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Exercises · Q5
Q.

The yield of wheat and rice per acre for 10 districts of a state is as under:

District12345678910
Wheat1210151921161892510
Rice22291223181512341812

Calculate for each crop: (i) Range, (ii) Q.D., (iii) Mean deviation about mean, (iv) Mean deviation about median, (v) Standard deviation, (vi) Which crop has greater variation?, (vii) Compare the values of the different measures for each crop.

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Wheat: Range 16, Q.D. 4.75, M.D.(mean) 4.3, M.D.(median) 4.3, S.D. 5.04. Rice: Range 22, Q.D. 6.25, M.D.(mean) 6.0, M.D.(median) 5.7, S.D. 7.16. Rice varies more.

WHEAT — sorted: 9, 10, 10, 12, 15, 16, 18, 19, 21, 25 (n = 10, sum = 155, mean = 15.5, median = (15+16)/2 = 15.5).

  • (i) Range =25−9=16= 25 - 9 = 16.
  • (ii) Q.D. — Q1Q_1 is the n+14=2.75th\dfrac{n+1}{4} = 2.75\text{th} value =10= 10; Q3Q_3 is the 3(n+1)4=8.25th\dfrac{3(n+1)}{4} = 8.25\text{th} value =19+0.25(21−19)=19.5= 19 + 0.25(21-19) = 19.5. Q.D.=19.5−102=4.75Q.D. = \dfrac{19.5 - 10}{2} = 4.75.
  • (iii) M.D. about mean (15.5) — ∑∣d∣=3.5+5.5+0.5+3.5+5.5+0.5+2.5+6.5+9.5+5.5=43\sum |d| = 3.5+5.5+0.5+3.5+5.5+0.5+2.5+6.5+9.5+5.5 = 43; M.D.=4310=4.3M.D. = \dfrac{43}{10} = 4.3.
  • (iv) M.D. about median (15.5) — the median equals the mean here, so M.D.=4310=4.3M.D. = \dfrac{43}{10} = 4.3.
  • (v) S.D. — ∑(X−Xˉ)2=254.5\sum (X-\bar{X})^2 = 254.5; σ=254.510=25.45=5.04\sigma = \sqrt{\dfrac{254.5}{10}} = \sqrt{25.45} = 5.04.

RICE — sorted: 12, 12, 12, 15, 18, 18, 22, 23, 29, 34 (n = 10, sum = 195, mean = 19.5, median = (18+18)/2 = 18).

  • (i) Range =34−12=22= 34 - 12 = 22.
  • (ii) Q.D. — Q1Q_1 is the 2.75th value =12= 12; Q3Q_3 is the 8.25th value =23+0.25(29−23)=24.5= 23 + 0.25(29-23) = 24.5. Q.D.=24.5−122=6.25Q.D. = \dfrac{24.5 - 12}{2} = 6.25.
  • (iii) M.D. about mean (19.5) — ∑∣d∣=2.5+9.5+7.5+3.5+1.5+4.5+7.5+14.5+1.5+7.5=60\sum |d| = 2.5+9.5+7.5+3.5+1.5+4.5+7.5+14.5+1.5+7.5 = 60; M.D.=6010=6.0M.D. = \dfrac{60}{10} = 6.0.
  • (iv) M.D. about median (18) — ∑∣d∣=4+11+6+5+0+3+6+16+0+6=57\sum |d| = 4+11+6+5+0+3+6+16+0+6 = 57; M.D.=5710=5.7M.D. = \dfrac{57}{10} = 5.7.
  • (v) S.D. — ∑(X−Xˉ)2=512.5\sum (X-\bar{X})^2 = 512.5; σ=512.510=51.25=7.16\sigma = \sqrt{\dfrac{512.5}{10}} = \sqrt{51.25} = 7.16. …

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