Understanding Mean Deviation About Median

The question asks us to find the new mean deviation about the median after all the original 10 observations are multiplied by -3. We are given that the original mean deviation about the median is 15.

Mean Deviation About Median Formula

The mean deviation about the median ($M$) for a set of $n$ observations ($x_1, x_2, \ldots, x_n$) is calculated using the formula:

$$MD_{median} = \frac{1}{n} \sum_{i=1}^{n} |x_i - M|$$

In this problem, we have $n=10$ observations, and the given mean deviation about the median is 15.

$$MD_{median} = 15$$

Effect of Multiplying Observations by a Constant

A key property of measures of dispersion like mean deviation is how they change when the data is transformed. If each observation $x_i$ is multiplied by a constant $a$, the new observation becomes $y_i = ax_i$. Consequently, the new median will be $M' = aM$.

The new mean deviation about the new median ($M'$) is:

$$MD'_{median} = \frac{1}{n} \sum_{i=1}^{n} |y_i - M'|$$ $$MD'_{median} = \frac{1}{n} \sum_{i=1}^{n} |ax_i - aM|$$ $$MD'_{median} = \frac{1}{n} \sum_{i=1}^{n} |a(x_i - M)|$$ $$MD'_{median} = \frac{1}{n} \sum_{i=1}^{n} |a| |x_i - M|$$ $$MD'_{median} = |a| \left( \frac{1}{n} \sum_{i=1}^{n} |x_i - M| \right)$$ $$MD'_{median} = |a| \times MD_{median}$$

It's important to note that mean deviation is always a non-negative value. Therefore, we use the absolute value of the constant multiplier.

Calculating the New Mean Deviation

In this specific problem:

• The original mean deviation about the median is $MD_{median} = 15$.
• Each observation is multiplied by the constant $a = -3$.

Using the property derived above, the new mean deviation about the median is:

$$MD'_{median} = |-3| \times MD_{median}$$ $$MD'_{median} = 3 \times 15$$ $$MD'_{median} = 45$$

Conclusion

Therefore, when each of the 10 observations is multiplied by $-3$, the new mean deviation about the median of the resulting observations is 45.