Standard deviation is a measurement used in statistics of the amount a number varies from the average number in a series of numbers. The standard deviation tells those interpreting the data, how reliable the data is or how much difference there is between the pieces of data by showing how close to the average all of the data is.

- A low standard deviation means that the data is very closely related to the average, thus very reliable.
- A high standard deviation means that there is a large variance between the data and the statistical average, thus not as reliable.

The standard deviation is determined by finding the square root of what is called the variance.

The variance is found by squaring the differences from the mean.

In order to determine standard deviation:

- Determine the mean (the average of all the numbers) by adding up all the data pieces and dividing by the number of pieces of data.
- Subtract each piece of data from the mean and then square it.
- Determine the average of all of those squared numbers calculated in #2 to find the variance.
- Find the square root of the numbers in #3 and that is the standard deviation.

Calculators are available online to quickly determine a standard deviation. For example, at Calculator.net and MathPortal.org.

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.

Now you see how standard deviation works.

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