Animated Solution for Mathematics - Statistics: The mean of a set of 30 observations is 75. If each observation is multiplied by a non-zero number $\lambda$ and then each of them is decreased by 25, their mean remains the same. Then $\lambda$ is equal to

Welcome, future engineer! Today, we are going to peel back the layers of a classic statistics problem. It is easy to get intimidated by the sheer volume of information in a JEE Advanced question, but I want you to take a deep breath.

Statistics is not just about crunching numbers; it is about understanding the 'center of gravity' of your data. Imagine you have a cloud of thirty data points, and their average position—the mean—is sitting at $\bar{x}=75$. Now, we are going to perform a transformation on every single one of these points: we multiply each by a constant $\lambda$ and then shift them all down by $25$. The question asks us to find $\lambda$ such that the mean remains unchanged.

First, let us address the elephant in the room: the number of observations, $n=30$. You might be tempted to start writing out sums like $\sum x_i=30\times 75$.

While that is mathematically correct, it is a trap! In competitive exams, examiners love to include extra information to test your confidence. The mean is a property of the distribution, and it behaves predictably under linear transformations regardless of the sample size. You can safely ignore the $30$ and focus on the transformation itself.

Here is the core principle you need in your toolkit: the mean is sensitive to both change of scale and origin. If you have a set of observations $x_i$ with mean $\bar{x}$, and you apply the transformation $y_i=a{x}_i+b$, the new mean $\bar{y}$ is simply:

Think of it like this: if you scale a map by a factor of $a$ and then shift the origin by $b$, the center of the map moves exactly the same way. In our problem, the transformation is $y_i=\lambda x_i-25$.

Therefore, our new mean must be $\bar{y}=\lambda \bar{x}-25$. This is the elegant shortcut that separates the masters from the novices.

Now that we have the relationship $\bar{y}=\lambda \bar{x}-25$, we simply plug in what we know. We are told the mean remains the same, so $\bar{y}=75$ and $\bar{x}=75$.

The equation becomes:

Now, let us solve this with precision. Add $25$ to both sides to get:

Finally, divide by $75$ to isolate $\lambda$:

Simplifying this fraction by dividing both the numerator and denominator by $25$, we arrive at:

There it is! We navigated the distractor, applied the fundamental property of the mean, and solved the algebra with confidence. Keep this mindset for your exam: identify the core principle, ignore the noise, and trust your tools. The final answer is $\lambda =\frac{4}{3}$.