Let the mean and the standard deviation of the observation 2, 3, 3, 4, 5, 7, a, b be 4 and √2 respec

Q: Wait, why do we need the variance formula? Can't we just use the mean?

Great question! The mean only gives us one constraint: $a+b=8$. With two unknowns, we need two independent equations to solve for them uniquely. The variance provides that crucial second constraint, allowing us to lock down the values of $a$ and $b$.

Q: How do I know which number is the mode?

The mode is simply the observation that appears most frequently in your dataset. Once you find $a=4$ and $b=4$, your full dataset becomes $2, 3, 3, 4, 4, 4, 5, 7$. By counting, you'll see that $4$ appears three times, which is more than any other number. Thus, $4$ is your mode.

Q: Is it possible for the mean deviation to be zero?

Only if all observations are identical to the mode! In this problem, the data is spread out, so the mean deviation must be a positive value. It measures the average 'distance' of your data points from the mode, so it acts like an average error.

Question

Let the mean and the standard deviation of the observation 2, 3, 3, 4, 5, 7, a, b be 4 and $\sqrt{2}$ respectively. Then the mean deviation about the mode of these observations is :

Options:

(A) 1

(B) $\frac{3}{4}$

(C) 2

(D) $\frac{1}{2}$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

The dataset consists of eight numbers: $2, 3, 3, 4, 5, 7, a, b$. We are given that the mean $\bar{x} = 4$ and the standard deviation $\sigma = \sqrt{2}$.

The mean is defined as:
$$\bar{x} = \frac{\sum x_i}{n}$$

Given $n=8$ and $\bar{x}=4$, the sum of all observations must be $8 	imes 4 = 32$. Summing the known values: $2 + 3 + 3 + 4 + 5 + 7 = 24$.

This leads to our first equation:
$$24 + a + b = 32 \Rightarrow a + b = 8$$

### The Variance: Measuring the Chaos

To find a second equation, we utilize the variance $\sigma^2 = 2$. The formula for variance is:
$$\sigma^2 = \frac{\sum x_i^2}{n} - (\bar{x})^2$$

Substituting our known values:
$$\frac{2^2 + 3^2 + 3^2 + 4^2 + 5^2 + 7^2 + a^2 + b^2}{8} - 4^2 = 2$$

Calculating the sum of the squares of the known terms: $4 + 9 + 9 + 16 + 25 + 49 = 112$. The equation becomes:
$$\frac{112 + a^2 + b^2}{8} - 16 = 2$$

Adding $16$ to both sides yields $18$, and multiplying by $8$ results in $144$. Subtracting $112$ gives us:
$$a^2 + b^2 = 32$$

### The Algebraic Bridge

We now have the system $a+b=8$ and $a^2+b^2=32$. We use the algebraic identity:
$$(a+b)^2 = a^2 + b^2 + 2ab$$

Substituting the known values:
$$8^2 = 32 + 2ab \Rightarrow 64 = 32 + 2ab \Rightarrow 2ab = 32 \Rightarrow ab = 16$$

We require two numbers that sum to $8$ and multiply to $16$. This identifies the values as **$a=4$** and **$b=4$**. The complete dataset is $2, 3, 3, 4, 4, 4, 5, 7$.

### The Grand Finale: Mean Deviation about the Mode

The mode is the most frequent observation. Since $4$ appears three times, the **$	ext{Mode} = 4$**.

The mean deviation about the mode is calculated as:
$$	ext{Mean Deviation} = \frac{\sum |x_i - 	ext{Mode}|}{n}$$

Calculating the absolute differences:
$|2-4|=2$, $|3-4|=1$ (twice), $|4-4|=0$ (three times), $|5-4|=1$, and $|7-4|=3$.

Summing these values:
$$2 + 1 + 1 + 0 + 0 + 0 + 1 + 3 = 8$$

Dividing by $n=8$:
$$	ext{Mean Deviation} = \frac{8}{8} = 1$$

The final result is **$1$**.

Animated Solution for Mathematics - Statistics: Let the mean and the standard deviation of the observation 2, 3, 3, 4, 5, 7, a, b be 4 and $2$ respectively. Then the mean deviation about the mode of these observations is :

Select Answer:

Visualized Solution

Identifying Given Data

Using the Mean Formula

Simplifying the Mean Equation

Setting up the Variance Equation

Substituting into Variance

Simplifying for $a^{2} + b^{2}$

Solving for $a$ and $b$

Finding the Values of $a$ and $b$

Visualizing the Complete Data

Adding the Missing Variables

Determining the Mode

Mean Deviation Setup

Final Calculation

The Sigma Insight: Mean Deviation

Analyzing the Setup

The Variance

The Algebraic Bridge

The Grand Finale

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