Standard deviation is one of the simplest measures applied in statistical analysis. The standard deviation, in statistics, refers to a measure of the spread of a data set in comparison to its mean. The formula is expressed as the positive square root of the variance, which is used to measure the spread of the data points of a set of data points. If the data points are a long distance away from the mean, i.e. there exists a high spread in the data set, then it suggests a high standard deviation. The low standard deviation indicates that the array of values of the data is close to the mean( or the expected value) of the data set. It is a good measure of variability as it is in the same unit as the data set's units. Standard deviation is all values-based. As a result, any shift in even a single value can contribute to the value of the standard deviation. In this article, we will learn about standard deviation, its importance and how it is associated with variance.

Standard Deviation

Now, we will discuss significant concepts revolving around standard deviation.

Definition

Standard deviation is the degree of dispersion or we can say the scatter of the data points relative to its mean. It tells how the values are spread across the data sample and it is the measure of the variation of the data points from the mean.

Standard Deviation Symbol

Standard deviation can be abbreviated as SD. The most commonly represented symbol of standard deviation used in mathematics is the lowercase Greek letter \(\sigma\), known as sigma, for the population standard deviation. Whereas the letter \(s\) is used for the sample standard deviation.

Steps To Calculate Standard Deviation

The steps to calculate standard deviation are as follows:

1. Find the arithmetic mean of the data values by summing all the data points together. Then, dividing them by the total number of data points involved.
2. For each data point, find the square of the distance to its mean.
3. Sum the values from step 2.
4. Divide the sum computed in step 3 by the number of data points.
5. Take the square root of the value calculated in step 4.

Since standard deviation is calculated as a square root, it cannot be negative. Therefore, the smallest value of the standard deviation is 0.

Standard Deviation Formula

• Population Standard Deviation Formula

\(\sigma=\sqrt{\frac{1}{N} \sum_{i=1}^{N}\left(X_{l}-\mu\right)^{2}}\)

Where,

\(\sigma=\) Population standard deviation

\(\mathrm{N}=\) Number of observations in population

\(X_{1}=\) ith observation in population

\(\mu=\) population mean

• Sample Standard Deviation Formula

\( s=\sqrt{\frac{1}{n-1} \sum_{l=1}^{n}\left(x_{i}-\bar{x}\right)^{2}}\)

Where,

\(s=\) Sample standard deviation

\(n=\) Number of observations in sample

\(x_{1}=\) ith observation in the sample

\(\overline{\boldsymbol{x}}=\) Sample mean

Significance of Standard Deviation

• It measures the deviation from the mean, which is a vital statistic. It shows a central tendency.
• It is significant in measuring the accuracy and precision of a single typical measurement. It is also a measure of spread of data around the mean.
• Since it is the positive square root, all the values we get are positive, making it easy for further interpretations and calculations.
• Standard deviation is highly used in the field of finance. Finance and banking are all about measuring risk, and standard deviation measures risk, i.e., volatility.
• Standard deviation is used by portfolio managers. The Sharpe ratio in portfolio management uses standard deviation.
• Standard deviation is used to carry out mathematical operations and statistical and data analysis.

Variance and Standard Deviation

Variance is the average squared deviations from the mean, while standard deviation is the square root of this number.

Relationship Between Variance and Standard Deviation

• Both measure volatility in a distribution.
• The variance equals the square of standard deviation, and the standard deviation is equal to the square root of variance.
• Standard deviation is the spread of a group of numbers from the mean. The variance measures the average degree to which each point differs from the mean.

Differences Between Variance and Standard Deviation

• The variance shows the average squared deviations from the mean, whereas the standard deviation represents the square root of variance.
• Both have different units. The standard deviation has similar units as that of the original data values, whereas variation has square units of the data values.
• The notations used for variance is σ2, s2, or Var(X) whereas the notations used for standard deviation is the lowercase Greek letter \(\sigma\), known as sigma, for the population standard deviation and the letter \(s\) is used for the sample standard deviation.

Advantages and Disadvantages of Standard Deviation

Some of the advantages and disadvantages of Standard Deviation are as follows:

Summary

• In statistics, the standard deviation is defined as a measure of the dispersion of a set of data relative to its mean.
• It is computed by finding the square root of the variance, which is a measure of how a collection of data is spread out.
• Standard deviation can be abbreviated as SD. The most commonly represented symbol of standard deviation used in mathematics is the Greek letter \(\sigma\), known as sigma, for the population standard deviation. For the sample standard deviation, the letter \(s\) is used.
• Population standard deviation formula:

\(\sigma=\sqrt{\frac{1}{N} \sum_{i=1}^{N}\left(X_{l}-\mu\right)^{2}}\)

• Sample standard deviation formula:

\( s=\sqrt{\frac{1}{n-1} \sum_{l=1}^{n}\left(x_{i}-\bar{x}\right)^{2}}\)

• The standard deviation has the similar units as that of the original data values.
• The standard deviation is used to carry out both mathematical operations and statistical analysis.
• The extreme values in the data values known as outliers can affect the standard deviation.

Standard Deviation Solved Examples

A class of students took an English test. The teacher wants to know whether most students are performing at the same level, or if there is a high standard deviation.

Solution:

First we calculate the mean of all the scores which is:

1279 ÷ 15 = 85.2

Now we subtract the mean from every test square to see how much they vary from the mean:

85 – 85.2 = -0.2

86 – 85.2 = 0.8

100 – 85.2 = 14.8

76 – 85.2 = -9.2

81 – 85.2 = -4.2

93 – 85.2 = 7.8

84 – 85.2 = -1.2

99 – 85.2 = 13.8

71 – 85.2 = -14.2

69 – 85.2 = -16.2

93 – 85.2 = 7.8

85 – 85.2 = -0.2

81 – 85.2 = -4.2

87 – 85.2 = 1.8

89 – 85.2 = 3.8

Now, we square each difference:

-0.2 x -0.2 = 0.04

0.8 x 0.8 = 0.64

14.8 14.8 = 219.04

-9.2 x -9.2 = 84.64

-4.2 x -4.2 = 17.64

7.8 x 7.8 = 60.84

-1.2 x -1.2 = 1.44

13.8 x 13.8 = 190.44

-14.2 x -14.2 = 201.64

-16.2 x -16.2 = 262.44

7.8 x 7.8 = 60.84

-0.2 x -0.2 = 0.04

-4.2 x -4.2 = 17.64

1.8 x 1.8 = 3.24

3.8 x 3.8 = 14.44

Now we find the average of these squares which gives us the variance:

830.64 ÷ 15 = 75.6

The standard deviation: Square root of 75.6 = 8.7

The standard deviation of these tests is 8.7 points out of 100. Since the variance and the standard deviation are low, the teacher can infer that most students are performing around the same level.

In summary, knowledge of standard deviation is important for students sitting for U.S.-based competitive exams like the SAT, ACT, GRE, and GMAT. Being one of the most basic statistical measures, standard deviation gives a clear picture of how data points are dispersed around the mean, which is very important in data analysis and interpretation. A large standard deviation shows the wide spread of the data, but a low standard deviation would point towards data being close together on the mean. As the calculation of standard deviation is dependent upon all data, even the minimum change in the value of a single item has the capability of affecting the end result. Mastering this idea not only improves problem-solving but also improves a student's capacity to analyze data correctly, making it an essential skill for academic achievement.