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Standard IX

Mathematics

Mean

If x 1, x 2, ...

Question

If $x_{1}, x_{2}, . . ., x_{n}$ are the means of n groups with $n_{1}, n_{2}, . . ., n_{n}$ number of observations respectively, then the mean $x$ of all the groups taken together is
(a) $\sum_{i = 1}^{n} n_{i} x_{i}$

(b) $\sum_{\frac{i = 1}{n^{2}}}^{n} n_{i} x_{i}$

(c) $\frac{\sum_{i = 1}^{n} n_{i} x_{i}}{\sum_{i = 1}^{n} n_{i}}$

(d) $\frac{\sum_{i = 1}^{n} n_{i} x_{i}}{2 n}$

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Solution

$(c) \frac{\sum_{i = 1}^{n} n_{i} x_{i}}{\sum_{i = 1}^{n} n_{i}}$

$Sum of all terms = n_{1} {\bar{x}}_{1} + n_{2} {\bar{x}}_{2} + . . . n_{n} {\bar{x}}_{n} Number of terms = (n_{1} + n_{2} + . . . n_{n}) ∴ Mean = \frac{\sum_{i = 1}^{n} n_{i} {\bar{x}}_{i}}{\sum_{i = 1}^{n} n_{i}}$

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Similar questions

Q. Question 17
If

_{1},_{2},_{3}, \dots \dots_{n}

, are the means of n groups with

n_{1}, n_{2}, n_{n}

number of observations, respectively. Then the mean

¯ x

of all the groups:

A)

\sum_{i = 1}^{n} n_{i}_{i}

\frac{\sum_{i = 1}^{n} n_{i}_{i}}{n^{2}}

\frac{\sum_{i = 1}^{n} n_{i}_{i}}{\sum_{i = 1}^{n} n_{i}}

\frac{\sum^{n} n_{i}_{i}}{2 n}

If $_{1},_{2} \dots,_{n}$ are the means of n groups with $n_{1}, n_{2}, \dots, n_{n}$ number of observations respectively then the mean $¯ x$ of all the groups taken together is
(a) $\sum_{n}^{i = 1} n,_{i}$

(b) $\frac{\sum_{i = 1}^{n} n_{i}_{i}}{n^{2}}$

(d) $\frac{\sum_{i = 1}^{n} n_{i} {¯ x}_{i}}{2 n}$

Q. If

{\bar{X}}_{1}, {\bar{X}}_{2}, . . ., {\bar{X}}_{k}

are the means of n group with n1,n2,...,nk number of observations respectively, then the mean

\bar{X}

of all the groups taken together is govern by

(a) \sum_{i = 1}^{k} n_{i} {\bar{X}}_{i} (b) \frac{1}{n^{2}} \sum_{i = 1}^{k} n_{i} {\bar{X}}_{i} (c) \frac{\sum_{i = 1}^{k} n_{i} {\bar{X}}_{i}}{\sum_{i = 1}^{k} n_{i}} (d) \frac{\sum_{i = 1}^{k} n_{i} {\bar{X}}_{i}}{2_{n}}

Q. If

_{1},_{2},_{3}, . . .,_{n}

are the means of n groups with

n_{1}, n_{2}, . . ., n_{n}

number of observations respectively, then the mean x of all the groups taken together is given by :

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,If the mean of a set of observations

x_{1}, x_{2}, . . . ., x_{n}

¯ x

then the mean of observations

x_{1} + 4 i

, i = 1, 2, 3, . . ., n

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If x1, x2, ..., xn are the means of n groups with n1, n2, ..., nn number of observations respectively, then the mean x of all the groups taken together is (a) ∑i=1nnixi (b) ∑i=1n2nnixi (c) ∑i=1nnixi∑i=1nni (d) ∑i=1nnixi2n

(c) ∑i=1nnixi∑i=1nni Sum of all terms=n1x¯1+n2x¯2+...nnx¯nNumber of terms=(n1+n2+...nn)∴Mean=∑i=1nnix¯i∑i=1nni

$(c) \frac{\sum_{i = 1}^{n} n_{i} x_{i}}{\sum_{i = 1}^{n} n_{i}}$

$Sum of all terms = n_{1} {\bar{x}}_{1} + n_{2} {\bar{x}}_{2} + . . . n_{n} {\bar{x}}_{n} Number of terms = (n_{1} + n_{2} + . . . n_{n}) ∴ Mean = \frac{\sum_{i = 1}^{n} n_{i} {\bar{x}}_{i}}{\sum_{i = 1}^{n} n_{i}}$