Tag: range and mean deviation

Questions Related to range and mean deviation

The mean annual salary of all employees in a company is Rs$25,000$. The mean salaries of male and female employees are Rs$27,000$ and $Rs17,000$ respectively, the percentage of males and females employees by the company is _______.

business economics and quantitative methods measures of dispersion and skewness shortcut method for calculating mean deviation about mean mean deviation about mean and median range and mean deviation

  1. $80,20$

  2. $20,80$

  3. $30,70$

  4. $70,30$

Show answer
Correct Option: A
Explanation:

Given, the mean annual salary of all employees in a company is $Rs\ 25,000$ & mean salaries of male and female employees are $Rs\ 27,000$ and $Rs\ 17,000$ respectively.


Let us consider mean salaries of male as $Rs. 27000$ and female as $17,000$

$\dfrac{27,000x+17,000y}{x+y}=25,000$
$27,000x+17,000y=25,000x+25,000y$
$2000x=8000y$

$x=4y$

Percentage of males $(4y/5y)\times 100=80\%$

Percentage of females $=20\%$

In distribution $25\%$ of the observations are less than $46$ and $25\%$ of the observations are more than $54$. The quartile deviation of the distribution is _______.

business economics and quantitative methods measures of dispersion and skewness shortcut method for calculating mean deviation about mean mean deviation about mean and median range and mean deviation

  1. $3$

  2. $7$

  3. $4$

  4. $6$

Show answer
Correct Option: C
Explanation:

Quartile deviation divides the series into four equal parts and measures the distance average between the third and the first quartile. The first quartile is denoted as Q1 and the third quartile is denoted as Q3 . 

Q1= 46 and Q3=54

Quartile deviation = (Q3-Q1) /2

                             = (54-46)/2

                             = 8/2

                             = 4

Probability is expressed as _______.

business economics and quantitative methods measures of dispersion and skewness shortcut method for calculating mean deviation about mean mean deviation about mean and median range and mean deviation

  1. percentage

  2. ratio

  3. proportion

  4. all (a), (b), (c)

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Correct Option: D
Explanation:
Probability shows the relationship between two variables in the form of ratio, percentage or proportion where there the chances of occurrence of one variable is expressed in terms of other variable. Since the value of one variable belongs to the range of value of another variable, the range o probability varies from 0 to 1.

If the standard deviation of $x _{1},x _{2},.....x _{n}$ is 3.5, then the standard deviatiuon of $-2x _{1}-3,-2x _{2}-3....,-2x _{n}-3$ is

maths statistics and probability coefficient of variance variance and standard deviation measures of dispersion range and mean deviation

  1. -7

  2. -4

  3. 7

  4. 1.75

Show answer
Correct Option: C

If $\sigma$ $f _i$ $x _i$  = 20 and $\sigma$ $f _i$ = 4, what is the mean of the data.

maths statistics and probability coefficient of variance variance and standard deviation measures of dispersion range and mean deviation

  1. $\dfrac{1}{5}$

  2. $80$

  3. $16$

  4. $5$

Show answer
Correct Option: A
Explanation:
$\sigma fixi=20$
and $\sigma fi=4$

Hence, Mean $=\dfrac{\sigma fi}{\sigma fixi}=\dfrac{4}{20}=\dfrac{1}{5}$

Hence Option $A$ is correct

The variance of the data $6,\ 8,\ 10,\ 12\,,14\,,\ 16,\ 18,\ 20,\ 22,\ 24$ is

maths statistics and probability coefficient of variance variance and standard deviation measures of dispersion range and mean deviation

  1. $15$

  2. $20$

  3. $30$

  4. $33$

Show answer
Correct Option: D
Explanation:
Mistake :$14$ is not given
Mean $\bar x=\dfrac{6+8+10+12+14+16+18+20+22+24}{10}=\dfrac{150}{10}=15$
Variance$=\dfrac{1}{n} \sum\limits _{i=1}^n(x _{i}-\bar x)^2$
$\implies \dfrac{1}{10}((6-15)^2+(8-15)^2+(10-15)^2+(12-15)^2+(14-15)^{2}+(16-15)^2+(18-15)^2+(20-15)^2$
$+(22-15)^2+(24-15)^2$

$\implies \dfrac{81+49+25+9+1+1+9+25+49+81}{10}$

$\implies \dfrac{330}{10}=33$

The variate x and u are related by $\displaystyle u= \frac{x-a}{h}$ then correct relation between $\displaystyle \sigma _{x}:and:\sigma _{u}$

maths statistics and probability coefficient of variance variance and standard deviation measures of dispersion range and mean deviation

  1. $\displaystyle \sigma _{x}= h\sigma _{u}$

  2. $\displaystyle \sigma _{x}= h+\sigma _{u}$

  3. $\displaystyle \sigma _{u}= h\sigma _{x}$

  4. $\displaystyle \sigma _{u}= h+\sigma _{x}$

Show answer
Correct Option: A
Explanation:

Given $\displaystyle u =\frac{x}{h}-\frac{a}{h}$
Since,S.D. is not depend on change of origin but it is depend on change of scale.
$\displaystyle \therefore \sigma _{u}=\frac{\sigma _{x}}{h}$
$\Rightarrow h\sigma _{u}=\sigma _{x}$

Standard deviation of a collection of data is $2\sqrt{2}$. If each value in a data set  is multipled by $3$, then the standard deviation of the new data is.

maths statistics and probability coefficient of variance variance and standard deviation measures of dispersion range and mean deviation

  1. $\sqrt{12}$

  2. $4\sqrt{2}$

  3. $6\sqrt{2}$

  4. $9\sqrt{2}$

Show answer
Correct Option: C
Explanation:
The standard deviation would also be multiplied by $3$.
Because the mean would also be $3x$ larger, the differences from the mean would be $3x$ larger too.
It is the same idea as if you were looking at your data set through an enlarging lens- everything would be $3x$ bigger, not only the data values, but also the mean, the differences from the mean, but just everything!
$\therefore$ the standard deviation becomes $2\sqrt{2}\times 3=6\sqrt{2}$

If the standard deviation of $x _1, x _2, .., x _n$ is $3.5$, then the standard deviation of $-2x _1-3, -2x _2-3$,....., -2x_n-3$ is?

maths statistics and probability coefficient of variance variance and standard deviation measures of dispersion range and mean deviation

  1. $-7$

  2. $-4$

  3. $7$

  4. $1.75$

Show answer
Correct Option: A
Explanation:
The Standard Deviation of a set remains unchanged if each data is increased or decreased by a constant however changes similarly when data is multiplied or divided by a constant.
$\therefore $ The SD for the new data set will be $=-2\times 3.5=-7$

Consider the following statements.Which of these is/are correct?

maths statistics and probability coefficient of variance variance and standard deviation measures of dispersion range and mean deviation

  1. Mode can be computed from histogram

  2. Median is not independent of change of scale

  3. Variance is independent of change of origin and scale

  4. none of these

Show answer
Correct Option: A,B
Explanation:

If we change scale by using x + h then median increases by h.
so median is not independent of change of scale.
From histogram we can see highest frequency so made.
Hence, options 'A' and 'B' are correct.