Core Logic
We have 8 observations: 2, 3, 3, 4, 5, 7, a, b$2, 3, 3, 4, 5, 7, a, b$. Given mean x = 4$\bar{x} = 4$:
(2 + 3 + 3 + 4 + 5 + 7 + a + b)/(8) = 4 24 + a + b = 32 a + b = 8$$\frac{2 + 3 + 3 + 4 + 5 + 7 + a + b}{8} = 4 \implies 24 + a + b = 32 \implies a + b = 8$$
Step 1: Using the Variance Property
Given standard deviation σ = √(2) Variance σ² = 2$\sigma = \sqrt{2} \implies \text{Variance } \sigma^2 = 2$.
The variance formula is σ² = (Σ xᵢ²)/(n) - ( x)²$\sigma^2 = \frac{\sum x_i^2}{n} - (\bar{x})^2$:
2 = (2² + 3² + 3² + 4² + 5² + 7² + a² + b²)/(8) - 4²$$2 = \frac{2^2 + 3^2 + 3^2 + 4^2 + 5^2 + 7^2 + a^2 + b^2}{8} - 4^2$$
2 = (4 + 9 + 9 + 16 + 25 + 49 + a² + b²)/(8) - 16 = (112 + a² + b²)/(8) - 16$$2 = \frac{4 + 9 + 9 + 16 + 25 + 49 + a^2 + b^2}{8} - 16 = \frac{112 + a^2 + b^2}{8} - 16$$
2 + 16 = (112 + a² + b²)/(8) 18 × 8 = 112 + a² + b²$$2 + 16 = \frac{112 + a^2 + b^2}{8} \implies 18 \times 8 = 112 + a^2 + b^2$$
144 = 112 + a² + b² a² + b² = 32$$144 = 112 + a^2 + b^2 \implies a^2 + b^2 = 32$$
Step 2: Solving for a and b and finding the Mode
We know (a+b)² = a² + b² + 2ab 8² = 32 + 2ab 64 = 32 + 2ab 2ab = 32 ab = 16$(a+b)^2 = a^2 + b^2 + 2ab \implies 8^2 = 32 + 2ab \implies 64 = 32 + 2ab \implies 2ab = 32 \implies ab = 16$.
Solving a+b=8$a+b=8$ and ab=16$ab=16$ gives a=4$a=4$ and b=4$b=4$.
The complete data set is 2, 3, 3, 4, 4, 4, 5, 7$\{2, 3, 3, 4, 4, 4, 5, 7\}$.
The value appearing with highest frequency is 4$4$ (occurs 3 times), so Mode = 4$\text{Mode} = 4$.
Step 3: Calculating Mean Deviation about Mode
Mean Deviation about Mode is:
M.D. = Σ |xᵢ - Mode|n$$\text{M.D.} = \frac{\sum |x_i - \text{Mode}|}{n}$$
= (|2-4| + |3-4| + |3-4| + |4-4| + |4-4| + |4-4| + |5-4| + |7-4|)/(8)$$= \frac{|2-4| + |3-4| + |3-4| + |4-4| + |4-4| + |4-4| + |5-4| + |7-4|}{8}$$
= (2 + 1 + 1 + 0 + 0 + 0 + 1 + 3)/(8) = (8)/(8) = 1$$= \frac{2 + 1 + 1 + 0 + 0 + 0 + 1 + 3}{8} = \frac{8}{8} = 1$$
Pattern Recognition
When a+b=2√(ab)$a+b=2\sqrt{ab}$ (here 8 = 2√(16)$8 = 2\sqrt{16}$), the roots are guaranteed to be equal, meaning a=b$a=b$. Recognizing this condition instantly avoids full quadratic polynomial substitution steps.
Chapter Mix
Class 11 Mathematics: Statistics
