The variance and the typical deviation, or standard, are two concepts that if we show them by themselves can be strange, however, when we stop to think, within our routine we refer to them with an amazing frequency, we just don’t realize bill.
How useful are these concepts to my life?
It is true that current daily life flows so fast that we barely have time to relax and think about ourselves, so we propose a little experiment so that you can see how many times a day you receive that concept without noticing it.
Sit at home and no matter how silly it may seem, make a summary in your mind of everything you did during the day. To understand us better let’s give an example:
You get up and while you are getting ready to go to work, you turn on the TV. If you put it on the news channel, it is very likely that while eating breakfast you will hear the phrase “it is forecast that for the next… period, year, elections” at least five times. Know that each of these sentences, however brief they may seem to you, carry a statistical analysis that implies, of course, a calculation of variance and deviation. But let’s get on with your day, which from now on will get much more interesting.
Utility at work?
When he arrives at work, he has between two and four reports on the table that he must review to deliver to his boss. The reports propose economic and commercial strategies that have only one objective, to increase income.
Once the documents have been reviewed, you must propose to your superior the strategy that you consider optimal. You must also be prepared for questions, so you must know well how to memorize these reports so as not to lose face. The boss for his part, will refute with different arguments the proposal that you present to him, now, what if your boss asks you, how do you know that this proposal is correct or not?
The answer to all these questions is summarized in two simple words, variance and deviation, and since you have understood how important they are in your life, we offer you a brief explanation of how to calculate them.
What is variance and how is it calculated?
Variance measures the spread within a data set. If the value of the variance is small, it means that the values in the set are fairly close together. If, on the contrary, the result of the variance is greater, it means that the elements within the set that is analyzed are scattered.
The variance is represented by the Greek letter Sigma (σ) raised to the square, that is, (σ²), and it is calculated as represented by the formula:
σ²= Σ (Xi-µ)²
no
Or what is the same
σ²= (Xi-µ)²+ (Xi-µ)²+ (Xi-µ)²+…………(Xn-µ)²
no
Given the level of abstraction that the concept of variance implies, as well as the difficulty to understand it, the variance is generally calculated as a starting point to know and quantify the standard deviation.
For a better interpretation of the concepts that we are dealing with today, we will give a practical, situational example, where we will calculate the variance and standard deviation as a whole. It is our goal that, once you have read this document, you have the clarity you need to interpret the concepts that are so frequent in our routine.
What is the standard deviation?
We are already clear that “variance is a measure of dispersion that calculates the deviations from the mean of a statistical distribution, or what is the same, a set of data.” Now let’s define the standard deviation, this represents the magnitude of the dispersion of the variables within a ratio interval. For its calculation we start from the variance and calculate its square root.
How to distinguish variance from standard deviation?
Although both are measures of dispersion and their definitions are similar, there are several aspects that will help us distinguish variance from standard deviation. It is important to note that the standard deviation measures the dispersion of a data set, while the variance measures the variability of this dispersion.
Below we list three quick ways to identify when we are in the presence of one or another indicator.
- The variance is measured in units squared and therefore its result will always have a positive value.
- The minimum value that the variance reaches is = 0.
- The variance is nothing more than the standard deviation raised to the square, and therefore, the standard deviation is summarized as the square root of the variance.
Two simple methods to calculate the variance
There are several methods to calculate the variance , the difference between them is mainly based on the size of the group selected for the study.
- If the group to be studied is small or medium, the data collection will be somewhat laborious, but doable, so it will be taken in its entirety and the variance of the group in general will be calculated, this is known as the variance of the population.
- If, otherwise, the group or universe to be studied is very large, a segment of data that is considered representative will be taken. This group selected for our study is known in statistics by the name of Sample, and the calculation method in this case will be the calculation of the sample variance or sample variance.
A practical example
We will give a practical example to reflect each of the methods, we will begin by calculating the variance of a population. We will list the procedure with steps to follow, hoping for a better understanding.
How to Calculate the Variance of a Population:
- Select the data set:
The first step is to select the data set, since we are going to analyze all of them, this method is suggested when the group is small. For example, if exactly 12 students study in a classroom, we can do an analysis of their ages. For this, it is known that the ages of the children are: 4,5,5,4,3,4,5,6,6,5,4,4
- Set up the formula for the population variance:
A characteristic of the population variance is that the result is exact, since all the data have been analyzed.
To calculate the population variance, the formula will be:
σ²= Σ(Xi-µ)²
no
Where:
σ² represents the variance.
Xi represents each of the values and µ the mean or average of the data.
n represents the amount of data.
- Calculate the population mean:
The arithmetic mean is nothing more than the average, which is calculated by adding all the data and dividing it by the amount of data, for the example that we intend to recreate the result of the average would be as follows:
µ=4+5+5+4+3+4+5+6+6+5+4+4= 55
Then:
µ=55:12=4.58
The average age of the children will be equal to 5.5
- Subtract the mean value from each data item.
If we know that µ=5.5 our results will be
4- 4.58= -0.58
5-4.58= 0.42
5-4.58= 0.42
And so on until the mean has been subtracted from the twelve ages of the children.
- Square all answers.
Likewise as in the previous step it will now take the results and square them. Following our example, the calculation will be as follows:
-0.58²= 0.3364
0.42²= 0.1764
0.42²= 0.1764
And again you will have to follow this step with each of the twelve results. Notice, that now all the numbers become positive.
- Recalculate the mean.
We will now calculate the average of the squared values, which will lead us to the final result, that is, the variance.
For our example, the calculation would be as follows:
0.3364+0.1764+0.1764+……0.3364= 8.5804
8.9168:12= 0.743
The variance of the population will be equal to 0.743
From this data it is very easy to obtain the standard deviation, since it is enough to find the square root of 0.743, which in our case would have a value of 0.86197448
Calculation of the variance of a sample
This method is used when the amount of data to be considered is extremely large, then a sample is selected and worked with. This, despite not yielding a result as exact as the population variance, is considered a very effective method. In our case we will continue with the previous example, so that you can later compare one method with the other.
Sample selection
Taking our example into account, we will select a sample of six children, which is equivalent to half the population. Now our data will be the following
4, 5, 5, 4, 3,4
Set up the variance formula
Now the variance formula is slightly different because we are looking at a sample.
σ²= Σ (Xi-µ)²
n-1
Calculate the sample mean
As in the previous example, the average is the result of the average, likewise the steps are 4, 5 and 6 are similar, so we will represent them in a table, in order not to make our article so extensive. For our sample, the average will be calculated as follows:
4+5+5+4+3+4=25
25:6=4.16666667
The steps that follow consist of subtracting the mean from each value, then squaring the result, and then adding these results, as shown in the previous example. The table below summarizes each of the steps.
Tips for Variance and Standard Deviation – calculate and formula
The study of statistical indicators can be tedious and long, but once you understand the meaning of these calculations, the analysis will be much easier and faster. Just look at the logical steps and you’ll see that one leads to the next, just like our feet when we walk.