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Sample Mean, Variance and Descriptive Estimation Notes for GATE DA

Sample Mean, Variance and Descriptive Estimation notes for GATE DA: 5 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice

sample mean variance and descriptive estimation notes

Chapter Roadmap: Statistical Inference

Chapter Journey: Statistical Inference 1 Descriptive Foundations (Current Topic) Sample Mean Updates & Variance Identities. Efficient calculation of basic statistics. 2 Point Estimation Method of Moments, MLE, Unbiasedness, Consistency. 3 Interval Estimation Confidence Intervals for Mean and Proportions. 4 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} @keyframes pulseUpdate { 0% { transform: scale(1); opacity: 1; } 50% { transform: scale(1.05); opacity: 0.8; } 100% { transform: scale(1); opacity: 1; } } .update-pulse { animation: pulseUpdate 2s infinite; display: inline-block; } + New Data Point \to Instant Update

Pattern: Updating the Mean

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

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Sample Mean, Variance and Descriptive Estimation Notes for GATE DA

Sample Mean, Variance and Descriptive Estimation notes for GATE DA: 5 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

Pattern: Decoding the Double Sum

Problem: For

n = 100

, it is known that:

\frac{1}{2000} i = 1 \sum n j = 1 \sum n (x_{i} - x_{j})^{2} = 99

Find the value of:

\frac{1}{99} i = 1 \sum n (x_{i} - \overset{x}{ˉ})^{2}

Analysis: The term we need to find is exactly the Sample Variance (

s^{2}

), since

n = 100

and the denominator is

n - 1 = 99

Step-by-Step Solution

1. Extract the Double Sum

Let

D = \sum_{i = 1}^{n} \sum_{j = 1}^{n} (x_{i} - x_{j})^{2}

.
Given:

\frac{D}{2000} = 99 ⟹ D = 99 \times 2000

2. Apply the Identity

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

Substitute

n = 100

D = 200 i = 1 \sum n (x_{i} - \overset{x}{ˉ})^{2}

3. Solve for Sum of Squared Deviations (SSD)

200 \cdot S S D = 99 \times 2000

S S D = \frac{99 \times 2000}{200} = 99 \times 10 = 990

4. Calculate Target (

s^{2}

)

s^{2} = \frac{S S D}{99} = \frac{990}{99} = 10

Answer: 10

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