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Statistics question

2022 · 26 Jul · Shift 2 · Q36
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Statistics question

2022 · 26 Jul · Shift 2 · Q36

JEE MainMathematicsStatisticsNumerical+4 / −1
The mean and standard deviation of 40 observations are 30 and 5 respectively. It was noticed that two of these observations 12 and 10 were wrongly recorded. If σ\sigmaσ is the standard deviation of the data after omitting the two wrong observations from the data, then 38σ238 \sigma^{2}38σ2 is equal to ‾\underline{\hspace{2cm}}​.
Numerical answer
View written solutionFree

Correct answer: 238

  1. Given data for 40 observations
  • Number of observations: n=40n=40n=40
  • Mean: xˉ=30\bar{x}=30xˉ=30
  • Standard deviation: 555

Using the standard formula,

σ2=∑xi2n−xˉ2\sigma^2=\frac{\sum x_i^2}{n}-\bar{x}^2σ2=n∑xi2​​−xˉ2

so

25=∑xi240−30225=\frac{\sum x_i^2}{40}-30^225=40∑xi2​​−302 25=∑xi240−90025=\frac{\sum x_i^2}{40}-90025=40∑xi2​​−900 ∑xi240=925\frac{\sum x_i^2}{40}=92540∑xi2​​=925 ∑xi2=925⋅40=37000\sum x_i^2=925\cdot 40=37000∑xi2​=925⋅40=37000

Also,

∑xi=nxˉ=40⋅30=1200\sum x_i = n\bar{x}=40\cdot 30=1200∑xi​=nxˉ=40⋅30=1200
  1. Omit the two wrongly recorded observations 121212 and 101010

After removing them:

  • New number of observations: 40−2=3840-2=3840−2=38
  • New sum:
∑xi′=1200−(12+10)=1200−22=1178\sum x_i' = 1200-(12+10)=1200-22=1178∑xi′​=1200−(12+10)=1200−22=1178
  • New mean:
xˉ′=117838=31\bar{x}'=\frac{1178}{38}=31xˉ′=381178​=31

Now for squares:

∑xi′2=37000−(122+102)\sum {x_i'}^2 = 37000-(12^2+10^2)∑xi′​2=37000−(122+102) =37000−(144+100)=37000−244=36756=37000-(144+100)=37000-244=36756=37000−(144+100)=37000−244=36756
  1. Find the new variance

Let the new standard deviation be σ\sigmaσ. Then

σ2=∑xi′238−(xˉ′)2\sigma^2=\frac{\sum {x_i'}^2}{38}-(\bar{x}')^2σ2=38∑xi′​2​−(xˉ′)2 σ2=3675638−312\sigma^2=\frac{36756}{38}-31^2σ2=3836756​−312

Therefore,

38σ2=36756−38⋅31238\sigma^2 = 36756-38\cdot 31^238σ2=36756−38⋅312

Since

312=961,31^2=961,312=961,

we get

38σ2=36756−38⋅96138\sigma^2=36756-38\cdot 96138σ2=36756−38⋅961 =36756−36518=238=36756-36518=238=36756−36518=238
  1. Final answer
38σ2=23838\sigma^2=23838σ2=238
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