Tag: variance and standard deviation

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Questions Related to variance and standard deviation

Multiple choice range and mean deviation statistics and probability maths coefficient of variance variance and standard deviation

Find the coefficient of range for the data $43,24,38,56,22,39,45$

  1. $0124$
  2. $0.212$
  3. $0.236$
  4. $0.436$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation
Given data is $43, 24, 38, 56, 2, 39,45$.
The largest value of data is $x _m=56$
The smallest value of data is $x _0=22$
Coefficient of data $=\dfrac{56-22}{56+22}=\dfrac{34}{78}=0.4359\approx 0.436$
Hence, option D is correct.
Multiple choice range and mean deviation statistics and probability maths coefficient of variance variance and standard deviation

The weight in Kg of 13 students in a class are $42.5,47.5,48.6,50.5,49,46.2,49.8,45.8,43.2,48,44.7,46.9,42.4$.Find the coefficient of range.

  1. $0.077$
  2. $0.213$
  3. $0.0803$
  4. $0.093$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The largest value of data is $x _m=49.8$

The smallest value of data is $x _0=42.4$
Coefficient of data$=\dfrac{49.8-42.4}{49.8+42.4}=\dfrac{7.4}{92.2}=0.08026\approx 0.0803$

Multiple choice range and mean deviation statistics and probability maths coefficient of variance variance and standard deviation

Find the coefficient of range for the given data
$59,46,30,23,27,40,52,35,29$

  1. $0.46$
  2. $0.44$
  3. $0.56$
  4. $0.124$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
Given data is $59, 46, 30, 23, 27, 40, 52, 35, 29$.
The largest value of data is $x _m=59$
The smallest value of data is $x _0=23$
Coefficient of data $=\dfrac{59-23}{59+23}=\dfrac{36}{82}=0.49\approx 0.44$
Multiple choice maths statistics and probability coefficient of variance variance and standard deviation measures of dispersion range and mean deviation

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

  1. -7

  2. -4

  3. 7

  4. 1.75
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Standard deviation is independent of the change of origin (adding or subtracting a constant) and is scaled by the absolute value of the change of scale (multiplying by a constant). If the original SD is 3.5, the new SD is | -2 | * 3.5 = 7.

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

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

  1. $15$
  2. $20$
  3. $30$
  4. $33$
Reveal answer Fill a bubble to check yourself
D Correct answer
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$=\dfrac{1}{n} \sum\limits _{i=1}^n(x _{i}-\bar x)^2$
$\implies \dfrac{1}{10}((6-15)^2+(8-15)^2+(10-15)^2+(12-15)^2+(14-15)^{2}+(16-15)^2+(18-15)^2+(20-15)^2$
$+(22-15)^2+(24-15)^2$

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

$\implies \dfrac{330}{10}=33$
Multiple choice maths statistics and probability coefficient of variance variance and standard deviation measures of dispersion range and mean deviation

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

  1. $\displaystyle \sigma _{x}= h\sigma _{u}$
  2. $\displaystyle \sigma _{x}= h+\sigma _{u}$
  3. $\displaystyle \sigma _{u}= h\sigma _{x}$
  4. $\displaystyle \sigma _{u}= h+\sigma _{x}$
Reveal answer Fill a bubble to check yourself
A Correct answer
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.
$\displaystyle \therefore \sigma _{u}=\frac{\sigma _{x}}{h}$
$\Rightarrow h\sigma _{u}=\sigma _{x}$

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

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.

  1. $\sqrt{12}$
  2. $4\sqrt{2}$
  3. $6\sqrt{2}$
  4. $9\sqrt{2}$
Reveal answer Fill a bubble to check yourself
C Correct answer
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 $2\sqrt{2}\times 3=6\sqrt{2}$
Multiple choice maths statistics and probability coefficient of variance variance and standard deviation measures of dispersion range and mean deviation

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?

  1. $-7$
  2. $-4$
  3. $7$
  4. $1.75$
Reveal answer Fill a bubble to check yourself
A Correct answer
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 $=-2\times 3.5=-7$
Multiple choice maths statistics and probability coefficient of variance variance and standard deviation measures of dispersion range and mean deviation

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

  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
Reveal answer Fill a bubble to check yourself
A,B Correct answer
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.