Introduction to Standard deviation with examples and applications

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The level of variability or dispersion in the data under observation is measured statistically using the standard deviation. It gives a crucial impression of how data points vary around the mean or average value.

The value of the standard deviation increases when the variability or dispersion is maximized. Standard Deviation may be represented by SD and is most commonly represented in mathematics by the small alphabet (a Greek letter) sigma σ, for the population SD. 

In this article, we will learn about the concept of Standard Deviation, its general formula, and examples as well.

 

Definition: Standard Deviation

The assessment of variability in statistical analyses is known as the Standard Deviation (SD). It is a phrase that is frequently and widely used while analyzing data. It shows the amount of departure from the mean (average) of the data.

Depending on one of two conditions, a low SD indicates that the data points tend to be close to the mean, whereas a high SD indicates that the data are distributed throughout a broad range of values. Its general formula is given by:

Formula:

For group data (Data with frequency distribution), the formula is given by:

 

For ungroup data (Data without frequency distribution), the formula is given by:

 

 

Types of Standard Deviation:

As we have discussed the process of taking Standard deviation has two types.

Population standard deviation:

The population standard deviation is a metric for gauging how widely distributed the individual data points are within a population. It is a means to measure how dispersed the data is from the mean, to put it simply. When the data you have is representative of the entire population, it is relevant. 

σ = √ {(xi - µ) 2 / N}

Sample standard deviation:

A sample standard deviation is a statistic that must be determined from a small sample of a given population. It indicated the sample standard deviation rather than the population standard deviation. 

S = √ {(xi - µ) 2 / (N – 1)}

How to calculate the standard deviation for grouped and ungrouped data?

The calculation varies depending on whether the data is grouped or ungrouped. Here are a few examples to learn how to calculate the standard deviation for both types.

Ungroup data:

Example 1: In the prominent area of Punjab, a cellular company publishes the usage of cellular data in GB (Gigabytes) of 30 houses are given as below 73,75,58,56,76,47,66,53,52,43

Compute the standard deviation of the given data.

Solution:

Here,

N (number of terms) = 31

∑X = 73 + 75 + 58 + 56 + 76 + 47 + 66 + 53 + 52 + 43

∑X= 599

Mean = ∑X/n = 599/10 = 59.9

Step 2: Now subtract each value from the mean.

X – Mean = 13.10,15.10,-1.90,-3.90,16.10,-12.90,6.10,-6.90,-7.90,-16.90

(X – Mean)2 = (13.10)2 , (15.10)2, (-1.90)2, (-3.90)2, (16.10)2, (-12.90)2, (6.10)2, (-6.90)2, (-7.90)2, (-16.90)2

(X – Mean)2 = 171.61, 228.01, 3.61, 15.21, 259.21, 166.41, 37.21, 47.61, 62.41, 285.61

∑(X – Mean)2 = 171.61 + 228.01 + 3.61 + 15.21 + 259.21 + 166.41 + 37.21 + 47.61 + 62.41 + 285.61

∑(X – Mean)2 = 1276.9

Step 3: Now put values to the formula and simplify it.

√∑(X – Mean)2/n-1 = √(1276.9/10-1)

√∑(X – Mean)2/n-1 = √(1276.9/9)

√∑(X – Mean)2/n-1 = √(141.9)

√∑(X – Mean)2/n-1 = 11.91

A standard deviation calculator could be a handy tool for calculating the sample or population standard deviation for ungrouped data with steps.

Group data:

Example 2: Find S.D for the following data

23, 44, 59, 58, 87, 38, 78, 39, 24, 74, 76, 47, 34, 38, 66, 12, 73, 44, 86

By taking class interval h =10

Solution:

Class interval = 10

Largest number = 87

Smallest number = 12

Step 1: To make a Frequency distribution

 

Classes	Tally Mark	Frequency	X	X2	FX	FX2
11-20	I	1	15.5	240.25	15.5	240.25
21-30	II	2	25.5	650.25	51	1300.5
31-40	IIII	4	35.5	1260.25	142	5041
41-50	III	3	45.5	2070.25	136.5	6210.75
51-60	II	2	55.5	3080.25	111	6160.5
61-70	I	1	65.5	4290.25	65.5	4290.25
71-80	IIII	4	75.5	5700.25	302	22801
81-90	II	2	85.5	7310.25	171	14620.50
		∑f=19			∑fX=1094	∑ fX2=57664.25

 

Formula:

 

S.D= √ [(57664.25/19)-(1094 /19)2]

S.D= 21.8715

Applications of Standard Deviation:

In a variety of contexts, including those related to academics, business, finance, forecasting, manufacturing, the medical field, polls, and demographic features, among others, standard deviation helps us to understand the diversity in data collection. 

It also teaches us how to utilize various methods for calculating the coefficient of variation, testing hypotheses, and calculating confidence intervals.

In Statistical Studies: We must be aware of things like the coefficient of variation, confidence intervals, and hypothesis testing. The standard deviation is important for interpreting statistics and assessing hypotheses. It helps with the standard error calculation, which is required to perform hypothesis testing and establish confidence intervals. The standard deviation is essential for determining the statistical significance of results and assessing the precision and dependability of estimators.

In Manufacturing: To ensure adequate size, we need to understand quality control, improvement logic, and precise machining of parts. By calculating the standard deviation for various components, manufacturers may identify the main contributors and determine the sources of variability. It supports the maintenance and monitoring of product quality requirements.

In Forecast Accuracy: Due to climate change, we must be aware of local weather patterns and natural calamities. We may access uncertainty and variability in weather forecasts by using several related concepts and measurements in the context of weather forecasting. These consist of probabilistic predictions, ensemble forecasting, and standard deviation of model output.

In Medicine: In the medical field, there is always a chance to provide a drug that is more suitable and effective in a positive way. To improve the reliability of suitable drugs or medical equipment, this term provides a helping hand to test data again and again. Through this technique, the most suitable product is provided in the market.

Conclusion

In the discussion above, we learned about the idea of standard deviation, its many forms and their respective formulae, as well as several words used in standard deviation data analysis. Its formulae make it simple for us to comprehend the subtle differences between group and ungroup data.

We may find standard deviation in various directions by using different sorts of instances. The details of how it is used in daily life serve as a reminder of its significance. After fully grasping the idea, we might be able to choose the best method for solving the data while determining its standard deviation.