# Standard Deviation (s) Calculator

getcalc.com's **Standard Deviation (s) Calculator** is an online statistics & *probability *tool to estimate the degree of uncertainty or linear variability of whole elements in a group from its *mean* or central location. It shows how many standard deviation, the whole elements of data distributed around the mean in statistical experiments. This calculator is featured to generate the complete standard deviation work with step by step calculation for any given valid input values to help grade school students to solve the similar worksheet problems or to know how to solve such calculations manually. Besides, the examples of real World problems may help users to understand how the uncertainty calculation is being used in various fields to improve the quality of service.

## What is Standard Deviation & its Usage?

**Standard Deviation** is a mathematical function or method used in the context of probability & statistics, represents the degree of uncertainty or linear variability from its mean or central location (μ) of data distribution. In other words, it defines how the whole elements or members in the sample deviates from its mean in the statistical surveys or experiments. The mean is the common behavioral characteristics of the data distribution whereas the standard deviation is the square root of *variance* and playing an important role to characterize the data distribution around the mean to take the scientific assessment in the event of uncertainty and variation. It's an most important & widely used function to measure the dispersion.

In the frequency distribution there is difference among the elements. Standard deviation represents how the elements of frequency distribution are spread around the mean. The lower dispersion indicates that there is relatively large amount of elements uniformly distributed around the mean and the higher dispersion indicates that there is relatively less amount of elements uniformly distributed around the mean. The statistical method used for finding the degree of uncertainty or dispersion of finite set of data is called as ** population standard deviation**, often denoted by the Greek symbol

**σ**, generally used in

**descriptive statistics**which deals with finite number of population elements. Where the statistical method used to estimate the degree of uncertainty or dispersion of sample set of data is called as

**sample standard deviation**, often denoted by an English alphabet

**s**and is one of the most frequently used methods in

**inferential statistics**which deals with infinite number of population to draw the conclusion about the population in the statistical experiments.

Users may use this standard deviation calculator to find how the whole data distributed around the mean or the spread of data distribution around the central location. Any business may analyze its quality of service or process by using the this uncertainty calculation. The summary of data which describes the deviation from the expected results indicates how quality your process or service is. The methods which are dealt with the finite number of population is called as Descriptive Statistics and the method which dealt with the infinite number of population is called as Inferential Statistics.

**Formula**

The below formula is the mathematical representation to estimate the sample standard deviation or to estimate the degree of uncertainty in the data distribution.

## Standard Deviation in Real World Problems?

Standard deviation is applicable in almost all scientific and engineering industries like chemical, mechanical, manufacturing, internet networks, digital computations, database, production etc to find the how the data is distributed around the mean or its central location. It's a repeated analysis based on the experiments with infinite or larger quantity of data to summarize the degree of uncertainty, or the quality of service or process. The below are the fewer examples where the sample standard deviation is effectively used in different practical scenarios or real World problems to analyze the observations of sample dataset to draw the conclusions about the entire population.

### In the Analysis of Manufacturing

The below example is one of a real World problems in manufacturing may useful for beginners to understand how to employ standard deviation in the manufacturing industry. For example, an industry is manufacturing pistons for four stroke engines, the theoretical or expected diameter of the piston is 15 inches but the actual measurement for 10 pistons are 15, 15, 14.99, 15.01, 15, 15, 15.05, 15, 14.98, & 15 inches. What is the standard deviation for the above data. What should be taken from the answer of standard deviation to improve the quality of process?

__Workout :__

For the above experiment, below are the input parameters & values

x_{1} = 15, x_{2} = 15, . . . . , x_{10} = 15

Number of samples n = 10

Degrees of freedom = n - 1

= (10 - 1) = 9

step 1 *Find the mean for 15, 15, 14.99, 15.01, 15, 15, 15.05, 15, 14.98, 15*

µ =
n
∑
i = 1
X_{i}n

=(15 + 15 + 14.99 + . . . . + 15)/10

µ = 15.003

step 2 Apply the samples, mean & degrees of freedom values in the standard deviation formula

=√{ (15 - 15.003)² + (15 - 15.003)² + (14.99 - 15.003)² + . . . . + (15 - 15.003)²}/9

= √(-0.003)² + (-0.003)² + (-0.013)² + . . . . + (-0.003)²/9

= √(0 + 0 + 0.0002 + . . . . + 0)/9

= √0.003/9

= √0.0003

s = 0.0173

The mean is 15.004

The standard deviation is 0.0173

The variation is 0.0003

The whole result deviates from mean by 0.0171σ. In other words the uncertainty of whole outcomes is 0.0171σ. The necessary changes are required in the manufacturing process to minimize the deviation as much as close to 0 to improve the quality of the process. Use this standard deviation calculator to summarize the data to analyze & improve the quality of process.

### In the Analysis of Material Science

The analysis of material is an essential part for any manufacturing industries. The standard deviation is used in the analysis of various materials to estimate how strong the material is. In material design & analysis the sample standard deviation is effectively used to measure the linear variability of material. The below example is one of a real World problems in material science, may helpful for engineers who are involved in the analysis of strength of materials.

For example, an industry is manufacturing connecting rods experimenting how strong is the connecting rod to measure the deviation or linear variability of results or the possibility of defective results. To measure the deviation of molecules for a particular connecting rod samples, the experiment require the molecules samples randomly collected at various points of the connecting rod. For example, if 15 inches connecting rod is involved in the analysis, the analysis may require the molecules count of random samples at 1 inch, 2 inches, 2.5 inches, 3 inches, 5 inches, 7 inches, 7.5 inches, 13 inches, 14 inches & 14.3 inches.

Random Samples Molecule Count | |
---|---|

Random Samples taken at | Molecules Count |

1 inch | 1655 |

2 inches | 1654 |

2.5 inches | 1654 |

3 inches | 1650 |

5 inches | 1655 |

7 inches | 1656 |

7.5 inches | 1657 |

13 inches | 1653 |

14 inches | 1655 |

14.3 inches | 1656 |

__Workout :__

For the above experiment, the below are the input parameters & values

x_{1} = 1655, x_{2} = 1654, . . . . , x_{10} = 1656

Number of samples n = 10

Degrees of freedom = n - 1

= (10 - 1) = 9

step 1 *Find the mean for 1655, 1654, 1654, 1650, 1655, 1656, 1657, 1653, 1655, 1656*

µ =
n
∑
i = 1
X_{i}n

=(1655 + 1654 + 1654 + . . . . + 1656)/10

µ = 1654.5

step 2Apply the samples, mean & degrees of freedom values in the standard deviation formula

=√{ (1655 - 1654.5)² + (1654 - 1654.5)² + (1654 - 1654.5)² + . . . . + (1656 - 1654.5)²}/9

= √(0.5)² + (-0.5)² + (-0.5)² + . . . . + (1.5)²/9

= √(0.25 + 0.25 + 0.25 + . . . . + 2.25)/9

= √34.5/9

= √3.8333

s = 1.9579

The central location of the data μ is 1,654.5

The sample standard deviation s is 1.9579σ

The collective variation between all the samples is 3.8333

From the results, what should be taken to improve the quality of process? The whole samples' molecules present within the standard deviation of 1.9579σ. Since the standard deviation is not more than 2 or 3σ, therefore the variation in the sample elements are within the range and is acceptable. The summary of data says that the molecules of connecting rod are almost equally distributed across its length. However, the corrective measures to be taken to brought the standard deviation as close to zero to improve the quality of process to have the molecules equally distributed across the rod to ensure the strength of material is equal at all points of the material. Using the increased number of sample points in sample standard deviation calculator would increase the quality of results or minimize the errors. Find the right number of samples to have the sound estimation in statistical experiments by using this *sample size calculator*.

### In the Analysis of E-commerce

Standard deviation is applicable in every field where the collection of data is involved in the process. It is applicable in e-commerce to analyze the orders, order rejections, orders in a particular time period, offers, profit, loss etc. In this below example, how the standard deviation is being used to analyze the order rejections. For example, an e-commerce website is receiving 1000 orders online everyday and delivering products at customers doorsteps. This experiment would include various data & circumstances to analyze the data. For learning purpose, lets do the simple analysis for the order rejections for every 100 orders. Finding the standard deviation for every successful orders for each 100 orders will bring the summary of data that help the examiner to improve the quality of process. If 93, 97, 99, 95, 87, 96, 100, 99, 93 & 92 are the successful deliveries for each 100 orders.

__Workout :__

for the above experiment, below are the input parameters & values

x_{1} = 93, x_{2} = 97, . . . . , x_{10} = 92

Number of samples n = 10

Degrees of freedom = n - 1

= (10 - 1) = 9

step 1 *Find the mean for 93, 97, 99, 95, 87, 96, 100, 99, 93, 92*

µ =
n
∑
i = 1
X_{i}n

=(93 + 97 + 99 + . . . . + 92)/10

µ = 95.1

step 2Apply the samples, mean & degrees of freedom values in the standard deviation formula

=√{ (93 - 95.1)² + (97 - 95.1)² + (99 - 95.1)² + . . . . + (92 - 95.1)²}/9

= √(-2.1)² + (1.9)² + (3.9)² + . . . . + (-3.1)²/9

= √(4.41 + 3.61 + 15.21 + . . . . + 9.61)/9

= √142.9/9

= √15.8778

s = 3.9847

The central location of the data μ is 95.1

The standard deviation s is 3.9847σ

The collective variation between all the samples is 15.8778

What should be taken from this summary of data? The average successful orders for the current process is 95.1 for every 100 orders. The variation in the orders is 15.8778 and the whole samples of data is fall within the range of 3.9847σ to -3.9847σ. Being the whole deviation of data is 3.9847σ, the process needs to take the necessary changes to brought the deviation as close to zero, to improve the quality of service. Use this standard deviation calculator to analyze the any collection of data in ecommerce to analyze & summarize the data to improve the quality of service.

This calculator, examples of real World problems and standard deviation work for any given input may helpful for users to understand, practice & verify various collection of data to improve the quality of any process or service.