Key Concepts and Formulas

1. Arithmetic Mean ($\overline{x}$): For a set of $n$ observations $x_1, x_2, \ldots, x_n$, the arithmetic mean is the sum of all observations divided by the number of observations. $\overline{x} = \frac{\sum_{i=1}^{n} x_i}{n}$
2. Properties of Summation: These properties are fundamental for manipulating sums:
   • Sum of a difference: The sum of differences can be split into differences of sums: $\sum_{i=1}^{n} (a_i - b) = \sum_{i=1}^{n} a_i - \sum_{i=1}^{n} b$.
   • Sum of a constant: The sum of a constant $c$ taken $n$ times is $nc$: $\sum_{i=1}^{n} c = nc$.
3. Relationship between Mean and Sum of Deviations from an Arbitrary Point ($A$): The sum of deviations of $n$ observations $x_i$ from an arbitrary constant $A$ is given by $\sum_{i=1}^{n} (x_i - A)$. Using the summation properties and the definition of the mean, we can derive a very useful relationship: $\sum_{i=1}^{n} (x_i - A) = \sum_{i=1}^{n} x_i - \sum_{i=1}^{n} A$ Since $\sum_{i=1}^{n} x_i = n\overline{x}$ and $\sum_{i=1}^{n} A = nA$: $\sum_{i=1}^{n} (x_i - A) = n\overline{x} - nA = n(\overline{x} - A)$ From this, we can directly solve for the mean: $\overline{x} = A + \frac{\sum_{i=1}^{n} (x_i - A)}{n}$ This formula is the basis of the "Assumed Mean Method" and provides a quick way to find the mean when the sum of deviations from an arbitrary point is known.

Step-by-Step Solution

The problem provides the following information:

• Number of observations ($n$) = 50.
• The sum of the deviations of these 50 observations from 30 is 50. This means the arbitrary constant $A = 30$, and $\sum_{i=1}^{50} (x_i - 30) = 50$.
• We need to find the mean ($\overline{x}$) of these observations.

Step 1: Translate the given information into a mathematical equation.

• What we are doing: We are converting the problem's verbal statement into a precise mathematical equation.
• Why we are doing it: This forms the fundamental equation that we will manipulate to determine the mean. The problem states, "the sum of the deviations of 50 observations from 30 is 50." If $x_i$ represents the $i$-th observation, then: $\sum_{i=1}^{50} (x_i - 30) = 50$

Step 2: Apply summation properties to expand the equation.

• What we are doing: We will use the properties of summation to separate the terms within the sum, specifically isolating the sum of all observations ($\sum x_i$).
• Why we are doing it: The definition of the arithmetic mean requires the total sum of all observations ($\sum x_i$). By expanding the given sum, we can work towards finding this crucial value. Using the property $\sum_{i=1}^{n} (a_i - b) = \sum_{i=1}^{n} a_i - \sum_{i=1}^{n} b$, we expand the left side of our equation: $\sum_{i=1}^{50} x_i - \sum_{i=1}^{50} 30 = 50$ Next, we evaluate the sum of the constant term. Using the property $\sum_{i=1}^{n} c = nc$, where $c=30$ and $n=50$: $\sum_{i=1}^{50} 30 = 50 \times 30 = 1500$ Substitute this value back into the expanded equation: $\sum_{i=1}^{50} x_i - 1500 = 50$

Step 3: Solve for the sum of observations ($\sum x_i$).

• What we are doing: We are algebraically isolating the term representing the sum of all observations.
• Why we are doing it: This value, $\sum x_i$, is a necessary component for calculating the arithmetic mean using its fundamental definition. To isolate $\sum x_i$, we add 1500 to both sides of the equation: $\sum_{i=1}^{50} x_i = 50 + 1500$ $\sum_{i=1}^{50} x_i = 1550$

Step 4: Calculate the arithmetic mean ($\overline{x}$).

• What we are doing: We are now applying the fundamental definition of the arithmetic mean using the sum of observations we just found.
• Why we are doing it: This is the final step to directly answer the question about the mean of the observations. We now have the sum of observations ($\sum x_i = 1550$) and the number of observations ($n=50$). Using the fundamental formula for the arithmetic mean: $\overline{x} = \frac{\sum_{i=1}^{n} x_i}{n}$ Substitute the values: $\overline{x} = \frac{1550}{50}$ Perform the division: $\overline{x} = \frac{155}{5}$ $\overline{x} = 31$

Alternative Approach (Using the Direct Formula from Key Concepts): We can also directly apply the formula $\overline{x} = A + \frac{\sum_{i=1}^{n} (x_i - A)}{n}$. Given:

• Arbitrary constant $A = 30$.
• Number of observations $n = 50$.
• Sum of deviations from $A$, $\sum_{i=1}^{n} (x_i - A) = 50$. Substitute these values into the formula: $\overline{x} = 30 + \frac{50}{50}$ $\overline{x} = 30 + 1$ $\overline{x} = 31$ This confirms the result obtained through the step-by-step derivation.

Common Mistakes & Tips

1. Misinterpreting "Deviation": Be careful to understand that "deviation of an observation from 30" means $(x_i - 30)$, not necessarily $(x_i - \overline{x})$. Deviation from the mean is a specific concept where the sum of such deviations is always zero.
2. Sum of a Constant: A common error is assuming $\sum_{i=1}^{n} c = c$ instead of $\sum_{i=1}^{n} c = nc$. When summing a constant $c$ for $n$ times, you are essentially multiplying $c$ by $n$. In this problem, $\sum_{i=1}^{50} 30 = 50 \times 30 = 1500$.
3. Utilize the Direct Formula: While understanding the step-by-step derivation is crucial for a deep conceptual grasp, for competitive exams like JEE, directly applying the formula $\overline{x} = A + \frac{\sum_{i=1}^{n} (x_i - A)}{n}$ can save valuable time.

Summary

This problem effectively illustrates how the arithmetic mean can be determined from the sum of deviations around an arbitrary point. By applying the basic properties of summation and the definition of the mean, we can systematically solve for the mean. The key takeaway is the powerful relationship $\overline{x} = A + \frac{\sum_{i=1}^{n} (x_i - A)}{n}$, which allows for a quick calculation. Understanding both the derivation and the direct application of this formula is essential for mastering such statistical problems. The calculation leads to an arithmetic mean of 31.

The final answer is $\boxed{\text{31}}$, which corresponds to option (A).