Standard deviation: what is it and what is it for?

Statistics serves to systematize, collect and organize the data of a phenomenon., with the purpose of deducing the laws that operate in it and thus make forecasts, make decisions and draw conclusions. Its operations, and its statistics, such as the standard deviation, allow us to transform numbers into conclusions. With it, we can describe the variables of a population or make inferences controlling the level of error.

Within the statistics, Measures of dispersion – such as the standard deviation – essentially serve two purposes. Constitute a reference to talk about the heterogeneity of a population or sample and to set the level of error when making inferences based on a level of confidence.

Before looking in more depth at the meaning of the standard deviation, we are going to describe other associated statistics that are also widely used in data analysis.

The mean and the variance

On the one hand, The mean is a measure of central tendency that, with complements, is intended to be a representation of a sample or population in a variable. It is the total sum of values ​​in a sample divided by the number of values ​​in your sample.

On the other hand, Variance is a measure of dispersion that represents the oscillation of data in relation to its mean. That is, it provides a measure of how closely the data is distributed around its center. It is calculated as the sum of the residuals—difference between value and mean—squared, divided by the total number of observations.

To understand it better, let’s go with an example. Below, we put a table in which the glasses manufactured by Jorge appear each day of the week.

WeekdaysNumber of glasses manufacturedMonday5Tuesday4Wednesday7Thursday3Friday6

To calculate the average we add the number of glasses manufactured per day: 5+4+7+3+6= 25

Then, we divide the result by the total number of data: 25/5= 5

The average number of glasses made by Jorge during the five days is five glasses.

To find the variance, it is necessary calculate the squared residuals and divide them by the total observations. In simpler words, we subtract the average (5) from the observations, which are the number of glasses made per day (5, 4, 7, 3 and 6), and square it. Then, we add and divide by the number of observations (5):

s²= (5-5) ²+ (4-5) ²+ (7-5) ²+ (3-5) ²+ (6-5) ² / 5= 2

The variation in the number of glasses manufactured from one day to the next, in relation to the average, is two. This information, said like this, is not useful for much. However, we can compare the variance obtained with that of other weeks and we will get an idea of ​​which weeks Jorge has been more constant in the production of glasses.

What is the standard deviation

The typical deviation It is a statistical measure that offers us information about the dispersion of data with respect to the average. It is the average of the individual deviations of each observation with respect to the mean of a distribution. This deviation is always greater than or equal to zero. When it is low it means that the data is close to the average and when it is high it indicates that they are further away from it, that is, they are more dispersed.

A standard deviation of 0, according to the example we have given, would occur if Jorge had produced a number of glasses every day of the week that coincided with the average.. This is a very rare case, since it is very rare for all people in a group to be the same height, weigh the same, or prefer the same thing. That is, what we expect is variability when analyzing the data of a variable.

How is it calculated?

Once we have found the value of the mean and the variance of the data, we can calculate the standard deviation, calculating the square root of the latter. Let’s put this into practice by finding the standard deviation of the number of glasses manufactured by Jorge in five days:

S= √ (5-5) ²+ (4-5) ²+ (7-5) ²+ (3-5) ²+ (6-5) ² / 5= 1.41

According to this result, we can say that, on average, the number of glasses manufactured per day deviates from the average (5 glasses) by 1.4142 glasses, that is, Jorge produces one glass more or one less than the average per day. This allows us to affirm that, in general, Jorge remains close to the average glass production.

This example illustrates how, Through this measure of dispersion, we can know, on average, what a population is like with respect to a variable. (more or less heterogeneous).

If you realize, on many occasions, the media only talks about socks. This causes us to assign a value to a population in a variable, when there may be great heterogeneity; So much so that, deep down, that average value does not represent anyone in the population.

For example, it could have been the case that Jorge did not produce five glasses on any given day. However, the headline of the news could be “Jorge makes five glasses a day.” Paradoxical, right?

What is the standard deviation for?

Serves for make an estimate about how spread out the data is with respect to the mean of the variable being studied. They allow us to know, on average, how the observations are concentrated around the mean.

In addition to serving as a reference to analyze the variability of a population, it also serves to set the width of the confidence interval when making inferences about the mean. The larger the standard deviation, the larger this interval will be.. This makes, for example, more complicated to say that there are significant differences between two populations.

At a psychometric level, it is related to reliability, which in some cases can be understood as the stability of the results obtained over time in longitudinal studies. Thus, a test would be reliable when it yields very similar measurements for the same real value.

The standard deviation reveals the reality of the data

The standard deviation allows us measure the dispersion of the data with respect to the mean. Likewise, it provides information about the heterogeneity of a sample and is useful for making inferences.

By calculating it, we obtain a measure of how much the observations spread from the mean, which helps us understand the fluctuations of the variable studied. The standard deviation is essential for avoid wrong conclusions based only on the average and gives us a more complete perspective of the data.

You might be interested…