For example: Suppose the true mean is 50

As the sample size increases:

n = 10 → Estimate = 54

n = 100 → Estimate = 51

n = 1,000 → Estimate = 50.4

n = 10,000 → Estimate = 50.1

The estimate is getting closer to the true value. This is an example of consistency.

🔹 17. Efficiency

Suppose two estimators are both unbiased.

Estimator A has variance 4

Estimator B has variance 9

Estimator A is generally considered more efficient because it has lower variance.

In simple terms: Among comparable unbiased estimators, the one with lower variance is more efficient.

Efficiency matters because we want accurate estimates without unnecessary uncertainty.

🔹 18. Mean Squared Error (MSE)

Another important concept is Mean Squared Error.

MSE combines both Bias and Variance

A useful relationship is: MSE = Variance + Bias²

This is extremely important in Machine Learning.

A model can have Low bias but high variance, or High bias but low variance

MSE helps evaluate the overall estimation error.

🔹 19. Why Squared Error?

Why do we square the bias and errors?

Because squaring:

Makes negative and positive errors positive

Penalizes larger errors more heavily

Gives us a convenient mathematical measure

For example:

Error = 2 → Squared Error = 4

Error = 5 → Squared Error = 25

A larger error gets a much larger penalty.

🔹 20. Example of MSE

Suppose: Bias = 2, Variance = 9

Then: MSE = Variance + Bias² = 9 + 2² = 9 + 4 = 13

So the total mean squared error is 13

🔹 21. Estimation in Data Science

Statistical estimation appears everywhere in Data Science.

📊 Business Analytics: Estimate Average revenue, Customer spending, Customer lifetime value

🛒 E-commerce: Estimate Conversion rates, Average order value, Customer retention

🤖 Machine Learning: Estimate Model parameters, Prediction errors, Expected performance

🧪 Experimentation: Estimate Treatment effects, Conversion-rate differences, Average outcome differences

📈 Finance: Estimate Expected returns, Risk, Volatility

🔹 22. A Practical Example

Suppose an online store has millions of users.

We want to estimate the average amount spent per user.

We randomly select 1,000 users and calculate: Sample Mean = ₹2,500

Therefore: Point Estimate = ₹2,500

Now suppose we calculate a 95% confidence interval: [₹2,350, ₹2,650]

We now have:

Point Estimate: ₹2,500

Interval Estimate: ₹2,350 to ₹2,650

This gives decision-makers both an estimate and an indication of uncertainty.

🔹 23. Python Example

We can calculate a sample mean as a point estimate using Python.

import numpy as np

data = np.array([2400, 2600, 2500, 2700, 2300])
point_estimate = np.mean(data)
print("Point Estimate:", point_estimate)