Created: 13 February 2023

Formula, Examples, Symbol, and Calculations for Standard Deviation

The idea of standard deviation, which assesses the degree of variation or dispersion of a set of data, is crucial in statistics.

It is an effective metric for describing data distribution since it tells us how far the data vary from the mean or average. In numerous disciplines, including finance, economics, psychology, and engineering, to mention a few, the standard deviation is a frequently used quantity.

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Formula

The standard deviation formula is a mathematical phrase used to determine how much a collection of data deviates from the mean. It is determined as the square root of variance and is denoted by the Greek letter (sigma). The average of the squared deviations between each data point and the mean is used to determine variance.

Symbols

The Greek letter σ (sigma) stands in for the symbol for standard deviation. Standard deviation is frequently used to represent it in mathematical and statistical computations.

Calculating

A simple procedure with numerous steps is calculating the standard deviation of a set of data. You must first determine the data set's mean. The difference will then be discovered by deducting the mean from each value in the data set. The discrepancies discovered in the preceding phase must then be squared. Divide the sum by the total number of values in the data set to get the sum of the squared differences. To determine the standard deviation, take the square root of the result from the previous step.

Exmple

You can better understand the idea of standard deviation by using examples.

Take into account the following data set of the heights of 10 people in centimeters: 170, 170, 175, 180, 182, 183, 185, 186, 190, 192.

To find the mean, we add up all the values and divide by the number of values, which is 10. The mean of the data set is calculated as:
μ = (170 + 170 + 175 + 180 + 182 + 183 + 185 + 186 + 190 + 192) / 10
μ = 180

Next, we subtract the mean from each value in the data set to find the difference. The differences are -10, -10, -5, 0, 2, 3, 5, 6, 10, and 12.

We then square the differences obtained in the previous step to get 100, 100, 25, 0, 4, 9, 25, 36, 100, and 144. Summing these squared differences gives us a sum of 543.

Dividing the sum by the number of values in the data set (10) gives us a result of 54.3. Finally, taking the square root of the result obtained in the previous step gives us the standard deviation, which is 7.3.

It is significant to remember that when comparing data sets with similar values, standard deviation is a relevant parameter. The standard deviation might not accurately reflect the data's dispersion if the data sets have wildly disparate values.

Conclusion

In conclusion, standard deviation is a crucial statistical concept that assesses how much a group of data vary or are dispersed. The standard deviation formula is a mathematical equation that quantifies how far away from the mean or average of the data each number in a set of data is. For people who work in fields where statistical analysis is required, understanding standard deviation's calculation and how it is denoted by the Greek letter (sigma) is crucial.

Frequently Asked Questions (FAQs)

• 1. What is Standard Deviation?
  The dispersion of a data set from its mean is quantified by calculating its standard deviation. It shows the dispersion of the data points relative to the mean.
• 2. How is Standard Deviation calculated?
  Take the square root of the variance to get the standard deviation. To calculate the variance, we simply take the mean square of all the deviations from the mean. For a population, the mathematical symbol for Standard Deviation is (sigma), while for a sample, the symbol is s.
• 3. What does a high or low Standard Deviation mean?
  The larger the Standard Deviation, the more dispersed the data points are. If the Standard Deviation is small, then the data points are concentrated in a narrow range around the mean.
• 4. Can Standard Deviation be negative?
  Since it is a measure of variability, which is always positive or zero, Standard Deviation can never be negative.
• 5. When is Standard Deviation most useful?
  When you need to know how far your data points are from the mean or average, standard deviation is your best friend. It's a standard tool for quantifying the degree of danger or unpredictability surrounding a given data set in fields as diverse as economics, engineering, science, and sociology.