Indirect questions - Incorrect observation

Chapter 13 Class 11 Statistics
Concept wise

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Example 16 The mean and standard deviation of 100 observations were calculated as 40 and 5.1, respectively by a student who took by mistake 50 instead of 40 for one observation. What are the correct mean and standard deviation? Given that number of observations (n) = 100 Incorrect mean ( ๐ฅ๏ทฏ ) = 40, Incorrect standard deviation (ฯ) = 5.1 We know that ๐ฅ๏ทฏ = 1๏ทฎ๐๏ทฏ 40 = 1๏ทฎ100๏ทฏ 40 ร 100 = 4000 = = 4000 โด Incorrect sum of observations = 4000 Finding correct sum of observations , 50 is taken in place of 40 So, Correct sum of observations = Incorrect sum โ 50 + 40 = 4000 โ 50 + 40 = 3990 Hence, Correct mean = Correct sum of observations๏ทฎNumber of observations ๏ทฏ = 3990๏ทฎ100๏ทฏ = 39.9 Now, Incorrect Standard deviation (ฯ) = 1๏ทฎN๏ทฏ ๏ทฎN ร(๐ผ๐๐๐๐๐๐๐๐ก ๏ทฎ๏ทฎxi2 ) โ ๐ผ๐๐๐๐๐๐๐๐ก ๏ทฎ๏ทฎxi๏ทฏ๏ทฏ2๏ทฏ๏ทฏ 5.1 = 1๏ทฎ100๏ทฏ ๏ทฎ100(๐ผ๐๐๐๐๐๐๐๐ก ๏ทฎ๏ทฎxi2 ) โ ๐ฐ๐๐๐๐๐๐๐๐ ๏ทฎ๏ทฎxi๏ทฏ๏ทฏ2๏ทฏ๏ทฏ 5.1 ร 100 = ๏ทฎ100(๐ผ๐๐๐๐๐๐๐๐ก ๏ทฎ๏ทฎxi2 ) โ ๐๐๐๐๏ทฏ2๏ทฏ๏ทฏ 510 = ๏ทฎ100(๐ผ๐๐๐๐๐๐๐๐ก ๏ทฎ๏ทฎxi2 ) โ 4000๏ทฏ2๏ทฏ๏ทฏ Squaring both sides (510)2 = ๏ทฎ100(๐ผ๐๐๐๐๐๐๐๐ก ๏ทฎ๏ทฎxi2 )โ 4000๏ทฏ2๏ทฏ๏ทฏ๏ทฏ๏ทฎ2๏ทฏ 260100= 100(๐ผ๐๐๐๐๐๐๐๐ก ๏ทฎ๏ทฎxi2 )โ 4000๏ทฏ2๏ทฏ 260100= 100(๐ผ๐๐๐๐๐๐๐๐ก ๏ทฎ๏ทฎxi2 ) โ๏ทฏ 16000000 260100 + 16000000 = 100(๐ผ๐๐๐๐๐๐๐๐ก ๏ทฎ๏ทฎxi2 ) ๏ทฏ 16260100 = 100(๐ผ๐๐๐๐๐๐๐๐ก ๏ทฎ๏ทฎxi2 ) ๏ทฏ 100(๐ผ๐๐๐๐๐๐๐๐ก ๏ทฎ๏ทฎxi2 ) ๏ทฏ = 16260100 ๐ผ๐๐๐๐๐๐๐๐ก ๏ทฎ๏ทฎxi2 ๏ทฏ = 1๏ทฎ100๏ทฏ ร 16260100 ๐ผ๐๐๐๐๐๐๐๐ก ๏ทฎ๏ทฎxi2 ๏ทฏ = 162601 Since 50 is removed and 40 is added, So, (๐ช๐๐๐๐๐๐ ๏ทฎ๏ทฎxi2 ) ๏ทฏ = 162601 โ (50)2 + (40)2 = 162601 โ 2500 + 1600 = 161701 Now, Correct Standard deviation (ฯ) = 1๏ทฎN๏ทฏ ๏ทฎN ร(๐ถ๐๐๐๐๐๐ก ๏ทฎ๏ทฎxi2 )โ ๐ถ๐๐๐๐๐๐ก ๏ทฎ๏ทฎxi๏ทฏ๏ทฏ2๏ทฏ๏ทฏ = 1๏ทฎ100๏ทฏ ๏ทฎ100 161701๏ทฏ โ 3990๏ทฏ2๏ทฏ = 1๏ทฎ100๏ทฏ ๏ทฎ16170100 โ15920100๏ทฏ = 1๏ทฎ100๏ทฏ ๏ทฎ250000๏ทฏ = 1๏ทฎ100๏ทฏ ๏ทฎ 500๏ทฏ2๏ทฏ = 1๏ทฎ100๏ทฏ ร 500 = 5 Hence, Correct mean = 39.9 & Correct standard deviation = 5