The problem in the image reads as follows:

18. The sum of deviations of a set of values $x_1, x_2, \dots, x_n$ measured from 50 is $-10$ and the sum of deviations of the same values from 46 is 70. Find the value of $n$ and the mean.

Let's break down the problem:

1. Sum of deviations from 50: $\sum_{i=1}^n (x_i - 50) = -10$

2. Sum of deviations from 46: $\sum_{i=1}^n (x_i - 46) = 70$

Step 1: Relationship Between the Two Equations

We can express the second sum of deviations as: $\sum_{i=1}^n (x_i - 46) = \sum_{i=1}^n (x_i - 50 + 4)$

This simplifies to: $\sum_{i=1}^n (x_i - 46) = \sum_{i=1}^n (x_i - 50) + \sum_{i=1}^n 4$

Since $\sum_{i=1}^n 4 = 4n$, we can write: $70 = -10 + 4n$

Step 2: Solve for $n$

Now, solve for $n$: $70 = -10 + 4n$ $70 + 10 = 4n$ $80 = 4n$ $n = 20$

Step 3: Find the Mean

Using the sum of deviations from 50: $\sum_{i=1}^n (x_i - 50) = -10$ This implies: $\sum_{i=1}^n x_i - 50n = -10$

Substitute $n = 20$: $\sum_{i=1}^n x_i - 50(20) = -10$ $\sum_{i=1}^n x_i - 1000 = -10$ $\sum_{i=1}^n x_i = 990$

The mean, $\bar{x}$, is given by: $\bar{x} = \frac{\sum_{i=1}^n x_i}{n} = \frac{990}{20} = 49.5$

Final Answer:

• The value of $n$ is 20.
• The mean is 49.5.

Would you like a more detailed explanation, or do you have any other questions?

Here are some related questions to deepen your understanding:

1. What is the significance of the sum of deviations being zero when measured from the mean?
2. How can we compute the variance using the deviations from the mean?
3. Why do we add 4n in the second equation involving deviations from 46?
4. Can the sum of deviations ever be greater than the sum of actual values?
5. How do we interpret the sum of deviations geometrically?

Tip: When solving problems with deviations, always use the fact that the sum of deviations from the mean is zero.