The standard deviation is one of the main tools in the analysis of the data and it is dispersion. When you are able to do the Standard deviation test by the Standard deviation calculator, then it is easy for us to analyze the whole data. When we are using the sd calculator, then we analyze what is the mean values and what are the upper and the lower limit of the whole population of the data. In reality, it is impossible to analyze the whole population of the data, and you always go for a sample evaluation. When you apply the test on the test on the sample by the standard deviation. Then the whole picture about the whole population would be quite clear for you. Calculate standard deviation and find the information about the whole population of the data, you can evaluate what is the highest values and what are the lowest values and between the lower the upper limit the whole set of the data is residing.

In the following article, we are discussing what is the importance of the standard deviation and what is its utilization.

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## Why do we use standard deviation?

The standard deviation is one of the main tools in defining the depth of the data in our observation. We are going to find all the variance in data by the mean and standard deviation calculator. When we are able to find the variance in the whole data, then we are able to predict the following calculation:

- The mean of the data, and the culture of the values around it. This would represent the whole population of the data and its resonating values.
- The upper and the lower limit of the data and clarify the picture of how the data is dispersed and what its variance is around the whole set of the data.

In the following example, we are defining the mean, variance, and standard deviation from the set of the data

*Question: Find the mean, variance, and standard deviation for the following data?*

Class Interval | 0-10 | 10-20 | 20-30 | 30-40 | 40-50 | 50-60 |

Frequency | 27 | 10 | 7 | 5 | 4 | 2 |

*Solution:*

Class Interval | Frequency (f) | Mid Value (xi) | fxi | fxi2 |

0 – 10 | 27 | 5 | 135 | 675 |

10 – 20 | 10 | 15 | 150 | 2250 |

20 – 30 | 7 | 25 | 175 | 4375 |

30 – 40 | 5 | 35 | 175 | 6125 |

40 – 50 | 4 | 45 | 180 | 8100 |

50 – 60 | 2 | 55 | 110 | 6050 |

∑f = 55 | ∑fxi = 925 | ∑fxi2 = 27575 |

*N = ∑f = 55*

*Mean = (∑fxi)/N = 925/55 = 16.818*

*Variance = 1/(N – 1) [∑fxi2 – 1/N(∑fxi)2]*

*= 1/(55 – 1) [27575 – (1/55) (925)2]*

*= (1/54) [27575 – 15556.8182]*

*= 222.559*

*Standard deviation = √variance = √222.559 = 14.918*

### Analysis of the concepts:

The complete analysis of the question is essential to understand the concept of the standard deviation and how we can find the standard deviation by the sample standard deviation calculator. The population standard deviation calculator can analyze the whole set of the population in a matter of seconds. But you should be familiar with the following concepts to make sure, that you are familiar with the whole result.

### Class intervals:

The first and the foremost concept is the class interval, in this case, the class intervals are **0-10,10-20, 20-30,30-40, 40-50, 50-60. **These are the class intervals, actually, we have divided the whole set of data into small and regular intervals to know their frequencies and how much data is residing in one specific interval.

### Frequencies:

The frequencies of various intervals are 27, 10, 7, 5, and 4,2, and these are the frequencies of various class intervals. For example, the interval 0-10 has a frequency of 27 and the class 10-20 has a frequency of 10, and the 7, 5, 4, and 2 for the proceeding class intervals. The class interval 0-10 has the highest frequency and the class interval 50-60 has the minimum frequency of 2.

**Mid values (xi):**

The mid values are the middle values of our data range, for example, the class interval 0-10 has a middle range of 5. The class interval 0-20 has a middle range of 15, 25, 35, 45, and 55 for the next class intervals.

### The frequency of middle values( fxi):

The Standard deviation calculator(fxi) can be calculated by multiplying the frequency (f) and the middle values. In the above question, the frequencies of the middle values are given as 135, 150, 175, 175,180,110. The sd calculator readily finds the frequencies of the middle values and we are easily able to evaluate the values.

### The total frequency of middle(fxi2):

The total frequency of the middle values is calculated by the frequency of the middle values. In the above question the the total frequencies are 675, 2250, 4375, 6125, 8100, 6050 for the class intervals 0-10, 0-20, 0-30, 0-40, 0-50, 0-60.The total frequency is telling how many times, the mean values are residing in our calculations. This can be extracted from the data, and it would be greatly helpful; in finding the frequency of the data.

### The Summations *∑f, ∑fxi, ∑fxi2 :*

The summation *∑f, ∑fxi, ∑fxi2 *is the simple addition of all the above calculations. Now the *∑f is the total frequency, ∑fxi is the total frequency of the middle values.* When we are able to find the sample standard deviation by the simple formula if we are able to find all the calculations.

### Conclusion:

The main thing for the students to learn is a concept like the Standard deviation, it is essential to learn the basics like the class interval, frequency, the mean frequency, and the total mean frequency. When you are able to extract all the data values, then it is easy for the students to calculate all the variance and the standard deviation calculation. It would be great for the students doing the research as the standard deviation is vastly used in the analysis of the data and the profile for your data.