Standard deviation

< Probability and statistics definitions

Standard deviation is a measure of the amount of variation in a set of values. A low standard deviation — close to zero — indicates that the values tend to be close to the mean of the set, while a high standard deviation indicates that the values are spread out over a wider range. For example, are all the data points close to the mean? Or are lots of scores far above (or far below) the mean score?

Standard deviation may be abbreviated SD, and is most commonly represented by the lower case Greek letter σ (sigma), for the population standard deviation, or the Latin letter s, for the sample standard deviation.

Contents:

1. Calculating standard deviation
2. Standard deviation and the normal distribution
3. History of the standard deviation

Calculating the standard deviation

The formula to find the sample standard deviation (s) is:

In this formula, s is the standard deviation, x is the data point we are solving for in the set, x̄ is the mean, and n is the total number of data points.

Example Problem: Calculate the standard deviation for the following data set: {12, 15, 17, 20, 30, 31, 43, 44, 54}

1. Sum the numbers: 12 + 15 + 17 + 20 + 30 + 31 + 43 + 44 + 54 = 266.
2. Square the sum from Step 1: 266 x 266 = 70756
3. Divide the result from Step 2 by the number of items (n) in the data set. In this case, there are 9 items: 70756 / 9 = 7861.777777777777. Temporarily set this number aside.
4. Individually square the original numbers {12, 15, 17, 20, 30, 31, 43, 44, 54}, then add them together:
   • (12 x 12) + (15 x 15) + (17 x 17) + (20 x 20) + (30 x 30) + (31 x 31) + (43 x 43) + (44 x 44) + (54 x 54) = 9620
5. Subtract the result from Step 3 from the result of Step 4: 9620 – 7861.777777777777 = 1758.2222222222226. Retain all decimal places without rounding until the final step. Set this number aside for now.
6. Subtract 1 from n (the number of items). In this case, n = 9: 9 – 1 = 8
7. Divide the result from Step 5 by the result from Step 6 to obtain the variance: 1758.2222222222226 / 8 = 219.77777777777783
8. Compute the square root of the result from Step 7: √(219.77777777777783) = 14.824903971958058. The standard deviation is approximately 14.825.

The formula for the population standard deviation is similar [1]:

Here, σ is the standard deviation, x₁ is the data point we are solving for in the set, µ is the mean, and N is the total number of data points. Notice that the formula is almost identical to the one for the sample standard deviation, except that the terminology is slightly different as you’re dealing with samples and not populations.

Standard deviation and the normal distribution

The bell curve, referred to by statisticians as a “normal distribution,” is often used in statistics as a tool to understand the standard deviation. The following graph of a normal distribution shows the mean (μ) at the center. Each segment (shaded from dark blue to light blue) signifies one standard deviation away from the mean. For instance, μ + 2σ indicates two standard deviations from the mean.

A normal distribution curve can depict numerous real-life scenarios. For instance, in a class where most students receive Cs and only a few get As or Fs, this distribution can be modeled with a bell curve. Similarly, people’s weights, heights, nutritional habits, and exercise routines can also be represented by graphs resembling this one. This understanding allows companies, schools, and governments to make predictions about future behavior. For behaviors that follow this type of bell curve (such as SAT performance), it is possible to predict that 34.1 + 34.1 = 68.2% of students will score close to the average score, or one standard deviation away from the mean.

A useful property of the standard deviation is that, unlike the variance, it is expressed in the same units as the data; the variance is a measure of how spread out a set of data is, but it is expressed in squared units. For example, let’s say you take measurements of people’s heights: the standard deviation will be expressed in feet while the variance will be expressed in squared feet.

History of the standard deviation

The journey of standard deviation’s development has been extensive and complex. Carl Friedrich Gauss, a German mathematician, first introduced the concept of standard deviation in the early 1800s. Considered one of the most influential figures in statistical history, Gauss created standard deviation as a method to measure data dispersion around the mean.

It wasn’t until the early 1900s that standard deviation gained widespread use. During this time, English statistician Ronald Fisher expanded upon the application of standard deviation in hypothesis testing. Fisher demonstrated how standard deviation could be used to compute the probability of obtaining a specific sample mean from a given population mean, making it an invaluable tool for statistical inference.

Today, standard deviation is among the most commonly used statistical measures, applied across various fields such as economics, finance, medicine, and social science. It serves as a potent instrument for understanding data spread and making population inferences.

Key contributors to the history of standard deviation include:

• Carl Friedrich Gauss (1777-1855): The German mathematician who conceived the idea of standard deviation.
• Ronald Fisher (1890-1962): The English statistician who pioneered the application of standard deviation in hypothesis testing.
• William Sealy Gosset (1876-1937): The Irish statistician who developed the student t-distribution, a frequently used distribution for hypothesis testing.
• Jerzy Neyman (1894-1980): The Polish-American statistician who established the Neyman-Pearson hypothesis testing framework.

As a versatile and influential measure, standard deviation has significantly impacted the field of statistics, proving invaluable for comprehending data distribution and making population inferences.

References

[1] Standard Deviation. Retrieved June 12, 2023 from: https://www.nlm.nih.gov/nichsr/stats_tutorial/section2/mod8_sd.html

[2] Ainali, CC BY-SA 3.0 https://creativecommons.org/licenses/by-sa/3.0, via Wikimedia Commons