For a statistical data x₁, x₂, …, x₁₀$x_1, x_2, \ldots, x_{10}$ of 10$10$ values, a student obtained the mean as 5.5$5.5$ and Σi=1¹⁰ xᵢ² = 371$\sum_{i=1}^{10} x_i^2 = 371$ . He later found that he had noted two values in the data incorrectly as 4$4$ and 5$5$, instead of the correct values 6$6$ and 8$8$, respectively. The variance of the corrected data is :

Related Formula

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:

Given mean of first four observations:

Substituting this back, we find the fifth observation:

Step 1: Finding the Sum of Squares

Using the variance of the 5 observations:

Step 2: Variance of First Four Observations

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

Related Formula

Core Logic

Given: Σ aᵢ = 50$\sum a_i = 50$ Σk < j ak aⱼ = 1100$\sum_{k < j} a_k a_j = 1100$ Number of observations, n = 10$n = 10$.

To find the standard deviation, we need the sum of squares, Σ aᵢ²$\sum a_i^2$. We use the algebraic identity for the square of a sum of n$n$ terms.

Step 1: Finding the Sum of Squares

Substitute the known values into the identity:

Step 2: Calculating Standard Deviation

Now, apply the variance formula:

The standard deviation σ$\sigma$ is the square root of variance:

Pattern Recognition

Whenever you see pairwise products Σ aᵢ aⱼ$\sum a_i a_j$ in a statistics problem, immediately bridge it to Σ aᵢ²$\sum a_i^2$ using the multinomial expansion identity. The variance formula handles the rest organically.

Chapter Mix

Class 11 Maths: Statistics

Related Formula

Core Logic

We are given 10 observations: 65, 68, 58, 44, 48, 45, 60, α, β, 60$65, 68, 58, 44, 48, 45, 60, \alpha, \beta, 60$. Total n=10$n=10$.

Sum of known observations:

The mean x = 56$\bar{x} = 56$:

Step 1: Use Variance Equation

The variance σ² = 66.2$\sigma^2 = 66.2$. Using the computational formula for variance:

Calculate the sum of squares of known observations:

= 25678$= 25678$

Insert this into the variance equation:

Step 2: Solve for Squares Sum

Isolate α² + β²$\alpha^2 + \beta^2$:

Pattern Recognition

If a question asks solely for α²+β²$\alpha^2+\beta^2$ given mean and variance, you do not need to solve the complex polynomial system to find the individual values of α$\alpha$ and β$\beta$. The variance equation isolates α²+β²$\alpha^2+\beta^2$ automatically as a single chunk.

Chapter Mix

Class 11 Mathematics: Statistics

Related Formula

Core Logic

Let's build the summation table for calculating mean and variance:

Step 1: Calculating the Mean

Step 2: Calculating the Variance

Rounding to the nearest integer as indicated by the official answer key gives 29.

Pattern Recognition

For grouped discrete data, the computational formula (Σ fᵢ xᵢ²)/(N) - μ²$\frac{\Sigma f_i x_i^2}{N} - \mu^2$ minimizes subtraction errors compared to tracking raw deviations point-by-point.

Chapter Mix

Class 11 Maths: Statistics

Related Formula

Core Logic

Constructing the Cumulative Frequency (CF) table:

Total frequency N = 36$N = 36$. Therefore, (N)/(2) = 18$\frac{N}{2} = 18$.

Step 1: Identifying median class

Since 18$18$ lies in the cumulative frequency interval > 12$> 12$ and ≤ 22$\leq 22$, the median class is 8-12$8-12$. Here, l = 8$l = 8$ (lower limit), cf = 12$cf = 12$ (CF of previous class), f = 10$f = 10$ (frequency of current class), h = 4$h = 4$ (class size).

Step 2: Calculating median

We need to find 20M$20M$:

Pattern Recognition

Finding the cumulative frequency sequence safely identifies the median class. Plugging into the standard linear interpolation formula yields the exact median.

Chapter Mix

Class 11 Maths: Statistics