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Calculating Probability using Mean and Standard Deviation

Added on -2019-09-23

This article explains the concepts of mean, standard deviation and probability. It provides step-by-step solution to calculate probability in given scenarios. The article also highlights the characteristics of standard deviation. The solution involves plotting a graph using the given data and calculating the required standard deviation to find the probability.
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Module 3 DiscussionFirst of all before discussing the solution one should be clear about certain facts like whatis mean, what is standard deviation and what is probability.Mean In the field of Mathematics and Statistics arithmetic mean is defined as the sum of allvalues giving in a series which is divided by the total number of values present in thatseries. It is a simple raw average and is also called as an unweighted average [1].Standard DeviationWhen the dispersion of a set of data from its mean is measured then it is called asstandard deviation. This has the notation as known as sigma. According to Robert Nilesit is a kind of “mean of mean”. This further helps to learn the normal distribution of data. The characteristics features of standard deviation are as follows:1) The value of standard deviation for a set of data tells us how tightly various examplesare bound together around the value of mean.2) If the examples are tightly clustered and the graph obtained is a bell shaped curve andis steep then the standard deviation is small.3) If the examples are spread apart or far from each other hence the bell shaped of thegraph obtained is flat which means that standard deviation is large.ProbabilityThis value in field of statistics us how often an event will happen after many repeatedtrials. The theory of probability generally describes an ideal situation where the chancesare known to people like in a game. The probability of an event describes themeasurement of degree to which one should believe that event will happen or how muchthe frequency of event will occur simultaneously [3].Thus if the question is observed it is seen that since no distribution table is given we haveto judge the standard deviation and probability in terms of a graph plotted using the datagiven. There are certain points that are required to reach the final answer and those arelisted below:Hence, in the given module it is given that mean is 14 days which is the numberof vacation days taken by one employees of a company and standard deviationthat is is 3 days. So if it is required to find the probability if the number ofvacation days taken are less than 10 days.If we move backward from 14 days to reach 10 days, we r going 4 days (14-10=4) in negative direction so the deviation is which is 3 days becomes as 1.33 sinceit is calculated as 4/3 which comes as 1.33. Thus the probability is the area underthe curve from negative infinity to 1.33 standard deviation and till 10 days in thegraph plotted.Now if it is required to find out the probability if number of vacation days takenare more than 21 days then in this case we are moving 7 days far from 14 days(21-14 =7) so the required standard deviation is 7/3 which is 2.33. Thus theprobability is the area under the curve plotted after 21 days till infinity withstandard deviation as 2.33.With the help of proper distribution table one can easily plot the graph and show thecalculations that are required to be done. As it is seen that standard deviation obtained issmall hence the graph drawn with the help of data will be a bell shaped curve which isfurther steep in shape.References[1] Tejvan Pettinger, Arithmetic Mean, November 28, 2012.[2] Robert Niles, Standard Deviation.[3] Glenn Shafer, The Meaning of Probability, Chapter 2.[4] Barnett, Vic (1973). Comparative Statistical Inference. Wiley. Second edition, 1982.[5]Jeffreys, Harold (1939). Theory of Probability. Oxford University Press. Secondedition, 1948; third edition, 1961.

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