uses of standard deviation

 Topic :uses of standard deviation

 Introduction:

The standard deviation of probability distribution,random variable, or population or multiset of values is a measure of the spread of its values. It is usually denoted with the letter σ (lower case sigma ). It is defined as the square root of the variance.

To understand standard deviation, keep in mind that variance is the average of the squared differences between data points and the mean. Variance is tabulated in units squared. Standard deviation, being the square root of that quantity, therefore measures the spread of data about the mean, measured in the same units as the data.

          Real-life 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).

1 Weather

As a simple example, consider average temperatures for cities. While two cities may each have an average temperature of 60 °F, it's helpful to understand that the range for cities near the coast is smaller than for cities inland, which clarifies that, while the average is similar, the chance for variation is greater inland than near the coast.

So, an average of 60 occurs for one city with highs of 80 °F and lows of 40 °F, and also occurs for another city with highs of 65 and lows of 55. The standard deviation allows us to recognize that the average for the city with the wider variation, and thus a higher standard deviation, will not offer as reliable a prediction of temperature as the city with the smaller variation and lower standard deviation.

3.1.2 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 be pretty good in most categories. The lower the standard deviation of their ratings in each category, the more balanced and consistent they might be. So, a team that is consistently bad in most categories will have a high standard deviation indicating it will probably lose more often than win. A team that is consistently good in most categories will also have a low standard deviation and will therefore end up winning more than losing. A team with a high standard deviation might be the type of team that scores a lot (strong offense) but gets scored on a lot too (weak defense); or vice versa, might have a poor offense, but compensate by being difficult to score on - teams with a higher standard deviation will be more unpredictable.

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.

3.1.3 Finance

In finance, standard deviation is a representation of the risk associated with a given security (stocks, bonds, property, etc.), or the risk of a portfolio of securities. Risk is an important factor in determining how to efficiently manage a portfolio of investments because it determines the variation in returns on the asset and/or portfolio and gives investors a mathematical basis for investment decisions. The overall concept of risk is that as it increases, the expected return on the asset will increase as a result of the risk premium earned - in other words, investors should expect a higher return on an investment when said investment carries a higher level of risk.

For example, you have a choice between two stocks: Stock A historically returns 5% with a standard deviation of 10%, while Stock B returns 6% and carries a standard deviation of 20%. On the basis of risk and return, an investor may decide that Stock A is the better choice, because the additional percentage point of return (an additional 20% in dollar terms) generated by Stock B is not worth double the degree of risk associated with Stock A. Stock B is likely to fall short of the initial investment more often than Stock A under the same circumstances, and will return only one percentage point more on average. In this example, Stock A has the potential to earn 10% more than the expected return, but is equally likely to earn 10% less than the expected return.

Calculating the average return (or arithmetic mean) of a security over a given number of periods will generate an expected return on the asset. For each period, subtracting the expected return from the actual return results in the variance. Square the variance in each period to find the effect of the result on the overall risk of the asset. The larger the variance in a period, the greater risk the security carries. Taking the average of the squared variances results in the measurement of overall units of risk associated with the asset. Finding the square root of this variance will result in the standard deviation of the investment tool in question. Use this measurement, combined with the average return on the security, as a basis for comparing securities.

Uses for Standard Deviation

Some examples of situations in which standard deviation might help to understand the value of the data:

A class of students took a math test. Their teacher found that the mean score on the test was an 85%. She then calculated the standard deviation of the other test scores and found a very small standard deviation which suggested that most students scored very close to 85%.

A dog walker wants to determine if the dogs on his route are close in weight or not close in weight. He takes the average of the weight of all ten dogs. He then calculates the variance, and then the standard deviation. His standard deviation is extremely high. This suggests that the dogs are of many various weights, or that he has a few dogs whose weights are outliers that are skewing the data.

A market researcher is analyzing the results of a recent customer survey. He wants to have some measure of the reliability of the answers received in the survey in order to predict how a larger group of people might answer the same questions. A low standard deviation shows that the answers are very projectable to a larger group of people.

A weather reporter is analyzing the high temperature forecasted for a series of dates versus the actual high temperature recorded on each date. A low standard deviation would show a reliable weather forecast.

A class of students took a test in Language Arts. The teacher determines that the mean grade on the exam is a 65%. She is concerned that this is very low, so she determines the standard deviation to see if it seems that most students scored close to the mean, or not. The teacher finds that the standard deviation is high. After closely examining all of the tests, the teacher is able to determine that several students with very low scores were the outliers that pulled down the mean of the entire class’s scores.

An employer wants to determine if the salaries in one department seem fair for all employees, or if there is a great disparity. He finds the average of the salaries in that department and then calculates the variance, and then the standard deviation. The employer finds that the standard deviation is slightly higher than he expected, so he examines the data further and finds that while most employees fall within a similar pay bracket, three loyal employees who have been in the department for 20 years or more, far longer than the others, are making far more due to their longevity with the company. Doing the analysis helped the employer to understand the range of salaries of the people in the department.

 References

1.https:///en.m.wikipedia.org/wiki/standard deviation.
2.mathematics education -Dr.k.sivarajan