Mean of Four Observations

Given:

- Mean of four observations is 10
- When a constant is added to each observation the mean becomes 13

We need to find the value of a.

Let's start by understanding the concept of mean.

Mean

Mean is a measure of central tendency that is calculated by adding up all the observations in a data set and dividing the sum by the total number of observations.

For example, if we have the following data set:

2, 4, 6, 8

The mean would be:

(2 + 4 + 6 + 8) / 4 = 5

Mean with Constant

When a constant is added to each observation in a data set, the mean also changes.

Let's take the same data set as before, but add a constant of 3 to each observation:

2 + 3, 4 + 3, 6 + 3, 8 + 3 = 5, 7, 9, 11

The new mean would be:

(5 + 7 + 9 + 11) / 4 = 8

Solving the Problem

Now, let's apply this concept to the problem at hand.

We are given that the mean of four observations is 10. Let's call these observations x1, x2, x3, and x4.

We can write an equation for the mean as:

(x1 + x2 + x3 + x4) / 4 = 10

Simplifying this equation:

x1 + x2 + x3 + x4 = 40

Next, we are told that when a constant a is added to each observation, the mean becomes 13.

We can write a new equation for the mean as:

(x1 + a + x2 + a + x3 + a + x4 + a) / 4 = 13

Simplifying this equation:

(x1 + x2 + x3 + x4 + 4a) / 4 = 13

Multiplying both sides by 4:

x1 + x2 + x3 + x4 + 4a = 52

We can substitute the first equation into the second equation:

40 + 4a = 52

Solving for a:

4a = 12

a = 3

Therefore, the value of a is 3.

Conclusion

In conclusion, we can use the concept of mean to solve problems where a constant is added to each observation in a data set. By setting up equations and solving for the unknown variable, we can find the value of the constant.