(1.1) Part A : Definition of Standard Deviation.

Standard Deviation:

The Standard Deviation is a measure of how spread out numbers are.

Deviation just means how far from the normal.

Its symbol is σ (the greek letter sigma)


What does it mean? :

It shows how much variation or "dispersion" exists from the average (mean, or expected value).

A low standard deviation indicates that the data points tend to be very close to the mean, whereas high standard deviation indicates that the data points are spread out over a large range of values.

Graphical illustration :

You can see from the illustration above that although all 3 graphs have the same mean or same average which is mean = 0 ,

the standard deviations are different.

The blue graph is less "spread out" while the red graph is the most "spread out".

We can say that the Blue graph has the lowest standard deviation while the Red Graph has the highest standard deviation.


Practical Application examples :

The practical value of understanding the standard deviation of a set of values is in appreciating how much variation there is from the "average" (mean).

Climate:
As a simple example, consider the average daily maximum temperatures for two cities, one inland and one on the coast. It is helpful to understand that the range of daily maximum temperatures for cities near the coast is smaller than for cities inland. Thus, while these two cities may each have the same average maximum temperature, the standard deviation of the daily maximum temperature for the coastal city will be less than that of the inland city as, on any particular day, the actual maximum temperature is more likely to be farther from the average maximum temperature for the inland city than for the coastal one.


Sports:
Another way of seeing it is to consider sports teams. In any set of categories, there will be teams that rate highly at some things and poorly at others.
Chances are, the teams that lead in the standings will not show such disparity but will perform well in most categories.
The lower the standard deviation of their ratings in each category, the more balanced and consistent they will tend to be.
Teams with a higher standard deviation, however, will be more unpredictable.
For example, a team that is consistently bad in most categories will have a low standard deviation.
A team that is consistently good in most categories will also have a low standard deviation. However, a team with a high standard deviation might be the type of team that scores a lot (strong offense) but also concedes a lot (weak defense), or, vice versa, that might have a poor offense but compensates by being difficult to score on.
Trying to predict which teams, on any given day, will win, may include looking at the standard deviations of the various team "stats" ratings, in which anomalies can match strengths vs. weaknesses to attempt to understand what factors may prevail as stronger indicators of eventual scoring outcomes.

In racing, a driver is timed on successive laps.
A driver with a low standard deviation of lap times is more consistent than a driver with a higher standard deviation.
This information can be used to help understand where opportunities might be found to reduce lap times.