Let P be the median, Q be the mean and R be the mode of observations x1, x2, x3, .....xn. Let \(S=\sum_{i=1}^n\left(2 x_i-a\right)^2\) S takes minimum value, when a is equal to
2Q
Minimizing a Sum of Squared Differences
The problem asks for the value of 'a' that minimizes the expression \(S=\sum_{i=1}^n\left(2 x_i-a\right)^2\), given that Q is the mean of the observations \(x_1, x_2, \dots, x_n\).
In statistics, a fundamental property states that the sum of squared deviations of a set of numbers from their mean is the minimum possible sum of squared deviations. That is, for any set of numbers \(y_1, y_2, \dots, y_n\), the sum \(\sum_{i=1}^n (y_i - c)^2\) is minimized when \(c\) is the mean of the \(y_i\)'s.
In our given expression \(S=\sum_{i=1}^n\left(2 x_i-a\right)^2\), the terms inside the square are of the form \((2x_i - a)\). Let's consider the values \(y_i = 2x_i\). Then the expression becomes \(S=\sum_{i=1}^n (y_i - a)^2\).
According to the property mentioned above, this sum \(S\) will be minimized when 'a' is equal to the mean of the values \(y_i\).
Let's calculate the mean of \(y_i\). The mean of \(y_1, y_2, \dots, y_n\) is given by:
\( \text{Mean of } y_i = \frac{\sum_{i=1}^n y_i}{n} \)
Substituting \(y_i = 2x_i\), we get:
\( \text{Mean of } (2x_i) = \frac{\sum_{i=1}^n (2x_i)}{n} \)
Using the property of summation \(\sum (c \cdot z_i) = c \cdot \sum z_i\), we can write:
\( \text{Mean of } (2x_i) = \frac{2 \sum_{i=1}^n x_i}{n} \)
We are given that Q is the mean of the observations \(x_1, x_2, \dots, x_n\). The mean of \(x_i\) is defined as:
\( Q = \frac{\sum_{i=1}^n x_i}{n} \)
Substituting this into the expression for the mean of \(2x_i\):
\( \text{Mean of } (2x_i) = 2 \left( \frac{\sum_{i=1}^n x_i}{n} \right) = 2Q \)
Therefore, the sum \(S=\sum_{i=1}^n\left(2 x_i-a\right)^2\) is minimized when 'a' is equal to the mean of the values \(2x_i\), which is \(2Q\).
Let's look at the given options:
- P (Median of \(x_i\))
- Q/2
- 2Q
- R (Mode of \(x_i\))
The value of 'a' that minimizes S is \(2Q\), which corresponds to the third option. The median (P) and mode (R) of \(x_i\) are not directly related to the minimization of this specific sum of squared differences.
Thus, S takes the minimum value when a is equal to 2Q.
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