In an experiment with 15 observations...

MathematicsMeasures of DispersionJEE Main 2003Easy

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In an experiment with 15 observations on x, the following results were available: $\sum x^{2} = 2830, \sum x = 170$ . One observation that was 20 was found to be wrong and was replaced by the correct value 30. The corrected variance is

(A)

8.33

(B)

78.00

(C)

188.66

(D)

177.33

Answer:B

Visualized Solution (English)

Identify Given Parameters

Number of observations $n = 15$
Given incorrect sum $\sum x = 170$
Given incorrect sum of squares $\sum x^{2} = 2830$

The Error Detection

Incorrect value $= 20$
Correct value $= 30$

Variance Formula Bridge

Variance formula: $σ^{2} = \frac{\sum x ^{2}}{n} - (\frac{\sum x}{n})^{2}$
We need corrected values for $\sum x$ and $\sum x^{2}$ .

Correcting the Sum .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } \sum x

Corrected $\sum x = (Old \sum x) - (Wrong Value) + (Correct Value)$
Corrected $\sum x = 170 - 20 + 30 = 180$

Correcting Sum of Squares .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } \sum x^{2}

Corrected $\sum x^{2} = (Old \sum x^{2}) - (Wrong Value)^{2} + (Correct Value)^{2}$
Corrected $\sum x^{2} = 2830 - 2 0^{2} + 3 0^{2}$

Calculating Corrected .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } \sum x^{2}

Corrected $\sum x^{2} = 2830 - 400 + 900$
Corrected $\sum x^{2} = 3330$

Calculating Corrected Mean

Corrected Mean $\overset{x}{ˉ} = \frac{Corrected \sum x}{n}$
$\overset{x}{ˉ} = \frac{180}{15} = 12$

Final Variance Setup

Corrected Variance $= \frac{3330}{15} - (12)^{2}$

Execution of Division

$\frac{3330}{15} = 222$
$1 2^{2} = 144$

The Final Result

Corrected Variance $= 222 - 144$
Corrected Variance $= 78$

Key Takeaway

Key Takeaway: Always correct $\sum x$ and $\sum x^{2}$ separately before calculating variance.
Next Challenge: Try solving this if two observations were incorrect instead of one!

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