Module 3 Discussion: Mean, SD, Probability

This discussion board post explains the concepts of mean, standard deviation, and probability. It defines each term and then applies them to a scenario involving employee vacation days. The post calculates the standard deviation for vacation days less than 10 and more than 21, explaining how to find the probability using the area under the curve. It also mentions that a distribution table would be helpful for plotting the graph and performing calculations. The post concludes that the standard deviation is small, resulting in a steep bell-shaped curve.

Module 3 Discussion
First of all before discussing the solution one should be clear about certain facts like what
is 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 all
values giving in a series which is divided by the total number of values present in that
series. It is a simple raw average and is also called as an unweighted average [1].
Standard Deviation
When the dispersion of a set of data from its mean is measured then it is called as
standard deviation. This has the notation as  known as sigma. According to Robert Niles
it 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 examples
are bound together around the value of mean.
2) If the examples are tightly clustered and the graph obtained is a bell shaped curve and
is steep then the standard deviation is small.
3) If the examples are spread apart or far from each other hence the bell shaped of the
graph obtained is flat which means that standard deviation is large.
Probability
This value in field of statistics us how often an event will happen after many repeated
trials. The theory of probability generally describes an ideal situation where the chances
are known to people like in a game. The probability of an event describes the
measurement of degree to which one should believe that event will happen or how much
the frequency of event will occur simultaneously [3].
Thus if the question is observed it is seen that since no distribution table is given we have
to judge the standard deviation and probability in terms of a graph plotted using the data
given. There are certain points that are required to reach the final answer and those are
listed below:
 Hence, in the given module it is given that mean is 14 days which is the number
of vacation days taken by one employees of a company and standard deviation
that is  is 3 days. So if it is required to find the probability if the number of
vacation 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 since
it is calculated as 4/3 which comes as 1.33. Thus the probability is the area under
the curve from negative infinity to 1.33 standard deviation and till 10 days in the
graph plotted.
 Now if it is required to find out the probability if number of vacation days taken
are 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 the
probability is the area under the curve plotted after 21 days till infinity with
standard deviation as 2.33.
With the help of proper distribution table one can easily plot the graph and show the
calculations that are required to be done. As it is seen that standard deviation obtained is
small hence the graph drawn with the help of data will be a bell shaped curve which is
further 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. Second
edition, 1948; third edition, 1961.