Let the median and the mean deviation about the median of 7 observation 170, 125, 230, 190, 210, a,

Q: Why do we assume $a, b \le 170$?

Because the median is 170 and there are already 3 values greater than 170. For 170 to be the 4th value in an ordered set of 7, the remaining values must be less than or equal to it.

Q: Can we just use the mean formula directly?

No, we need the sum of $a$ and $b$ first, which we get from the mean deviation about the median. The mean formula alone would leave us with two unknowns.

Q: Why don't we need to find $a$ and $b$ individually?

Because the mean and mean deviation formulas only depend on the sum $(a+b)$, not the individual values. We only need the sum to solve the problem.

Question

Let the median and the mean deviation about the median of 7 observation 170, 125, 230, 190, 210, a, b be 170 and $\frac{205}{7}$ respectively. Then the mean deviation about the mean of these 7 observations is :

Options:

(A) 31

(B) 28

(C) 30

(D) 32

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Statistics is not just about crunching numbers; it is about finding the hidden structure within them. We are given seven observations: $170, 125, 230, 190, 210, a, b$.

We know the median is $170$ and the mean deviation about the median is $\frac{205}{7}$. Our goal is to find the mean deviation about the mean.

### The Modulus Trap

For seven observations, the median is the fourth term when arranged in ascending order. We already have three values greater than $170$: $190, 210,$ and $230$.

For $170$ to sit perfectly in the fourth position, the remaining three values must be less than or equal to it. This means $a$ and $b$ must be $\le 170$.

This constraint allows us to resolve the modulus brackets in the formula for mean deviation about the median, $MD_M = \frac{\sum |x_i - M|}{n}$, where $M = 170$. Because $a, b \le 170$, the terms $|a-170|$ and $|b-170|$ become $(170-a)$ and $(170-b)$ respectively.

### The Power of the Sum

We set up the equation for the mean deviation about the median:

$$
\frac{205}{7} = \frac{|125-170| + |170-170| + |190-170| + |210-170| + |230-170| + (170-a) + (170-b)}{7}
$$

Simplifying the numerator, we obtain:

$$
205 = 45 + 0 + 20 + 40 + 60 + 340 - (a+b)
$$

This reduces to $205 = 505 - (a+b)$, which yields the critical sum:

$$
a+b = 300
$$

### The Final Stretch

With $a+b = 300$, we calculate the arithmetic mean $\bar{x}$:

$$
\bar{x} = \frac{125 + 170 + 190 + 210 + 230 + (a+b)}{7} = \frac{925 + 300}{7} = \frac{1225}{7} = 175
$$

Now, we calculate the mean deviation about the mean, $MD_{\bar{x}} = \frac{\sum |x_i - 175|}{7}$. Since $a, b \le 170$, they are also $< 175$, meaning $|a-175| = 175-a$ and $|b-175| = 175-b$.

The sum becomes:

$$
MD_{\bar{x}} = \frac{|125-175| + |170-175| + |190-175| + |210-175| + |230-175| + (175-a) + (175-b)}{7}
$$

Substituting the values:

$$
MD_{\bar{x}} = \frac{50 + 5 + 15 + 35 + 55 + 350 - (a+b)}{7} = \frac{510 - 300}{7} = \frac{210}{7} = 30
$$

The final answer is **30**.

Animated Solution for Mathematics - Statistics: Let the median and the mean deviation about the median of 7 observation 170, 125, 230, 190, 210, a, b be 170 and $\frac{205}{7}$ respectively. Then the mean deviation about the mean of these 7 observations is :

Select Answer:

Visualized Solution

Identify the Given Data

Arrange Observations Using Median

Define Mean Deviation about Median

Calculate Absolute Differences

Solve for $(a + b)$

Calculate the Arithmetic Mean

Setup Mean Deviation about Mean

Final Atomic Computation

Conclusion & Key Takeaway

The Sigma Insight: Measures of Dispersion

Analyzing the Setup

The Modulus Trap

The Power of the Sum

The Final Stretch

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