# How to Calculate Standard Deviation ( with Formulas )

How to Calculate Standard Deviation. Statistician in the world could calculate standard deviation by manualy . Calculations are complicated and the risk of making mistakes is very high. Besides this manual calculation is very troublesome. This is why statisticians need electronic spreadsheets and computer programs to do calculations.

Probability theory and Calculate Standard Deviation is used to summarize the distributions of population, sample, probability distribution and random variable which are included in the scope of statistical science.

## How to Calculate Standard Deviation

Calculate Standard Deviation is the square root of the sum of the squares of the differences of the numbers in a series from the arithmetic mean of the series to the subtraction of the number of elements in the array.

To open a little, to calculate stantart deviation;

- The arithmetic mean of the numbers is calculated.
- Each number differs from the arithmetic mean.
- The square of each of the differences found is calculated.
- The squares of the differences are summed.
- The sum obtained is divided by an exponent of the number of elements in the series.
- The square of the found number is taken.

With Calculate Standard Deviation we find how much of the data is close to the average. If the standard deviation is small, the data are scattered near the average. Conversely, if the deviation is large, the data are scattered in the distant places. If all values are the same, the deviation is zero.

## Let’s explain it with an example

Let’s have two data sets. The elements of one of the data strings

90, 70, 80 and 80

the other

10, 30, 80 and 100 . The average of these two indexes is 80. However, it is clear that the distribution of the first sequence is more uniform. The deviation of this index used for any measurement will be small if many elements are close to the average.

In the second line, a majority of the elements seem to have received far away values. The standard deviation of this index will be large. Therefore, we can say that the data in the first row is more reliable and stable. The deviation of the first index in the example is 8,16 .. and the standard deviation of the second index is 42,03 ….

If we assume that these series are grades of the students in the first grade, the grades of the first grade students are more stable and we can say that the students are more successful. Already when you subtract 100 values in the second class (class), you see that the average falls to 40 (half of the other class).

Again, if we assume that the values in these ratios are the daily sales amounts of two grocery stores, we can say that the sales of the first grocery store are better than those of the second grocery store.

The standard deviation shows how uniform and balanced the data are distributed.

As you can see, we can use the stantard deviation to make a diagnosis about the sequence and therefore make some decisions.

Of course the standard deviation will be more useful when used in larger data sets.

## What is the formula for Calculate Standard Deviation ?

Let’s give two simple formula.

The first formula is:

To explain the formula:

σ: Standard Deviation

N: Number of elements in the index

xi: Index of the x. element.

x: Arithmetic Mean of the Numbers in the Sequence

(xi – x) 2: i. Get the average difference of the element.

N

Σ: Collect the results of the operation for the first element (x = 1) to the last element (N) for each element you repeat.

x = 1

1: Divide the adjacent value by N-1.

N-1

√: And finally calculate the square of the value obtained.

### Our simple formula is as follows

In this form, x is the arithmetic mean, and n is the number of elements. According to the form we sum the squares of the differences of all the elements from the arithmetic mean, divide by one of the number of the resultant elements, and take the square root of the resulting value.

## What is Standard Deviation?

The representation of the standard slicer used mathematically in terms of population, probability distribution, and random variable is done by sigma, which is an ancient Greek letter. Sigma is expressed by the σ sign. Besides, the deviation in the standard deviation calculations for the sample is expressed as s or s’.

Calculate Standard Deviation with the above mentioned data is equal to the square root value of the variance. In other words, the standard deviation is expressed as: the deviation is the square root of the result of dividing the total value of the squares of difference values of the data values by the arithmetic mean, divided by the number of data minus one.

Briefly, the deviation is the square of the average of the squares of the deviations of the data values in the mean.

For the Calculate Standard Deviation used as a spreading measure, the unit of measurement should be viewed. Another definition of diffusion, defined as variance, is the average of the squares of differences in the mean of the data.

The square of the data units must be taken to calculate the variance measure. Calculate Standard Deviation made in this way can reach different results. For the deviation, the square root of the variance can be avoided because the result obtained by the calculation is different.

This, Calculate Standard Deviation becomes unit data units. Also, the propagation of the data is measured by the data units.

Calculate Standard Deviation can also be expressed as a statistical measure which indicates the spread of the data according to the average received and is used very frequently in quantitative scale numbers.

The fact that the data included in the measurement is close to the average causes the standard deviation to be small and the standard deviation of the data included in the measurement is large.

However, the fact that all of the data included in the measurement are at exactly the same value means that the standard deviation is zero.