Tag: means

Questions Related to means

The arithmetic mean (average) of the first $n$ positive integers is

maths average arithmetic mean of ap introduction to averages means

  1. $\dfrac {n}{2}$

  2. $\dfrac {n^{2}}{2}$

  3. $n$

  4. $\dfrac {n - 1}{2}$

  5. $\dfrac {n + 1}{2}$

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Correct Option: E
Explanation:

The sum of the first $n$ positive integers is $s = \dfrac {n(n + 1)}{2}$. Since $A.M. = \dfrac {s}{n}$, the correct choice is (e).

The mean of the cubes of the first $n$ natural numbers is :

maths average arithmetic mean of ap introduction to averages means

  1. $\displaystyle \frac{n(n+1)^2}{4}$

  2. $n^2$

  3. $\displaystyle \frac{n(n+1)(n+2)}{8}$

  4. $(n^2+n+1)$

Show answer
Correct Option: A
Explanation:

Sum of the cubes of first $n$ natural numbers
$=\displaystyle \left [ \frac{n(n+1)}{2}\right]^2 = \frac{n^2 (n+1)^2}{4}$
$\therefore $ Mean $= \displaystyle \frac{n(n+1)^2}{4}$

If the arithmetic mean of $n$ numbers of a series is $\bar{x}$ and sum of the first $(n - 1)$ numbers is $k$, then which one of the following is the nth number of the series ?

maths average arithmetic mean of ap introduction to averages means

  1. $\bar{x} - nk$

  2. $n\bar{x} - k$

  3. $k\bar{x} - n$

  4. $nk\bar{x}$

Show answer
Correct Option: B
Explanation:

Mean of n terms $=\bar{x}$
Sum of n terms  $=n\bar{x}$
Sum of $(n-1)$  terms $=k$

$n^{th}$ term $=$ Sum of n terms $-$ Sum of $(n-1)$ terms $=n\bar{x} - k$

The mean marks got by $300$ students in the subject of statistics was $45$. The mean of the top $100$ of them was found to be $70$ and the mean of the last $100$ was known to be $20$, then the mean of the remaining $100$ students is 

maths average arithmetic mean of ap introduction to averages means

  1. $45$

  2. $58$

  3. $68$

  4. $88$

Show answer
Correct Option: A
Explanation:
Let mean marks of remaining student be $M$
then Mean $=\dfrac{\displaystyle\sum (fx)}{\displaystyle\sum f}=45$        $\displaystyle\sum f=300$
$\therefore \dfrac{\displaystyle\sum (fx)}{\displaystyle\sum f}=100\times 70+100\times M+100\times 20$
$\dfrac{\displaystyle\sum (f)(x)}{\displaystyle\sum f}=\dfrac{100(90+M)}{\displaystyle\sum f}$
$45=\dfrac{100(90+M)}{300}$
$135=90+M$
$\Rightarrow M=135-90$
$=45$

If $a _{1}=0$ and $a _{1}, a _{2}, a _{3}, ...., a _{n}$ are real numbers such that $|a _{i}|=|a _{i-1}+1|$ for all $i$ then the Arithmetic mean of the numbers $a _{1}, a _{2}, ..., a _{n}$ has value $x$ where

maths average arithmetic mean of ap introduction to averages means

  1. $x<-1$

  2. $x<-\dfrac{1}{2}$

  3. $x>-\dfrac{1}{2}$

  4. $x=-\dfrac{1}{2}$

Show answer
Correct Option: C