Within mathematics, and more precisely within statistics, variance is frequently used, a very specific concept of this last discipline, and yet very important given its wide application in many aspects where statistics intervenes, either in calculations of the given effects of a hospital treatment, either in the electoral results to choose the government of a city council, or to ponder the learning capacities in a specific institute.
The variance is a measure by which the relationship is calculated in a group of data on which one works. If this variance is minimal, this means that you are working with highly grouped values. If it is the contrary, obviously, the data is scattered.
In this article we also give you a lot of information to learn how to calculate the variance.
It is a concept invented by the English mathematician Ronal Fisher, in the middle of the 20th century, with which the average of the squared calculation of a variable or random number is identified, based on the average value of the variable.
Variance as a measure
The variance measures the dispersion or variability of variable quantities and is officially defined as the expectation of the deviation of a variable squared compared to its mean. If the variable, for example, deals with a distance calculated in kilometers, the variance will be given in square kilometers.
Variance is a concept that is used to compare two groups of data that you want to relate. For example, you can compare the results of a group of men who are trying a drug with another equal but of women, the variance allows you to check how far the variations in the results of one group with respect to the other. Another example is statistical models, in which a small variance will show the data is being overfitted.
The data sample
In general, when a statistical study is going to be carried out, the data of each one of the individuals or members of the group that is studied is not taken. For example, if you want to find out the price of mobile phones purchased in Spain in the last year, it is practically impossible to get that information unless you deploy a huge effort and resources,
For this reason, a sample of this set is made to find out the most common price of the total number of mobile phones purchased, and for this, the price of several thousand phones can be found.
With another example, let’s take a bar where dozens of coffees are poured every day, in all its varieties, with milk, plain, with or without sugar, etc. To get the variance that will give us an average image of the coffees that are taken in a month, or in a year, we are going to take a sample of seven days. Thus, the random sample will give us that the coffees that have been sold every day of that almost entire week are the following: 13, 23, 9, 17, 7, 22 and 15.
Calculating the variance
First we will obtain the average, by adding all the values we have -the coffees served each day-, that is, 12+23+9+17+7+22+15= 105. We divide this result by the number of days for which we have data, seven. We will get the value of 15.
Now we will subtract each unit value -the coffees sold each day- from the previously obtained average, which is 15. This would offer the following result: -3; 7; -6; 2; -8; 7; 0. So, to find out the variance, we will have to square each of the resulting values, in order not to have negative numbers, add all the results and we would end up dividing this sum by 7, the number of sampling days. Thus, the sum of the squares gives 125 and its division by 7 would result in 17.86.
Covariance and sample variance
The sample variance is the name of the variance when the calculation is made from a population, a community or a group, from a sample of that type of social organization. On the other hand, there is covariance, which is the name that specialists give to the measurement of the joint dispersion of two variables. It should not be forgotten that the variance translates into recognizable measures the variability that random variables can have. Variance is a measurement method that joins others used by statisticians and other mathematicians, and each of them its most appropriate application.
Standard deviation
The standard or typical deviation, as it is also known, is another concept related to variance, and it helps us to find the scope of the dispersion of certain variables, such as the interval and the ratio. It is a very important and necessary magnitude in the area of descriptive statistics.
However, it is easy to obtain the standard deviation, since we only need to know the variance as we have described it above, and having this data, calculate its square root.