A statistic is any real measurable function of the sample of a random variable.
The concept of statistic is a concept of advanced statistics . The definition is short and certainly abstract. It is a very broad concept, but, as we will see below, very simple.
Given the difficulty of the term we will make the description in parts. Thus, in the first place, it will be necessary to describe what we mean by real measurable function. And, in the second instance, define what we understand as a sample of a random variable.
A statistician is a real measurable function
When we refer to function, we are talking about a mathematical function. For example:
Y = 2X
Depending on the values X takes, then Y will take one value or another. Suppose that X is worth 2. Then, Y will be worth 4, the result of multiplying 2 by 2. If X is worth 3, then Y will be worth 6. Result of multiplying 2 by 3.
Of course, a statistician is not just any function. It is a real and measurable function. This mathematical concept is frankly simple. Real, because it gives rise to real numbers and measurable because it can be measured.
Statistics has innumerable applications to daily life. Therefore, it makes sense that the values that a statistic can give rise to are real and can be measured.
Sample of a random variable
Many times we have heard the concept of sample. Or the concept of a representative sample. For this case, we will not distinguish between the different types of sample. Thus, we will use the concept of sample in the broad sense.
Imagine that we want to know the average expenditure of Mexican families in buying clothes. Obviously, we don’t have enough resources to ask the entire Mexican population. What do we do? We estimate it through a sample. A sample of, for example, 50,000 families.
That sample, everything is said, will have to meet specific characteristics. That is, it must be representative and contain in it many families from different geographical areas, different tastes, religions or purchasing power. If not, we will not obtain a reliable value.
A random variable
Now, it is a sample, but a sample of a random variable. What do we understand by random variable? A random variable, in simple words, is a difficult variable to predict. That is, under similar conditions, it takes different values.
For example, the number that will come out when rolling a dice is a random variable. Although we always release it in very similar conditions, we will obtain different results.
Now that we understand the technical definition of the concept, we have to gather everything we have learned. We know what a real and measurable function is. And, we also know what the sample of a random variable is.
How in spite of everything, the concept is still abstract, the best way to understand it will be with an example.
Suppose there are 100 students in a school. A teacher, proposed as an activity, try to estimate what is the average grade of the students of that school in the subject of mathematics.
Since we don’t have time or resources to ask the 100 students, we decided to ask 10 students. From there, we will try to estimate the average grade. We have the following data:
Before calculating the average grade, following the purpose of this article, we will apply what we have learned about the statistics on this example.
We know that a statistic is a real and measurable function of the sample of a random variable. We have the sample of a random variable (the table above). Thus, any real and measurable function of said sample will be a statistic. For example:
Statistic 1: Student 1 + Student 2 + Student 3 +…. + Student 10 = 60
Statistical 2: Student 1 – Student 2 + Student 3 – Student 4 +… – Student 10 = 2
Statistic 3: – Student 1 – Student 2 – Student 3 -… .- Student 10 = -60
These three statistics are real and measurable functions of the sample. With which, they are statistical. On a theoretical level, all this makes sense. The sense is that not all statistics will be valid to estimate according to what parameters.
At this point, the concept of estimator enters . An estimator is a statistic that will be required certain conditions so that it can reliably calculate the desired parameter.
For example, to estimate the parameter we know as “Average grade” or “Average grade” we need an estimator. We know that estimator as “mean.” The average is an estimator. That is, a statistician who requires certain conditions to be able to calculate the average grade with certain guarantees.
If we want to know the average grade, we will have to add all the grades and divide by the total number of students. That is to say:
Average grade = (4 + 8 + 6 + 7 + 9 + 9 + 7 + 2 + 5 + 3) / 6 = 6
The average formula is the same, whatever the sample. Always use all the data that the sample contains. In this case we have data from 10 students and the average formula uses the 10 data. If we had 20 data from 20 students, I would use the 20. Statisticians who meet this characteristic are known as sufficient statisticians .
In conclusion, a statistician is any real and measurable function of a sample. Once you have several possible statistics, certain conditions are required to be considered as estimators. And, thanks to the estimators, we can try to “predict” certain values from smaller samples.