Related Formula
μ = (Σ xᵢ)/(N)$$\mu = \frac{\sum x_i}{N}$$
σ² = (Σ xᵢ²)/(N) - μ²$$\sigma^2 = \frac{\sum x_i^2}{N} - \mu^2$$
Core Logic
Calculate x+y$x+y$ using the mean formula:
(-10 - 7 - 1 + x + y + 9 + 2 + 16)/(8) = (7)/(2)$$\frac{-10 - 7 - 1 + x + y + 9 + 2 + 16}{8} = \frac{7}{2}$$
(9 + x + y)/(8) = (7)/(2) ⇒ 9 + x + y = 28 ⇒ x + y = 19 (1)$$\frac{9 + x + y}{8} = \frac{7}{2} \Rightarrow 9 + x + y = 28 \Rightarrow x + y = 19 \dots (1)$$
Step 1: Calculate sum of squares using Variance
Using the variance formula:
((-10)² + (-7)² + (-1)² + x² + y² + 9² + 2² + 16²)/(8) - ((7)/(2))² = (293)/(4)$$\frac{(-10)^2 + (-7)^2 + (-1)^2 + x^2 + y^2 + 9^2 + 2^2 + 16^2}{8} - \left(\frac{7}{2}\right)^2 = \frac{293}{4}$$
(100 + 49 + 1 + x² + y² + 81 + 4 + 256)/(8) - (49)/(4) = (293)/(4)$$\frac{100 + 49 + 1 + x^2 + y^2 + 81 + 4 + 256}{8} - \frac{49}{4} = \frac{293}{4}$$
(491 + x² + y²)/(8) = (293)/(4) + (49)/(4) = (342)/(4) = (684)/(8)$$\frac{491 + x^2 + y^2}{8} = \frac{293}{4} + \frac{49}{4} = \frac{342}{4} = \frac{684}{8}$$
491 + x² + y² = 684 ⇒ x² + y² = 193 (2)$$491 + x^2 + y^2 = 684 \Rightarrow x^2 + y^2 = 193 \dots (2)$$
Step 2: Solve for x and y
From (1), y = 19 - x$y = 19 - x$. Substitute into (2):
x² + (19 - x)² = 193$$x^2 + (19 - x)^2 = 193$$
x² + 361 - 38x + x² = 193$$x^2 + 361 - 38x + x^2 = 193$$
2x² - 38x + 168 = 0 ⇒ x² - 19x + 84 = 0$$2x^2 - 38x + 168 = 0 \Rightarrow x^2 - 19x + 84 = 0$$
(x - 12)(x - 7) = 0$$(x - 12)(x - 7) = 0$$
Thus, x = 12$x = 12$ and y = 7$y = 7$ (or vice-versa, which doesn't affect absolute differences).
Step 3: Calculate the New Mean
We need the mean of 4$4$ numbers: x, y, x + y + 1, |x - y|$x, y, x + y + 1, |x - y|$.
Substitute x = 12, y = 7$x = 12, y = 7$:
The numbers are 12, 7, (12+7+1), |12-7| = 12, 7, 20, 5$12, 7, (12+7+1), |12-7| = 12, 7, 20, 5$.
New Mean = (12 + 7 + 20 + 5)/(4) = (44)/(4) = 11$\frac{12 + 7 + 20 + 5}{4} = \frac{44}{4} = 11$.
Pattern Recognition
Statistics questions mapping x+y$x+y$ and x²+y²$x^2+y^2$ inherently hide a quadratic symmetric system (x+y)² - 2xy = x²+y²$(x+y)^2 - 2xy = x^2+y^2$. Solving for the roots immediately grants the discrete terms.
Chapter Mix
Class 11 Maths: Statistics
