**Mean and Standard Deviation in Statistics**

Mean and standard deviation are important measures of central tendency and variability, respectively, in statistics. The mean represents the average value of a set of data, while the standard deviation measures the dispersion or spread of the data around the mean.

**Given Information**

In this question, we are given a set of 25 observations. If 40 is added to each observation, the coefficient of variation (C.V.) is 40%. If 40 is subtracted from each observation, the C.V. increases to 80%.

**Coefficient of Variation (C.V.)**

Let's first understand the concept of the coefficient of variation (C.V.). It is a relative measure of variability that is used to compare the dispersion of different data sets. It is calculated as the ratio of the standard deviation to the mean, expressed as a percentage.

C.V. = (Standard Deviation / Mean) * 100

**Analysis**

1. Addition of 40 to Each Observation

When 40 is added to each observation, the C.V. is 40%. This means that the standard deviation is 40% of the mean. Mathematically, we can write it as:

C.V. = (Standard Deviation + 40) / (Mean + 40) * 100

2. Subtraction of 40 from Each Observation

When 40 is subtracted from each observation, the C.V. increases to 80%. This means that the standard deviation becomes twice the mean. Mathematically, we can write it as:

C.V. = (Standard Deviation - 40) / (Mean - 40) * 100

**Calculating the Mean and Standard Deviation**

To find the mean and standard deviation, we need to solve the above two equations simultaneously.

Let's assume the mean as 'M' and the standard deviation as 'S'.

1. From the first equation, we have:

40 = 0.4M

2. From the second equation, we have:

80 = 2S

**Solving the Equations**

1. Solving for the Mean:

From the first equation, we get:

M = 40 / 0.4 = 100

Therefore, the mean is 100.

2. Solving for the Standard Deviation:

From the second equation, we get:

S = 80 / 2 = 40

Therefore, the standard deviation is 40.

**Summary**

The mean of the set of observations is 100, and the standard deviation is 40. Adding or subtracting 40 from each observation does not affect the mean, but it changes the dispersion of the data. In the given scenario, when 40 is added, the C.V. is 40%, indicating moderate variability. However, when 40 is subtracted, the C.V. increases to 80%, indicating higher variability or spread of the data around the mean.