__Definition of Standard Deviation Calculator__

The standard deviation is a statistic that measures the dispersion of a dataset relative to its mean. In simple words, standard deviation tells us how much the given number from a group of data deviates from the mean value of the group. If the value of standard deviation is low, it means that the data is closer to the mean value. Higher standard deviation means more spread out data. It is denoted by and is square root of variance. Variance describes how much a random variable differs from its expected value. It is calculated as the average of the squares of the differences between the individual (observed) and the mean/expected value. Since there are different ways in which standard deviation can be used, it has different formulas.

__Population Standard Deviation__

As the name indicates, this type of standard deviation is used when an entire population can be measured and every member of the population is considered as an individual value. This deviation is more reliable. Since every single value is used while calculating the mean, the mean value is unbiased. It is denoted by . The formula for population standard deviation is as follows:

σ=1Ni=1N(xi-μ)2

Where

N is the total number values of in the given set of data.

xi is the individual value or the number under observation

is the mean/expected value

The first step is to find the mean value . It is obtained by adding all the values and then dividing the sum by total number of values.

Now the mean value is subtracted from each individual value . The difference is called deviation.

Square each deviation to get a positive value.

Add all the deviations. The summation operator i=1N(xi-)2 indicates that each individual value starting from 1 to N (N can be any number) is taken one by one. Moreover, the square of difference between mean and individual value at each step is added to the value obtained in the previous step.

Find average by dividing the total sum by total number of values N. This is called variation.

Lastly, take the square root of variance to get population standard deviation.

The concept will be clear once we do this with actual numbers. The following examples will help.

__Example1__

Set of data= {3, 4, 7, 5, 9, 2}

Total number of values= N= 6

Mean value = = (3+4+7+5+9+2)/6= 30/6= 5

i=1N(xi-μ)2 = (3-5)2+ (4-5)2+ (7-5)2+ (5-5)2+ (9-5)2+ (2-5)2

= (-2)2+ (-1)2+ (2)2+ (0)2+ (4)2+ (-3)2

= 4+1+4+0+16+9=34

1Ni=1N(xi-μ)2=1634=5.667

= 5.667=2.380

Population Standard deviation of 3, 4, 7, 5, 9, 2} is 2.380. It is evident that the value of standard deviation is low. This is because the difference between the individual values in the data set is low.

__Example2 __

Let’s try another example with higher values this time.

Set of data = {34, 8, 15, 39, 12, 10, 15}

Total number of values = N = 7

Mean value = = (34+ 8+15+39+12+10+15)/7= 133/7= 19

i=1N(xi-μ)2 = (34-19)2+ (8-19)2+ (15-19)2+ (39-19)2+ (12-19)2+ (10-19)2+ (15-19)2

= (15)2+ (-11)2+ (-4)2+ (20)2+ (-7)2+ (-9)2+ (-4)2

=225+121+16+400+49+81 +16=908

1Ni=1N(xi-μ)2=9087=9087=129.714

=129.714= 11.389

Population Standard deviation of {34, 8, 15, 39, 12, 10, 15} is 11.389. It is evident that the value of standard deviation is high. This is because the difference between the individual values in the data set is high.

__Sample Standard Deviation__

In some cases, it is not possible to have all the values. This means that the entire population cannot be measured. In such situations, we use sample standard deviation. The set of data we have is not the entire data but only a sample of the entire data. Sample standard deviation is denoted by s and has the following formula

s=1N-1i=1N(xi-x)2

Where

N is the sample size

xi is one sample vale

x is the sample mean

There are, however, many equations for calculating sample standard deviation. The issue is that the mean value is not completely unbiased as in case of population standard deviation because the selected sample might not represent the entire population correctly. The above equation looks complicated but it is very simple.

The first step is to find the mean value x. It is obtained by adding all the values and then dividing the sum by total number of values.

Now the mean value is subtracted from each individual value xi. The difference is called deviation.

Square each deviation to get a positive value.

Add all the deviations. The summation operator i=1N(xi-x )2 indicates that each individual value starting from 1 to N (N can be any number) is taken one by one. Moreover, the square of difference between mean and individual value at each step is added to the value obtained in the previous step.

Find average by dividing the total sum by one less than the total number of values (N-1). This is called variation.

Lastly, take the square root of variance to get sample standard deviation.

The concept will be clear once we do this with actual numbers. The following examples will help.

__Example1__

Let’s use the same set of data as in the previous examples to notice the difference in both types of standard deviation.

Set of data = {3, 4, 7, 5, 9, 2}

Sample size= N = 6

Sample mean = x = (3+4+7+5+9+2)/6= 30/6= 5

i=1N(xi-x)2 = (3-5)2+ (4-5)2+ (7-5)2+ (5-5)2+ (9-5)2+ (2-5)2

= (-2)2+ (-1)2+ (2)2+ (0)2+ (4)2+ (-3)2

= 4+1+4+0+16+9=34

1N-1i=1N(xi-x)2=346-1=345=6.8

s=6.8= 2.067

Sample Standard deviation of {3, 4, 7, 5, 9, 2} is 2.067. It is evident that the value of standard deviation is low. This is because the difference between the individual values in the data set is low.

__Example2__

Set of data = {34, 8, 15, 39, 12, 10, 15}

Sample size= N = 7

Sample mean = x = (34+ 8+15+39+12+10+15)/7= 133/7= 19

i=1N(xi-x)2 = (34-19)2+ (8-19)2+ (15-19)2+ (39-19)2+ (12-19)2+ (10-19)2+ (15-19)2

= (15)2+ (-11)2+ (-4)2+ (20)2+ (-7)2+ (-9)2+ (-4)2

= 225+121+16+400+49+81 +16=908

1N-1i=1N(xi-x)2=9087-1=9086=151.33

s=151.33= 12.3017

Sample Standard deviation of {34, 8, 15, 39, 12, 10, 15} is 12.3017. It is evident that the value of standard deviation is high. This is because the difference between the individual values in the data set is high.

__Applications of Standard Deviation__

Standard deviation is a statistical measurement that is widely used in finance and industries. It is used to test real world data. Calculating standard deviation is useful whenever we want to know how far from the mean a typical value from a distribution can be.

__In weather forecasting__

As we already discussed that sample standard deviation is used when we don’t have the complete information. This means that it can be used to predict future values using a sample from presently available values. This phenomena finds application in weather forecasting. It can also be used to determine differences in regional climate. Mean temperature alone cannot give proper information. Coastal cities tend to have far more stable temperatures due to regulation by large bodies of water and have low standard deviation while inland cities have high values of standard deviation. This means that even with the same mean value of 75°C, the temperature in inland cities ranges between 60°F and 85°F while that of coastal cities ranges from 30°F to 110°F.

__In finance__

Standard deviation is commonly used to determine volatility of stocks by using price data. Greater value of standard deviation of securities means a larger price range. It is a useful tool in investing and trading strategies as it helps measure market and security volatility—and predict performance trends. Standard deviation can also be used to measure risk associated with price fluctuations of assets. This helps the investors to decide if they should invest in those assets based on the chances of return. For example, if stock A has an average return of 5% with a standard deviation of 20% and stock B has the same average return but a standard deviation of 55%, the first stock is the safer option for the investors because standard deviation of stock B is significantly larger, for the exact same return. This means that there is uncertainty in stock B. Now, this uncertainty might as well result in even better rates of return. So there’s no fix rule. It depends upon the investor. If he is willing to take the risk, he can choose high standard deviation.

__In industries__

One of the most common applications of standard deviation in industry is quality control. When standard deviation is very high and values fall outside the calculated range, it shows that a change is needed in the production or designing process.

__In sports__

People often bet on their favorite teams. Teams with a higher standard deviation are less predictable. Teams with a lower standard deviation are consistent. Trying to know ahead of time which teams will win may include looking at the standard deviations of the various team statistics.

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