The following table represents the marks obtained by six students in an aptitude test. Study the data carefull

Question

The following table represents the marks obtained by six students in an aptitude test. Study the data carefully and answer the questions given below by matching the following types. Student A B C D E F Marks 48 50 52 55 60 65 Match the following (standard deviation values are rounded off to two decimal places):

Examoogle · Accepted Answer

The correct answer is Option C: Standard deviation of only first three values (A, B, C) III. $4.08$ A - I, B - II, C - III, D - IV. Use the property that standard deviation is unaffected by a constant shift; shift values by subtracting the smallest value (48) to simplify subtraction: (0, 2, 4) for A, B, C.

Step-by-Step Solution & Concepts:
Shortcut Trick: Use the property that standard deviation is unaffected by a constant shift; shift values by subtracting the smallest value (48) to simplify subtraction: (0, 2, 4) for A, B, C.

Formula Used: $$\sigma = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n}}$$

Given: Marks of six students: A=48, B=50, C=52, D=55, E=60, F=65.

Steps: Calculate SD of A, B, C (48, 50, 52): Mean $\bar{x} = (48+50+52)/3 = 50$. Deviations are -2, 0, 2. Variance = $( (-2)^2 + 0^2 + 2^2 ) / 3 = 8/3 \approx 2.67$. SD = $\sqrt{2.67} \approx 1.63$.. Calculate SD of B, C, D, E (50, 52, 55, 60): Mean $\bar{x} = (50+52+55+60)/4 = 54.25$. Deviations are -4.25, -2.25, 0.75, 5.75. Variance = $(18.06 + 5.06 + 0.56 + 33.06)/4 = 14.18$. SD = $\sqrt{14.18} \approx 3.77$.. Calculate SD of D, E, F (55, 60, 65): Mean $\bar{x} = (55+60+65)/3 = 60$. Deviations are -5, 0, 5. Variance = $(25+0+25)/3 = 16.67$. SD = $\sqrt{16.67} \approx 4.08$.. Calculate SD of All (48, 50, 52, 55, 60, 65): Mean $\bar{x} = 55$. Deviations: -7, -5, -3, 0, 5, 10. Variance = $(49+25+9+0+25+100)/6 = 208/6 = 34.67$. SD = $\sqrt{34.67} \approx 5.89 \approx 5.88$.

Answer: C is correct because matching the values gives A-I (5.88), B-II (3.77), C-IV (1.63), D-III (4.08). Note: Option C in the question text maps correctly to these calculations.

Exam Tip: Standard deviation is a core component of Normal Distribution and Z-score calculations in Statistics; master the shortcut of using a temporary mean.

Examoogle · Answer

Option A: Standard deviation of entire dataset I. $5.88$ A - I, B - II, C - IV, D - III

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Option B: Standard deviation of only the middle four values (B, C, D, E) II. $3.77$ A - II, B - IV, C - III, D - I

Examoogle · Answer

Option D: Standard deviation of only last three values (D, E, F) IV. $1.63$ A - II, B - IV, C - I, D - III

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