🚀 Data Science Roadmap 2026

📘 Phase 2: Mathematics & Statistics for Data Science

📖 Topic 14: Statistical Estimation — Point Estimation, Bias & Variance

In Data Science, we often want to estimate something about a population using only a sample.

For example:
What is the average income of customers?
What percentage of users will purchase a product?
What is the average delivery time?
How much revenue does the average customer generate?

Usually, we don't have access to the entire population.
So we use statistical estimation.

🔹 1. What Is Statistical Estimation?

Statistical estimation is the process of using sample data to estimate an unknown population parameter.

For example:
Suppose a company has 1 million customers.
We want to know their true average annual spending.
It may be impractical to collect spending data from all 1 million customers.
Instead, we randomly select 5,000 customers and calculate:
Sample Mean = ₹18,500
We can use ₹18,500 to estimate the population's average spending.
This is statistical estimation.

🔹 2. Parameter vs Statistic

This distinction is fundamental.

Population Parameter
A numerical value describing the entire population.
Examples: Population mean, Population proportion, Population variance
Usually, the parameter is unknown.

Sample Statistic
A numerical value calculated from a sample.
Examples: Sample mean, Sample proportion, Sample variance
We use the statistic to estimate the parameter.

Simple relationship:
Population → Parameter
Sample → Statistic
Statistic → Estimate of Parameter

🔹 3. What Is an Estimator?

An estimator is a rule or mathematical procedure used to estimate an unknown population parameter.

For example:
Sample Mean = Sum of observations / Number of observations
The sample mean is an estimator of the population mean.

Suppose the sample contains: 20, 30, 40, 50, 60
Then: Sample Mean = (20 + 30 + 40 + 50 + 60) / 5 = 40
So: 40 is the estimate.
The procedure used to calculate the sample mean is the estimator.
The result, 40, is called the estimate.

🔹 4. Estimator vs Estimate

These terms are easy to confuse.

Estimator: The method or rule used to estimate a parameter. Example: Sample Mean
Estimate: The actual numerical result obtained from a particular sample. Example: 40

Think of it like:
Estimator = Formula/Method
Estimate = Result

🔹 5. Point Estimation

A point estimate provides a single value as the estimate of an unknown population parameter.

For example:
Population Mean → estimated using Sample Mean
If Sample Mean = ₹50,000 then Point Estimate of Population Mean = ₹50,000

Point estimates are simple and easy to communicate, but they don't tell us how uncertain the estimate is.
That's why confidence intervals are also important.

🔹 6. Interval Estimation

Instead of providing one value, interval estimation provides a range.

For example:
Point Estimate = 50
But instead of simply reporting 50, we might report:

95% Confidence Interval = [47, 53]

This gives us information about uncertainty.

So:
Point Estimation = One value
Interval Estimation = Range of plausible values

🔹 7. What Makes a Good Estimator?

A good estimator should have desirable statistical properties.
The most important ones include:
Unbiasedness, Consistency, Efficiency, Low variance
Let's understand them.

🔹 8. Unbiased Estimator

An estimator is unbiased if its expected value equals the true population parameter.
In simple terms: An unbiased estimator does not systematically overestimate or underestimate the parameter.

For example, suppose the true population mean is 100
If we repeatedly take samples and calculate the sample mean, an unbiased estimator will have an average close to 100
It may produce 98 for one sample, 103 for another, 99 for another, and so on.
Individual estimates can differ.
But across repeated samples, the average of the estimates approaches the true parameter.

🔹 9. Bias