation if the incorrect observation are omitted

Q. The mean of five observations is $5$ and their variance is $9.20$. If three of the given five observations are $1,3$ and $8$, then a ratio of other two observations is :

1. $10:3$
2. $4:9$
3. $5:8$
4. $6:7$