Determining 'y' from Algebraic Sum of Deviations

This problem requires us to find a specific value, y, based on information about the algebraic sum of deviations of a set of values $\{x_1, x_2, ..., x_n\}$ from different reference points. We are given two scenarios and asked to solve for a third.

Understanding the Algebraic Sum of Deviations

The algebraic sum of deviations of a set of values from a specific point 'a' is calculated by subtracting 'a' from each value in the set and then summing up these differences. Mathematically, this is represented as:

$$ \sum_{i=1}^{n} (x_i - a) $$

This expression can be expanded using the properties of summation:

$$ \sum_{i=1}^{n} (x_i - a) = \sum_{i=1}^{n} x_i - \sum_{i=1}^{n} a $$

Since $a$ is a constant, $\sum_{i=1}^{n} a = n \cdot a$. Let $S$ represent the sum of all values in the set, $S = \sum_{i=1}^{n} x_i$. The formula becomes:

$$ \sum_{i=1}^{n} (x_i - a) = S - n \cdot a $$

This formula connects the sum of deviations to the total sum ($S$) and the number of values ($n$).

Calculating Unknown Values using Given Deviations

Step 1: Formulating Equations from the Given Data

We are provided with the following details:

• The algebraic sum of deviations from 100 is -20. Using our formula ($a=100$):

  $$ S - n \cdot 100 = -20 \quad \quad (1) $$

• The algebraic sum of deviations from 92 is 140. Using our formula ($a=92$):

  $$ S - n \cdot 92 = 140 \quad \quad (2) $$

Step 2: Solving the System of Equations for 'S' and 'n'

We have two equations and two unknowns ($S$ and $n$). We can solve them by elimination. Subtracting Equation (2) from Equation (1):

$$ (S - 100n) - (S - 92n) = -20 - 140 $$

$$ S - 100n - S + 92n = -160 $$

$$ -8n = -160 $$

To find $n$, divide both sides by -8:

$$ n = \frac{-160}{-8} $$

$$ n = 20 $$

Now, substitute the value $n=20$ back into Equation (1) to find $S$:

$$ S - (20 \cdot 100) = -20 $$

$$ S - 2000 = -20 $$

$$ S = 2000 - 20 $$

$$ S = 1980 $$

So, the total sum of the values is $S = 1980$, and the count of values is $n = 20$.

Step 3: Calculating the Value of 'y'

The problem asks for the value of $y$ where the algebraic sum of deviations is 180.

$$ \sum_{i=1}^{n} (x_i - y) = 180 $$

Using the formula $S - n \cdot y$:

$$ S - n \cdot y = 180 $$

Substitute the values we found: $S = 1980$ and $n = 20$:

$$ 1980 - 20 \cdot y = 180 $$

Now, we solve for $y$:

$$ -20y = 180 - 1980 $$

$$ -20y = -1800 $$

Divide both sides by -20:

$$ y = \frac{-1800}{-20} $$

$$ y = 90 $$

Final Answer

The value of $y$ that satisfies the condition is 90.