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Calculation of Standard Deviation

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Added on  2021-07-05

Calculation of Standard Deviation

   Added on 2021-07-05

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Standard DeviationList of Contents1.Definition:...........................................................................................................................................22.Scope of Standard Deviation:.............................................................................................................23.Calculation of Standard Deviation:..................................................................................................33.1.SD for Individual Series:........................................................................................................33.2.SD for Discrete Series:............................................................................................................43.3.SD for Frequency Distribution:..............................................................................................44.Population and Sample standard deviation:......................................................................................5FAQs:.........................................................................................................................................................6References..................................................................................................................................................71
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1.Definition:Standard deviation is used to assess the behavior of data in term of dispersion. It measures averageinterval in all values of a data set. The low standard deviation (less than zero) indicates that there isclose association in the obtained values of data while high value (greater than 1) indicates that thereis great degree of dispersion in data values. 2.Scope of Standard Deviation:In statistics, standard deviation has a pivotal role as a more powerful analytical tool to demonstratethe level of dispersion in data. Basically, dispersion indicates stretch and squeeze in datadistribution. Difference among the values from a certain value is assessed with the help ofdispersion. Standard deviation is a primary and most preferable measure of dispersion in statisticalanalysis. A Greek symbol “σ” is used to denote standard deviation. It has two main characteristics asdescribed below;1.Arithmetic mean plays a critical role to achieve the value of deviation. It is calculated bytaking the mean value of data as a reference point. 2.It deals with optimistic values. We follow the assumption to have positive values ofdeviations. 3.The standard deviation must be measured in the same unit. For instance, if the data is relatedto distance in meters then standard deviation must also be in meters.Critical analysis of standard deviation concluded that it is the best measure of change in distributionof values and more easy to understand. However, there are also some limitations in the standarddeviation as its calculation is difficult if the observations are large in number then it highly affectsits value.Standard deviation also provides a clear guideline for probability. In this regard, statisticiansdiscovered “Chebyshev’s theorem”. This theorem states that 75% units of any population havetwo standard deviations related to the population’s mean. For example, if the mean score of apopulation (Shoe size) is 9.6 and SD is 1.1, then it indicates that 75 percent shoe size fallbetween 7.4 (two SDs less than mean value) and 11.8 (two SDs greater than mean value).So inthis example, theorem is aimed to explain probability of a population’s behavior. 3.Calculation of Standard Deviation:The standard deviation is calculated with an objective to measure the extent to which observations are differed from the mean value. If these differences are added together then 2
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positive numbers must balance the negative ones and hence ultimate sum must be in term of zero. Statisticians described different approaches to calculate standard deviation. Before understandingthese approaches, it is very important to get insights about types of data distribution. These main three types are explained as below: Individual seriesDiscrete seriesFrequency distribution3.1. SD for Individual Series:It includes observations in only one column. Following table will help to provide clearunderstanding about this data type:Score15303548566274For this data type, first of all, arithmetic mean is find out by employing the formula as: ́x=∑XNAfter that deviation is calculated for all recorded observations with the help of:D = x- ́xNext, square of all deviations is find out individually and their sum is divided by total number ofobservations. At last, square root of previous value is called standard deviation. Its formula is listedbelow:σ=√(∑D²)NIn this formula;3
Calculation of Standard Deviation_3

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