Question

Find the mean deviation from the mean for the following data:
(i)

xi	5	7	9	10	12	15
fi	8	6	2	2	2	6

(ii)

xi	5	10	15	20	25
fi	7	4	6	3	5

(iii)

xi	10	30	50	70	90
fi	4	24	28	16	8

(iv)

Size	20	21	22	23	24
Frequency	6	4	5	1	4

[NCERT EXEMPLAR]
(v)

Size	1	3	5	7	9	11	13	15
Frequency	3	3	4	14	7	4	3	4

[NCERT EXEMPLAR]

Answer 1

i)

xi	fi	fixi	$\left|x_i-\overline{x}\right|$	$f_i\left|x_i-9\right|$
5	8	40	4	32
7	6	42	2	12
9	2	18	0	0
10	2	20	1	2
12	2	24	3	6
15	6	90	6	36
	$N=\Sigma f_i=26$	$\sum _{i=1}^n{f}_i{x}_i=234$		$\sum _{i=1}^n{f}_i\left|x_i-9\right|=88$

$\overline{x}=\frac{{\displaystyle \underset{i=1}{\overset{n}{\sum f_i}}}{x}_i}{N}=\frac{234}{26}=9$
$M.D.=\frac{1}{N}\sum _{i=1}^n{f}_i\left|x_i-\overline{x}\right|=\frac{1}{26}\times 88=3.39$


ii)

xi	fi	fixi	$\left|x_i-\overline{x}\right|$	$f_i\left|x_i-14\right|$
5	7	35	9	63
10	4	40	4	16
15	6	90	1	6
20	3	60	6	18
25	5	125	11	55
	$N=25$	$\sum _{i=1}^n{f}_i{x}_i=350$		$\sum _{i=1}^n{f}_i\left|x_i-14\right|=158$

$\overline{x}=\frac{{\displaystyle \underset{i=1}{\overset{n}{\sum f_i}}}{x}_i}{N}=\frac{350}{25}=14$
$MD=\frac{1}{N}\sum _{i=1}^n{f}_i\left|x_i-\overline{x}\right|=\frac{1}{25}\times 158=6.32$


iii)

xi	fi	fi​xi	$\left|x_i-\overline{x}\right|$	$f_i\left|x_i-50\right|$
10	4	40	40	160
30	24	720	20	480
50	28	1400	0	0
70	16	1120	20	320
90	8	720	40	320
	$N=80$	$\sum _{i=1}^n{f}_i{x}_i=4000$		$\sum _{i=1}^n{f}_i\left|x_i-50\right|=1280$

$\overline{x}=\frac{{\displaystyle \underset{i=1}{\overset{n}{\sum f_i}}}{x}_i}{N}=\frac{4000}{80}=50$
$MD=\frac{1}{N}\sum _{i=1}^n{f}_i\left|x_i-\overline{x}\right|=\frac{1}{80}\times 1280=16$

(iv)

Size(xi)	Frequency (fi)	fi​xi	$$\begin{gathered} \left|x_i-\overline{x}\right| \\ =\left|x_i-21.65\right| \end{gathered}$$	$$\begin{gathered} f_i\left|x_i-\overline{x}\right| \\ =f_i\left|x_i-21.65\right| \end{gathered}$$
20	6	120	1.65	9.9
21	4	84	0.65	2.6
22	5	110	0.35	1.75
23	1	23	1.35	1.35
24	4	96	2.35	9.4
	$N=20$	$\sum _{i=1}^n{f}_i{x}_i=433$		$\sum _{i=1}^n{f}_i\left|x_i-\overline{x}\right|=25$

$\overline{x}=\frac{{\displaystyle \underset{i=1}{\overset{n}{\sum f_i}}}{x}_i}{N}=\frac{433}{20}=21.65$
$MD=\frac{1}{N}\sum _{i=1}^n{f}_i\left|x_i-\overline{x}\right|=\frac{1}{20}\times 25=1.25$

(v)

Size(xi)	Frequency (fi)	fi​xi	$$\begin{gathered} \left|x_i-\overline{x}\right| \\ =\left|x_i-8\right| \end{gathered}$$	$$\begin{gathered} f_i\left|x_i-\overline{x}\right| \\ =f_i\left|x_i-8\right| \end{gathered}$$
1	3	3	7	21
3	3	9	5	15
5	4	20	3	12
7	14	98	1	14
9	7	63	1	7
11	4	44	3	12
13	3	39	5	15
15	4	60	7	28
	$N=42$	$\sum _{i=1}^n{f}_i{x}_i=336$		$\sum _{i=1}^n{f}_i\left|x_i-\overline{x}\right|=124$

$\overline{x}=\frac{{\displaystyle \underset{i=1}{\overset{n}{\sum f_i}}}{x}_i}{N}=\frac{336}{42}=8$
$MD=\frac{1}{N}\sum _{i=1}^n{f}_i\left|x_i-\overline{x}\right|=\frac{1}{42}\times 124=2.95$