Understanding Deviations and Their Sum

The problem provides information about the sum of deviations of a set of observations measured from two different points. Let's denote the set of $n$ observations as $x_1, x_2, \dots, x_n$. The deviation of an observation $x_i$ from a constant value $a$ is given by $(x_i - a)$. The sum of deviations of all $n$ observations from $a$ is $\sum_{i=1}^{n} (x_i - a)$.

Setting Up Equations from Given Information

We are given two pieces of information:

1. The sum of deviations of the $n$ observations measured from 2.5 is 50. $$\sum_{i=1}^{n} (x_i - 2.5) = 50$$
2. The sum of deviations of the same set of observations measured from 3.5 is -50. $$\sum_{i=1}^{n} (x_i - 3.5) = -50$$

Expanding the Summation

We can expand the summations using the property $\sum (a_i - b_i) = \sum a_i - \sum b_i$ and $\sum c = nc$ for a constant $c$.

From the first piece of information:

Let $\sum x_i$ denote the sum of all observations. This gives us Equation (1):

From the second piece of information:

This gives us Equation (2):

Solving the System of Equations to Find n

We now have a system of two linear equations with two unknowns, $\sum x_i$ and $n$:

Equation (1): $\sum x_i - 2.5n = 50$

Equation (2): $\sum x_i - 3.5n = -50$

We can solve this system by subtracting Equation (2) from Equation (1):

The $\sum x_i$ terms cancel out:

So, the value of $n$, the number of observations, is 100.

Understanding the Relationship Between Sum of Deviations

It is a property that the sum of deviations from the mean ($\bar{x}$) is always zero: $\sum_{i=1}^{n} (x_i - \bar{x}) = 0$. The sum of deviations from any other point 'a' is given by $\sum_{i=1}^{n} (x_i - a) = \sum_{i=1}^{n} x_i - na = n\bar{x} - na = n(\bar{x} - a)$.

Using this property for the given problem:

Sum of deviations from 2.5: $n(\bar{x} - 2.5) = 50$

Sum of deviations from 3.5: $n(\bar{x} - 3.5) = -50$

Dividing the first equation by the second (assuming $n \neq 0$ and $\bar{x} \neq 3.5$):

The mean of the observations is 3. Now substitute $\bar{x}=3$ into either deviation equation. Using the first one:

This confirms our earlier result for $n$. Both methods yield the same answer.

Final Answer Derivation

By setting up and solving the system of linear equations based on the given sum of deviations, we found that the number of observations, $n$, is 100.

Revision Table: Key Concepts in Statistics

Additional Information on Sum of Deviations

The sum of deviations is a fundamental concept in statistics. While the sum of deviations from any arbitrary point isn't necessarily zero, the sum of deviations from the mean is always zero. This property is a direct result of how the mean is defined.

Let's consider the sum of deviations from the mean, $\bar{x}$:

Since $\bar{x} = \frac{\sum x_i}{n}$, we have $\sum x_i = n\bar{x}$. Substituting this back into the equation:

This fundamental property is crucial in many statistical formulas, including the calculation of variance and standard deviation, which are based on squared deviations from the mean.