The standard deviation or standard deviation is a measure that provides information on the average dispersion of a variable. The standard deviation is always greater than or equal to zero.
To understand this concept we need to analyze 2 fundamental concepts.
- Mathematical expectation, expected or average value:It is the average of our data series.
- Deviation:The deviation is the separation that exists between any value of the series and the mean.
Now, understanding these two concepts, the standard deviation will be calculated in a similar way to the average. But taking deviations as values. And although this reasoning is intuitive and logical, it has a flaw that we will check with the following graph.
In the previous image we have 6 observations, that is, N = 6. The average of the observations is represented by the black line located in the center of the graph and is 3. We will understand by deviation, the difference that exists between any of the observations and The black line So, we have 6 deviations.
- Deviation -> (2-3) = -1
- Deviation -> (4-3) = 1
- Deviation -> (2-3) = -1
- Deviation -> (4-3) = 1
- Deviation -> (2-3) = -1
- Deviation -> (4-3) = 1
As we can see if we add the two deviations 6 deviations and divide by N (6 observations), the result is zero. The logic would be that the average deviation was 1. But a mathematical characteristic of the mean with respect to the values that form it is precisely that the sum of the deviations is zero. How do we fix this? Squareing the deviations
Formulas to calculate the standard deviation
The first is to square the deviations, divide by the total number of observations and finally make the square root to undo the square, such that:
Alternatively there would be another way to calculate it. It would be averaging the sum of the absolute values of the deviations. That is, apply the following formula:
However, this formula is not an alternative to the standard deviation as it yields different results. Actually, the above formula is the deviation from the mean. The standard or standard deviation and the deviation from the measure
Example of calculation of the standard deviation
We will check how, with any of the two formulas exposed, the result of the standard deviation or average deviation is the same.
According to the variance formula (square root):
According to the absolute value formula:
As the intuitive calculation dictated. The mean deviation is 1. But, had we not said that the formula of absolute value and standard deviation gave different values? That’s right, but there is an exception. The only case in which the standard deviation and the deviation from the mean offer the same result is the case in which all deviations are equal to 1.
The relationship of the standard deviation with the variance
In short, the variance is no more than the standard deviation squared. Or what is the same, the standard deviation is the square root of the variance. They are related as follows:
After this image, it is clear that all the formula that is within the square root is the variance. The reason why it is necessary to understand that this part is known as variance is that it is used in other formulas to calculate other measures. Thus, although the standard deviation is more intuitive to interpret results, it is imperative how the variance is calculated.
