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]

Solution

## 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   $N=\Sigma {f}_{i}=26$   $\sum _{i=1}^{n}{f}_{i}\left|{x}_{i}-9\right|=88$ $\overline{)x}=\frac{\underset{i=1}{\overset{n}{\sum {f}_{i}}}{x}_{i}}{N}=\frac{234}{26}=9$ $M.D.=\frac{1}{N}\sum _{i=1}^{n}{f}_{i}\left|{x}_{i}-\overline{)x}\right|=\frac{1}{26}×88=3.39$ 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   $N=25$ $\sum _{i=1}^{n}{f}_{i}{x}_{i}=350$   $\sum _{i=1}^{n}{f}_{i}\left|{x}_{i}-14\right|=158$ $\overline{)x}=\frac{\underset{i=1}{\overset{n}{\sum {f}_{i}}}{x}_{i}}{N}=\frac{350}{25}=14$ $MD=\frac{1}{N}\sum _{i=1}^{n}{f}_{i}\left|{x}_{i}-\overline{)x}\right|=\frac{1}{25}×158=6.32$ 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   $N=80$ $\sum _{i=1}^{n}{f}_{i}{x}_{i}=4000$   $\sum _{i=1}^{n}{f}_{i}\left|{x}_{i}-50\right|=1280$ $\overline{)x}=\frac{\underset{i=1}{\overset{n}{\sum {f}_{i}}}{x}_{i}}{N}=\frac{4000}{80}=50$ $MD=\frac{1}{N}\sum _{i=1}^{n}{f}_{i}\left|{x}_{i}-\overline{)x}\right|=\frac{1}{80}×1280=16$ (iv) Size(xi) Frequency (fi) fi​xi $\left|{x}_{i}-\overline{x}\right|\phantom{\rule{0ex}{0ex}}=\left|{x}_{i}-21.65\right|\phantom{\rule{0ex}{0ex}}$ ${f}_{i}\left|{x}_{i}-\overline{)x}\right|\phantom{\rule{0ex}{0ex}}={f}_{i}\left|{x}_{i}-21.65\right|$ 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   $N=20$ $\sum _{i=1}^{n}{f}_{i}{x}_{i}=433$   $\sum _{i=1}^{n}{f}_{i}\left|{x}_{i}-\overline{)x}\right|=25$ $\overline{)x}=\frac{\underset{i=1}{\overset{n}{\sum {f}_{i}}}{x}_{i}}{N}=\frac{433}{20}=21.65$ $MD=\frac{1}{N}\sum _{i=1}^{n}{f}_{i}\left|{x}_{i}-\overline{)x}\right|=\frac{1}{20}×25=1.25$ (v) Size(xi) Frequency (fi) fi​xi $\left|{x}_{i}-\overline{x}\right|\phantom{\rule{0ex}{0ex}}=\left|{x}_{i}-8\right|$ ${f}_{i}\left|{x}_{i}-\overline{)x}\right|\phantom{\rule{0ex}{0ex}}={f}_{i}\left|{x}_{i}-8\right|$ 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   $N=42$ $\sum _{i=1}^{n}{f}_{i}{x}_{i}=336$   $\sum _{i=1}^{n}{f}_{i}\left|{x}_{i}-\overline{)x}\right|=124$ $\overline{)x}=\frac{\underset{i=1}{\overset{n}{\sum {f}_{i}}}{x}_{i}}{N}=\frac{336}{42}=8$ $MD=\frac{1}{N}\sum _{i=1}^{n}{f}_{i}\left|{x}_{i}-\overline{)x}\right|=\frac{1}{42}×124=2.95$MathematicsRD Sharma XI (2019)All

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