The median of a set of 9 distinct observations is 20.5. If each of the largest 4 observations of the

Q: Wait, if we increase the values of the observations, shouldn't the median also increase?

That is a common trap! The median is a positional statistic, not an arithmetic one like the mean. It only cares about the middle position. As long as the values being increased remain on the same side of the median, the median itself remains anchored to its original position.

Q: What if we increased the smallest 4 observations instead of the largest 4?

Great question! If you increased the smallest 4 observations, they would still remain smaller than the median $x_5$. Thus, the median would still be the 5th term, and it would remain unchanged. The median is remarkably robust to changes in the extreme values!

Q: Why does the problem specify that the observations are 'distinct'?

The 'distinct' property ensures that the inequality $x_1 < x_2 < \dots < x_9$ is strict. This guarantees that when we add 2 to the largest 4 terms, they definitely remain greater than the median, preventing any ambiguity about their new positions relative to the median.

Question

The median of a set of 9 distinct observations is 20.5. If each of the largest 4 observations of the set is increased by 2, then the median of the new set

Options:

(A) remains the same as that of the original set

(B) is increased by 2

(C) is decreased by 2

(D) is two times the original median.

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are standing in a line of nine people, arranged strictly by height from shortest to tallest. In this scenario, the person standing exactly in the middle is the median.

In our problem, we have nine distinct observations, which we can represent as $x_1 < x_2 < x_3 < x_4 < x_5 < x_6 < x_7 < x_8 < x_9$.

Because there are nine observations, the median is the middle-most term. Using the formula for the position of the median, which is $\frac{n+1}{2}$, we find that for $n=9$, the position is:

$$ \frac{9+1}{2} = 5 $$

Thus, the fifth term, $x_5$, is our median, and we are given that $x_5 = 20.5$. This value is the anchor of our data set, perfectly splitting the observations into two groups of four.

### The Transformation: A Shift in Perspective

Now, the problem introduces a transformation: each of the largest four observations is increased by 2. In our sorted sequence, the largest four observations are clearly $x_6, x_7, x_8,$ and $x_9$.

These are the four individuals standing to the right of our median, $x_5$. When we add 2 to each of these values, we are essentially shifting them further to the right on the number line.

The new values become:

$$ x'_6 = x_6 + 2, \quad x'_7 = x_7 + 2, \quad x'_8 = x_8 + 2, \quad x'_9 = x_9 + 2 $$

The crucial realization here is that because $x_6, x_7, x_8,$ and $x_9$ were already strictly greater than $x_5$, adding 2 to them only makes them even larger. They do not 'jump over' the median; they simply move further away from it.

### The Stability of the Median

Let us look at the new sequence: $x_1 < x_2 < x_3 < x_4 < x_5 < x'_6 < x'_7 < x'_8 < x'_9$. Notice that the first five terms, including our median $x_5$, have not changed at all.

The total number of observations is still nine. Therefore, the median of this new set is still the fifth term.

Since the fifth term remains $x_5 = 20.5$, the median of the new set is identical to the median of the original set. This is the beauty of the median—it is a measure of central tendency that is resistant to changes in the extreme values of a dataset.

While the mean would have shifted significantly due to the increase in the largest values, the median stands firm, unmoved by the transformation of the outliers.

**The median remains 20.5, and it remains the same as the original set.**

Animated Solution for Mathematics - Statistics: The median of a set of 9 distinct observations is 20.5. If each of the largest 4 observations of the set is increased by 2, then the median of the new set

Select Answer:

Visualized Solution

Visualizing the Observations

Understanding the Median Formula

Locating the Median Position

Identifying the Median Value

Identifying the Largest Four Observations

Applying the Transformation

Analyzing the Order of the New Set

Formulating the New Sequence

Determining the New Median

Final Conclusion

The Sigma Insight: Measures of Central Tendency (Mean, Median, Mode)

Analyzing the Setup

The Transformation

The Stability of the Median

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