Sample Mean, Variance and Descriptive Estimation Notes for GATE DA: Concepts, Formulas, Worked Examples & Practice

Sample Mean, Variance and Descriptive Estimation notes for GATE DA: 7 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

Chapter Roadmap: Statistical Inference

Chapter Journey: Statistical Inference

Descriptive Foundations (Current Topic)

Sample Mean Updates & Variance Identities. Efficient calculation of basic statistics.

Point Estimation

Method of Moments, MLE, Unbiasedness, Consistency.

Interval Estimation

Confidence Intervals for Mean and Proportions.

Hypothesis Testing

Null vs Alternative, Errors, Z/T/Chi-square tests, P-values.

Exam Weightage Hint: Hypothesis Testing and MLE are the highest yield areas. However, the identities in this first topic are often the hidden step required to solve complex numerical questions quickly.

Concept: The Running Mean Update

The Running Mean Update

Standard definition of sample mean for $n$ observations:

\overset{x}{ˉ}_{n} = \frac{1}{n} i = 1 \sum n x_{i}

If a new observation $x_{n + 1}$ arrives, recalculating from scratch requires summing $n + 1$ terms. For large $n$ or streaming data, this is costly.

The Update Formula

\overset{x}{ˉ}_{n + 1} = \frac{n x ˉ _{n} + x _{n + 1}}{n + 1}

1 Total sum of first

n

items:

S_{n} = n \overset{x}{ˉ}_{n}

2 New total sum:

S_{n + 1} = S_{n} + x_{n + 1}

3 New mean:

\overset{x}{ˉ}_{n + 1} = \frac{S _{n + 1}}{n + 1}

+ New Data Point → Instant Update

Pattern: Updating the Mean

Problem: The sample average of 50 data points is 40. The updated sample average after including a new data point taking the value of 142 is ______.

Step-by-Step Solution

1. Identify Given Values

n = 50

\overset{x}{ˉ}_{50} = 40

x_{n e w} = 142

2. Calculate Old Sum (

S_{50}

)

S_{50} = n \times \overset{x}{ˉ}_{50} = 50 \times 40 = 2000

3. Calculate New Sum (

S_{51}

)

S_{51} = S_{50} + x_{n e w} = 2000 + 142 = 2142

4. Calculate New Mean (

\overset{x}{ˉ}_{51}

)

\overset{x}{ˉ}_{51} = \frac{S _{51}}{n + 1} = \frac{2142}{51} = 42

Answer: 42

Formula: The Pairwise Difference Identity

The Pairwise Difference Identity

For a dataset ${x_{1}, \dots, x_{n}}$ , the sum of squared differences between all distinct pairs relates directly to the variance.

i = 1 \sum n j = 1 \sum n (x_{i} - x_{j})^{2} = 2 n i = 1 \sum n (x_{i} - \overset{x}{ˉ})^{2}

Why is this useful?

Exam questions often provide the double sum $\sum\sum (x_{i} - x_{j})^{2}$ and ask for the sample variance $s^{2}$ or vice versa.

Recall Sample Variance definition:

s^{2} = \frac{1}{n - 1} i = 1 \sum n (x_{i} - \overset{x}{ˉ})^{2}

Combining these gives the master identity:

i = 1 \sum n j = 1 \sum n (x_{i} - x_{j})^{2} = 2 n (n - 1) s^{2}

More notes in this unit

Sampling Distributions and Estimator Properties Notes

chapter

Sample Mean, Variance and Descriptive Estimation Notes for GATE DA: Concepts, Formulas, Worked Examples & Practice

Sample Mean, Variance and Descriptive Estimation notes for GATE DA: 7 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice

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