Statistic vs Parameter. What is the difference?

Course Content
1) Exploratory Data Analysis
2) Probability Theory
3) Random Variables
4) Distributions
5) Generating Functions
6) Joint Distributions
7) Conditional Expectations
8) CLT
9) Sampling & Inference
10) Point Estimation
11) Confidence Intervals
12) Hypothesis Testing
13) Regression and Correlation
14) ANOVA

For exams:
Exam P
CS1
Formerly CT3

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What is the difference between a Parameter and a Statistic?

Statistics is a subject that attempts to extract information from data. The information we seek is the parameters and distribution of the random variable that generated the data. If we know this information we can optimize processes and answer questions.

We use mathematics to create a statistic which we then use to estimate parameters.

Lets say we have a random variable from the Normal Distribution but we don’t know what the values of the parameters are. We know that a Normal Distribution has two parameters. [mew] and [sigma squared]. The mean of the Normal Distribution is equal to [mew] and the variance of the Normal Distribution is equal to [sigma squared].

Our random variable generates data points, these are also called observations. A group of observations is known as a random sample. A function of the random sample is a statistic. The most well known statistic is X bar. Otherwise known as the sample mean. Add all the observations together and divide this sum by the number of observations to get X bar. We can then use the result of this statistic to estimate the parameter of our random variable.

Where things get interesting is that the statistic is a function of a sample generated by a random variable. This means that the statistic is also a random variable, meaning that it has its own distribution and set of parameters. Fortunately we have something called the Central Limit Theorem which tells us what the distribution and parameters are for X bar.

But we are getting ahead of ourselves, for now just remember that stats is all about extracting information from data. The information we want is the distribution and parameters of the random variable which generated the data in the first place. We use mathematics to create a function out of the data which we call a statistic. We can then use a statistic to estimate the parameter.

This naturally leads to more advanced topics in the subject of Statistics. We can use the final result of a statistic to estimate a parameter. This is known as Point Estimation. Or we can incorporate the fact that the statistic is a random variable and use its variance to help us construct an Interval Estimation for our parameter. This is known as Confidence Intervals.

If you want to learn more about statistics, I have a course on Udemy called Actuarial Statistics as well as an online school called Statistics by MJ. Links in the description below, feel free to comment.