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Weighted Average

If n number o...

Question

If $n$ number of data points are $x_{1}, x_{2}, x_{3}, . . ., x_{n}$ , and the corresponding weighted factors are $w_{1}, w_{2}, w_{3}, . . ., w_{n}$ , then which of the following is incorrect?

Given notation: $n_{1} + n_{2} + n_{3} + . . . + n_{l} = \sum_{i = 1}^{l} n_{i}$

Average of data set=

\frac{\sum_{i = 1}^{n} x_{i}}{n}

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Weighted average of data set=

\frac{\sum_{i = 1}^{n} w_{i} x_{i}}{\sum_{i = 1}^{n} w_{i}}

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Weighted average of data set=

\frac{\sum_{i = 1}^{n} w_{i} x_{i}}{n}

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Solution

The correct option is C Weighted average of data set= $\frac{\sum_{i = 1}^{n} w_{i} x_{i}}{n}$
The given data set is ${x_{1}, x_{2}, x_{3}, . ., x_{n}}$ .
Number of data points in the given data set is $n$ .

$∙$ The average of the data set

$= \frac{Sum of all data points}{Total number of data points}$

$= \frac{x_{1} + x_{2} + x_{3} + . . . + x_{n}}{n}$

$= \frac{\sum_{i = 1}^{n} x_{i}}{n}$

$∴$ Option (a.) is correct.

$∙$ The weighted average of the data set

$= \frac{w_{1} x_{1} + w_{2} x_{2} + w_{3} x_{3} + . . . + w_{n} x_{n}}{w_{1} + w_{2} + w_{3} + . . . w_{n}}$

$= \frac{\sum_{i = 1}^{n} w_{i} x_{i}}{\sum_{i = 1}^{n} w_{i}}$

$Weighted Average: - ---------------------- -$ The average resulting from the multiplication of each value by a weighted factor.

$∴$ Option (b.) is correct, and option (c.) is incorrect.

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Similar questions

If $x_{1}, x_{2}, x_{3}, . . . .$ are n values of the variable x, then mathematical averages, Arithmetic Mean (A.M.), Geometric Mean (G.M) and Harmonic Mean (H.M) are count by the following formulas's.

$A . M . = \frac{x_{1} + x_{2} + x_{3} + . . . + x_{n}}{n} = \frac{1}{n} {n \sum i = 1 x_{1}} ≮ x_{1} > 0$

$G . M . = {(x_{1} \cdot x_{2} . . . x_{n})}_{1}^{1 / n}$ if each $x_{1} (i = 1, 2, . . . ., n)$ is positive and

$H . M = \frac{n}{\frac{1}{x_{1}} + \frac{1}{x_{2}} + . . . + \frac{1}{x_{n}}} = \frac{1}{\frac{1}{n} \sum_{i = 1}^{n} (\frac{1}{x_{1}})}$

* In case of frequency distribution $x / f_{1} (i = 1, 2, . . ., n)$ where $f_{1}$ is the frequency of the variable $x_{r}$ then the calculation of A.M.is counted as $A . M = n \sum i = 1 f_{1} x_{1} / N w h e r e N = f_{1} + f_{2} . . . . + f_{n}$

** If $w_{1}, w_{2}, . . . ., w_{_{n}}$ be the weight assigned to the n values

$x_{1}, x_{2}, . . . ., x_{_{n}}$ then weighted A.M is counted by $\frac{\sum_{i = 1}^{n} w_{1} x_{1}}{\sum_{i = 1}^{n} w_{1}}$

*** if $G_{1}, G_{2}$ are the G. M's of two series of sizes $n_{1}$ & $n_{2}$ respectively, then the geometric mean (G M.) of the combined series is counted by $log (G . M) = \frac{n_{1} log G_{1} + n_{2} log G_{2}}{n_{1} + n_{2}}$

On the basis of above information answer the following questions,The weighted A.M of first n natural numbers whose weights are squares the corresponding numbers, is equal to

$A . M . = \frac{x_{1} + x_{2} + x_{3} + . . . + x_{n}}{n} = \frac{1}{n} {n \sum i = 1 x_{1}} ≮ x_{1} > 0$

$G . M . = {(x_{1} \cdot x_{2} . . . x_{n})}_{1}^{1 / n}$ if each $x_{1} (i = 1, 2, . . . ., n)$ is positive and

$H . M = \frac{n}{\frac{1}{x_{1}} + \frac{1}{x_{2}} + . . . + \frac{1}{x_{n}}} = \frac{1}{\frac{1}{n} \sum_{i = 1}^{n} (\frac{1}{x_{1}})}$

** If $w_{1}, w_{2}, . . . ., w_{_{n}}$ be the weight assigned to the n values

$x_{1}, x_{2}, . . . ., x_{_{n}}$ then weighted A.M is counted by $\frac{\sum_{i = 1}^{n} w_{1} x_{1}}{\sum_{i = 1}^{n} w_{1}}$

On the basis of above information answer the following questions,The mean of a set of n numbers is

¯ x

If each number is divided by 4 then new mean is

Q. If

A = (w_{1}, w_{2} \dots . . w_{n})

, where

w_{1}, w_{2} \dots . . w_{n}

are the outcomes, then

P (A) = P (w_{1}) + P (w_{2}) \dots . . P (w_{n})

Q. A gas is compressed from some initial volume

V_{1}

to final volume

V_{f}

along two different paths as shown in figure.

W_{1}, W_{2}

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n_{1}, n_{2}

respectively, then which of the following statements is/are correct ?

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1, w, w^{2}, . . . w^{n - 1}

are the

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