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Misc 7 - Mean, standard deviation of a group of 100 observations - Indirect questions - Incorrect observation

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  1. Chapter 15 Class 11 Statistics
  2. Serial order wise
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Misc 7 The mean and standard deviation of a group of 100 observations were found to be 20 and 3, respectively. Later on it was found that three observations were incorrect, which were recorded as 21, 21 and 18. Find the mean and standard deviation if the incorrect observations are omitted Given that number of observations (n) = 100 Incorrect mean ( ๐‘ฅ๏ทฏ ) = 20, Incorrect standard deviation (ฯƒ) = 3 We know that ๐‘ฅ๏ทฏ = 1๏ทฎ๐‘›๏ทฏ 20 = 1๏ทฎ100๏ทฏ 20 ร— 100 = 2000 = = 2000 โˆด Incorrect sum of observations = 2000 Finding correct sum of observations , incorrect observations 21, 21 and 18 are removed So, Correct sum of observations = Incorrect sum โ€“ 21 โ€“ 21 โ€“ 18 = 2000 โ€“ 21 โ€“ 21 โ€“ 18 = 1940 Hence, Correct mean = Correct sum of observations๏ทฎNumber of observations ๏ทฏ = 1940๏ทฎ100 โˆ’ ๐Ÿ‘๏ทฏ = 1940๏ทฎ๐Ÿ—๐Ÿ•๏ทฏ = 20 Now, Incorrect Standard deviation (ฯƒ) = 1๏ทฎN๏ทฏ ๏ทฎN ร—(๐ผ๐‘›๐‘๐‘œ๐‘Ÿ๐‘Ÿ๐‘’๐‘๐‘ก ๏ทฎ๏ทฎxi2 ) โ€“ ๐ผ๐‘›๐‘๐‘œ๐‘Ÿ๐‘Ÿ๐‘’๐‘๐‘ก ๏ทฎ๏ทฎxi๏ทฏ๏ทฏ2๏ทฏ๏ทฏ 3 = 1๏ทฎ100๏ทฏ ๏ทฎ100(๐ผ๐‘›๐‘๐‘œ๐‘Ÿ๐‘Ÿ๐‘’๐‘๐‘ก ๏ทฎ๏ทฎxi2 ) โ€“ ๐‘ฐ๐’๐’„๐’๐’“๐’“๐’†๐’„๐’• ๏ทฎ๏ทฎxi๏ทฏ๏ทฏ2๏ทฏ๏ทฏ 3 ร— 100 = ๏ทฎ100(๐ผ๐‘›๐‘๐‘œ๐‘Ÿ๐‘Ÿ๐‘’๐‘๐‘ก ๏ทฎ๏ทฎxi2 ) โ€“ ๐Ÿ๐ŸŽ๐ŸŽ๐ŸŽ๏ทฏ2๏ทฏ๏ทฏ 300 = ๏ทฎ100(๐ผ๐‘›๐‘๐‘œ๐‘Ÿ๐‘Ÿ๐‘’๐‘๐‘ก ๏ทฎ๏ทฎxi2 ) โ€“ 2000๏ทฏ2๏ทฏ๏ทฏ Squaring both sides (300)2 = ๏ทฎ100(๐ผ๐‘›๐‘๐‘œ๐‘Ÿ๐‘Ÿ๐‘’๐‘๐‘ก ๏ทฎ๏ทฎxi2 )โ€“ 2000๏ทฏ2๏ทฏ๏ทฏ๏ทฏ๏ทฎ2๏ทฏ 90000 = 100(๐ผ๐‘›๐‘๐‘œ๐‘Ÿ๐‘Ÿ๐‘’๐‘๐‘ก ๏ทฎ๏ทฎxi2 ) โ€“ 2000๏ทฏ2๏ทฏ 90000 = 100(๐ผ๐‘›๐‘๐‘œ๐‘Ÿ๐‘Ÿ๐‘’๐‘๐‘ก ๏ทฎ๏ทฎxi2 ) โ€“๏ทฏ 4000000 90000 + 4000000 = 100(๐ผ๐‘›๐‘๐‘œ๐‘Ÿ๐‘Ÿ๐‘’๐‘๐‘ก ๏ทฎ๏ทฎxi2 ) ๏ทฏ 4090000 = 100(๐ผ๐‘›๐‘๐‘œ๐‘Ÿ๐‘Ÿ๐‘’๐‘๐‘ก ๏ทฎ๏ทฎxi2 ) ๏ทฏ 100(๐ผ๐‘›๐‘๐‘œ๐‘Ÿ๐‘Ÿ๐‘’๐‘๐‘ก ๏ทฎ๏ทฎxi2 ) ๏ทฏ = 4090000 ๐ผ๐‘›๐‘๐‘œ๐‘Ÿ๐‘Ÿ๐‘’๐‘๐‘ก ๏ทฎ๏ทฎxi2 ๏ทฏ= 1๏ทฎ100๏ทฏ ร— 4090000 ๐ผ๐‘›๐‘๐‘œ๐‘Ÿ๐‘Ÿ๐‘’๐‘๐‘ก ๏ทฎ๏ทฎxi2 ๏ทฏ = 40900 Since 21, 21 and 18 are removed So, (๐‘ช๐’๐’“๐’“๐’†๐’„๐’• ๏ทฎ๏ทฎxi2 ๏ทฏ) = 40900 โ€“ (21)2 โ€“ (21)2 โ€“ (18)2 = 40900 โ€“ 441 โ€“ 441 โ€“ 364 = 39694 Now, Correct Standard deviation (ฯƒ) = 1๏ทฎN๏ทฏ ๏ทฎN ร—(๐ถ๐‘œ๐‘Ÿ๐‘Ÿ๐‘’๐‘๐‘ก ๏ทฎ๏ทฎxi2 )โ€“ ๐ถ๐‘œ๐‘Ÿ๐‘Ÿ๐‘’๐‘๐‘ก ๏ทฎ๏ทฎxi๏ทฏ๏ทฏ2๏ทฏ๏ทฏ = 1๏ทฎ๐Ÿ—๐Ÿ•๏ทฏ ๏ทฎ๐Ÿ—๐Ÿ• 39694๏ทฏ โˆ’ 1940๏ทฏ2๏ทฏ = 1๏ทฎ97๏ทฏ ๏ทฎ3850318 โˆ’3763600๏ทฏ = 1๏ทฎ97๏ทฏ ๏ทฎ๐Ÿ–๐Ÿ”๐Ÿ•๐Ÿ๐Ÿ–๏ทฏ = 1๏ทฎ97๏ทฏ ร— 294.4 = 3.036 Hence, correct Standard deviation = 3.036

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