Animated Solution for Mathematics - Statistics: If the mean deviation about the median of the numbers a, 2a,.......,50a is 50, then | a| equals

Welcome, future engineer! Today, we are not just solving a statistics problem; we are embarking on a journey to understand the 'heartbeat' of a data set. When you look at the sequence $a,2a,3a,\dots ,50a$, what do you see?

To the untrained eye, it is just a list of numbers. But to a physicist or a mathematician, this is a perfectly uniform distribution—a beautiful, linear progression. Let us break this down step by step.

Before we can measure deviation, we must find our anchor: the median. In statistics, the median is the 'center of gravity' of your data.

Because our sequence has $n=50$ terms—an even number—there is no single middle term. Instead, the median $M$ lies right in the gap between the $25^{th}$ and $26^{th}$ terms.

We know the $25^{th}$ term is $25a$ and the $26^{th}$ term is $26a$. To find the median, we calculate the average:

Think of this as the balance point on a seesaw. Everything to the left is balanced by everything to the right.

Now, we define Mean Deviation about the median ($M{D}_M$). It is the average of the absolute distances of every point from our median.

Mathematically, this is expressed as:

This formula is elegant. It asks: 'On average, how far is each data point from the center?' We substitute our values: $n=50$, $x_i=ia$, and $M=25.5a$.

This gives us:

Here is where the magic happens. Look at the term inside the summation: $|ia-25.5a|$. We can factor out the $|a|$ immediately.

Why the absolute value? Because distance is always positive, regardless of whether $a$ is positive or negative. We are left with:

Now, let us expand that sum. It looks like this: $|1-25.5|+|2-25.5|+\cdots +|50-25.5|$.

Notice the symmetry! The distance of the $1^{st}$ term ($1$) from $25.5$ is $24.5$. The distance of the $50^{th}$ term ($50$) from $25.5$ is also $24.5$.

The $2^{nd}$ term ($2$) and the $49^{th}$ term ($49$) share the same distance of $23.5$. We can group these pairs! Instead of summing 50 terms, we sum the first 25 and multiply by 2:

We are looking at an arithmetic progression inside the parentheses. The sum of an AP is $\frac{n}{2}(first+last)$.

Here, $n=25$, the first term is $0.5$, and the last is $24.5$:

Don't forget our symmetry multiplier of $2$! The total sum is $2\times 312.5=625$.

Now, we plug this back into our equation for $M{D}_M$:

We are given that the mean deviation is $50$. So, we set $12.5|a|=50$.

Solving for $|a|$, we get:

And there it is! Through logic, symmetry, and a bit of algebraic discipline, we have arrived at the solution. Never fear the complexity of a summation; look for the symmetry, and the problem will solve itself. The final answer is $|a|=4$.