How To Calculate Standard Deviation

The standard deviation is one of the best-known formulas in mathematics today, a formula that will allow us to very easily calculate how much a product has deviated from the mean.

This belongs to the mathematics of statistics, a mathematics that is responsible for analyzing all data for purposes of classification and analysis. The standard deviation is responsible for telling you exactly the relationship with the average of the products, seeing what usually deviates from what is the trend.

The more standard deviation there is, the more extremes we will have, since there is more standard deviation when we have data from 1 to 20 than when we have data from 1 to 10.

That if, the bad thing about the standard deviation is that it brings them, since although the formula is simple, you have to first calculate other things that are not so simple, such as the class mark and the variance, which have long formulas.

Today we are going to teach you how to calculate all this, that is, that today you will learn to do variance, class mark, standard deviation and coefficient of variation, all in full. In addition, you will have a solved exercise to be able to solve all your doubts easily.

Instructions for calculating the standard deviation

1. Calculate the mean of the data: The first thing we must do to be able to calculate the standard deviation correctly is to calculate the mean of the data. The average of the data is nothing more than trying to reach a middle point between everything there is. If, for example, there are two people, one who measures 1.50 and the other who measures 1.60, the average will be 1.55, which is the midpoint between the two. The way to calculate it is easy, since you add all of them and divide by the number of people there are or the data there are. However, in this case it gets a bit complicated, since sometimes they do not give us exact data, but instead they give us data by intervals, which must be obtained by the so-called class mark.
   1. Calculate the class mark if we have separate data: If, for example, we have that the exercise gives us intervals instead of real data, that is, instead of telling me that Jaime measures 2.03, it tells us that he measures between 2 and 2.05 meters, we must calculate the class mark. This is an average between the two values ​​and is calculated in the same way as the normal average, so surely you know how to do it without any problem. For example, if we have people between 2 and 4 years old, the class mark is 3 This will undoubtedly allow us to solve any type of standard deviation problem, since the only thing that needs to be done to solve them is to start making the class mark of all the values ​​and leave it as a real value with which we later have to deal with it. average.
2. Variance; The most important formula of the exercise is the standard deviation, however, there is no standard deviation if we do not have variance, so we have to learn first of all to calculate it in the correct way. The formula is exactly like this: Sum of the class mark minus the mean of the data squared by the number of data that corresponds to each data item (f sub i) divided by the number of data items that are taken (n). Here you have to subtract the class mark minus the average that we have calculated before. Once we have it, we are going to do the square (otherwise it could be negative) and multiply each one by each frequency of each data. We add it all (they are sums) and divide it by the number of data that there are. I know that it is a bit complicated to understand now, however, later I will present you an example, an example that will allow you to understand this right away.
3. Standard Deviation Formula: Now it’s time to calculate the standard deviation, since that’s what we’re here for and it’s the most important formula in the whole exercise, hands down. The standard deviation is the root of the variance, neither more nor less. A square root is the opposite of a square number and it is easily done with any calculator, in which we will click on the square root icon that everyone knows and it will automatically calculate that root. Once we have done it, all that remains is to write down the deviation that has come out on a piece of paper, something that will tell us that we already have the exercise completely finished and ready to deliver.
4. Variance coefficient: Yes, there are teachers who are going to go a little further, since they are not only going to ask us for the standard deviation, but they are going to ask us for the so-called coefficient of variance. This coefficient is nothing more than a percentage of the variance and the mean of the data, which will tell us the amount by which these vary. This is calculated using the formula for the Standard Deviation between the mean of X. This gives us a number that always has to be less than one, otherwise we would be doing something wrong. People usually express it multiplied by 100 to be able to express it as a percentage, since it is how it should be delivered and how it works best for teachers, apart from the fact that it takes a second to do the percentage and it is always much better than putting decimals.
5. Example exercise: Statement: We have 10 people in a class, of which one has between 1 and 2 euros, 3 people have between 3 and 4 euros, 5 have between 4 and 5 euros and 1 has between 6 and 7 euros. Calculate the standard deviation and the coefficient of variation:
6. The first thing we must do is average the data, something that we can only do by calculating the respective class marks of the euros. I’ve made it easy for all of them to come out right, since the first is 1.5(1+2/2), the second is 3.5, the third is 4.5, and the fourth is 6.5. Then we can do the average by multiplying each one by the number of people there, leaving 1.5 the first, 10.5 the second, 22.5 the third and 6.5 the fourth, leaving a total of 41/10 and 4.1 euros on average. Once we have the mean, we must proceed to calculate the variance, something that we will be able to do by subtracting the mean from each class mark first. In the first it comes out -2.6, the second comes out .0.6, the third comes out 0.1 and the fourth comes out 2.4. We do all this with a square, which gives us 6.76 for the first, 0.36 for the second, 0.01 for the third and 5, 76 the fourth and last. Now you have to multiply by the number of people who have this money, coming out in the first 6.76 (1 person), in the second 1.08, in the third 0.05 and in the fourth 5.76. We add all this up and it gives us 13.65, which we have to divide among 10 people in the class, leaving us with a variance of 1.365. If we want the standard deviation, we must take the root of this, which is approximately 1.168. The coefficient of variation is going to be 1.168 (typical deviation) between the mean, which is going to be 4.1. This gives 0.28, which if multiplied by 100 gives 28% percent. As you can see, the exercise is simple, it’s just that it’s very complicated to explain it in words, however, seeing it on paper is much easier. 76(1 person), in the second 1.08, in the third 0.05 and in the fourth 5.76. We add all this up and it gives us 13.65, which we have to divide among 10 people in the class, leaving us with a variance of 1.365. If we want the standard deviation, we must take the root of this, which is approximately 1.168. The coefficient of variation is going to be 1.168 (typical deviation) between the mean, which is going to be 4.1. This gives 0.28, which if multiplied by 100 gives 28% percent. As you can see, the exercise is simple, it’s just that it’s very complicated to explain it in words, however, seeing it on paper is much easier. 76(1 person), in the second 1.08, in the third 0.05 and in the fourth 5.76. We add all this up and it gives us 13.65, which we have to divide among 10 people in the class, leaving us with a variance of 1.365. If we want the standard deviation, we must take the root of this, which is approximately 1.168. The coefficient of variation is going to be 1.168 (typical deviation) between the mean, which is going to be 4.1. This gives 0.28, which if multiplied by 100 gives 28% percent. As you can see, the exercise is simple, it’s just that it’s very complicated to explain it in words, however, seeing it on paper is much easier. If we want the standard deviation, we must take the root of this, which is approximately 1.168. The coefficient of variation is going to be 1.168 (typical deviation) between the mean, which is going to be 4.1. This gives 0.28, which if multiplied by 100 gives 28% percent. As you can see, the exercise is simple, it’s just that it’s very complicated to explain it in words, however, seeing it on paper is much easier. If we want the standard deviation, we must take the root of this, which is approximately 1.168. The coefficient of variation is going to be 1.168 (typical deviation) between the mean, which is going to be 4.1. This gives 0.28, which if multiplied by 100 gives 28% percent. As you can see, the exercise is simple, it’s just that it’s very complicated to explain it in words, however, seeing it on paper is much easier.
7. Graphic representations: If requested or if you simply want to give a little touch of quality to the exercise, you can opt for graphic representations, which are used to express various things about the exercise. You can draw a graph in which each class mark is the X and the frequency is the Y. Draw a line that represents it and then try to express there data such as the mean (which is usually close to the middle) and other data. If you look closely, you probably missed the famous Gaussian bell (which by the way was everywhere, because I can’t stop hearing its name), which comes out when there is little frequency of low values, high frequency of medium values, and low frequency of high values. This is happening right now in the Spanish population, since there are few retirees, few young people and many adults. However, this is going to change and it will surely end up being an inverted pyramid. In short, the graphical representation will help us to better see what the average is and to better implement the exercise in practice and in real life.

Tips for Calculating Standard Deviation

• Learn the exercise better: This example that I have given you can help you to begin to understand it, however, it may be that you still have your little doubts, since it is perfectly normal in this world. What you have to do is try to learn the exercise in the correct way, seeing the statement and then doing it yourself in the correct way, putting the statement on a piece of paper, with all the data clear, and then trying to do it yourself, without help. from nobody and on a piece of paper. You can also search the internet for statements of the standard deviation, since there are billions of exercises in the world that will allow you to maintain all this in a very simple way and be able to do these exercises to the sack, until you can get a lot of ease here.
• Learn more mathematics: In this section of educating we have good news for you, since we are going to learn mathematics in all possible ways. First of all, we are going to have mathematics of statistics, that is, mathematics that will help you to calculate data, work with graphs and so on. We also have other mathematics, such as engineering, which help you learn more advanced things. Without a doubt, it is worth visiting all these articles, since in the end you will be able to easily find all this and much more without having to leave here.
• Comments with doubts: If you have any doubts about how to make the standard deviation, you should take into account that it is something normal, however, nothing happens, since we are here to solve all your doubts without any type of problem.