Assertion (A): The mean deviation about the mean for the data 4, … - Vidyadip

MCQ

Assertion $(A):$ The mean deviation about the mean for the data $4, 7, 8, 9, 10, 12, 13, 17$ is $3.$
Reason $(R):$ The mean deviation about the mean for the data $38, 70, 48, 40, 42, 55, 63, 46, 54, 44$ is $8.5.$

A
Both $A$ and $R$ are true and $R$ is the correct explanation of $A.$
B
Both $A$ and $R$ are true but $R$ is not the correct explanation of $A.$
✓
$A$ is true but $R$ is false.
D
$A$ is false but $R$ is true.

✓

Answer

Correct option: C.

$A$ is true but $R$ is false.

Assertion: Mean of the given series
$\overline{\bar{x}}=\frac{\text { Sum of terms }}{\text { Number of terms }}=\frac{\sum x_i}{n}$
$=\frac{4+7+8+9+10+12+13+17}{8}=10$

$xi$	$\| x i -\bar{x}\|$
$4$	$\|4-10\|=6$
$7$	$\|7-10\|=3$
$8$	$\|8-10\|=2$
$9$	$\|9-10\|=1$
$10$	$\|10-10\|=0$
$12$	$\|12-10\|=2$
$13$	$\|13-10\|=3$
$17$	$\|17-10\|=7$
$\sum x_i=80$	$\sum\left\|x_i-\bar{x}\right\|=24$

$\therefore$ Mean deviation about mean
$=\frac{\Sigma\left|x_i-\bar{x}\right|}{n}=\frac{24}{8}=3$
Reason Mean of the given series
$\bar{x}=\frac{\text { Sum of terms }}{\text { Number of terms }}=\frac{\sum x_i}{n}$
$=\frac{38+70+48+40+42+55}{+63+46+54+44}=50$
$\therefore$ Mean deviation about mean
$=\frac{\Sigma\left|x_i-\bar{x}\right|}{n}$
$=\frac{84}{10}=8.4$
Hence, Assertion is true and Reason is false.

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