Related Formula
Mean X = (Σ xᵢ)/(n)$$\text{Mean } \overline{X} = \frac{\sum x_i}{n}$$
Variance σ² = (Σ xᵢ²)/(n) - ( X)²$$\text{Variance } \sigma^2 = \frac{\sum x_i^2}{n} - (\overline{X})^2$$
Core Logic
Let the five observations be x₁, x₂, x₃, x₄, x₅$x_1, x_2, x_3, x_4, x_5$.
Given total mean:
(x₁ + x₂ + x₃ + x₄ + x₅)/(5) = (24)/(5) x₁ + x₂ + x₃ + x₄ + x₅ = 24$$\frac{x_1 + x_2 + x_3 + x_4 + x_5}{5} = \frac{24}{5} \implies x_1 + x_2 + x_3 + x_4 + x_5 = 24$$
Given mean of first four observations:
(x₁ + x₂ + x₃ + x₄)/(4) = (7)/(2) x₁ + x₂ + x₃ + x₄ = 14$$\frac{x_1 + x_2 + x_3 + x_4}{4} = \frac{7}{2} \implies x_1 + x_2 + x_3 + x_4 = 14$$
Substituting this back, we find the fifth observation:
14 + x₅ = 24 x₅ = 10$$14 + x_5 = 24 \implies x_5 = 10$$
Step 1: Finding the Sum of Squares
Using the variance of the 5 observations:
σ² = (194)/(25) = (x₁² + x₂² + x₃² + x₄² + x₅²)/(5) - ((24)/(5))²$$\sigma^2 = \frac{194}{25} = \frac{x_1^2 + x_2^2 + x_3^2 + x_4^2 + x_5^2}{5} - \left(\frac{24}{5}\right)^2$$
(194)/(25) = (x₁² + x₂² + x₃² + x₄² + 100)/(5) - (576)/(25)$$\frac{194}{25} = \frac{x_1^2 + x_2^2 + x_3^2 + x_4^2 + 100}{5} - \frac{576}{25}$$
(194 + 576)/(25) = (x₁² + x₂² + x₃² + x₄² + 100)/(5)$$\frac{194 + 576}{25} = \frac{x_1^2 + x_2^2 + x_3^2 + x_4^2 + 100}{5}$$
(770)/(5) = x₁² + x₂² + x₃² + x₄² + 100 154 = x₁² + x₂² + x₃² + x₄² + 100$$\frac{770}{5} = x_1^2 + x_2^2 + x_3^2 + x_4^2 + 100 \implies 154 = x_1^2 + x_2^2 + x_3^2 + x_4^2 + 100$$
x₁² + x₂² + x₃² + x₄² = 54$$x_1^2 + x_2^2 + x_3^2 + x_4^2 = 54$$
Step 2: Variance of First Four Observations
Variance₄ = Σi=1⁴ xᵢ²4 - ( Σi=1⁴ xᵢ4)²$$\text{Variance}_{4} = \frac{\sum_{i=1}^4 x_i^2}{4} - \left(\frac{\sum_{i=1}^4 x_i}{4}\right)^2$$
Variance₄ = (54)/(4) - ((7)/(2))² = (54)/(4) - (49)/(4) = (5)/(4)$$\text{Variance}_{4} = \frac{54}{4} - \left(\frac{7}{2}\right)^2 = \frac{54}{4} - \frac{49}{4} = \frac{5}{4}$$
Pattern Recognition
Isolate the missing elements sequentially. Use the sum of elements first to find x₅$x_5$, then use the sum of squares equation to find the squared sum of the subset.
Chapter Mix
Class 11 Mathematics: Statistics
