Standard deviation is a statistical measure that calculates the amount of dispersion or variation within a set of data. It is a measure of how much the data is spread out from the mean or average value. The standard deviation is an important tool in statistics that helps in analyzing and understanding the data. In this article, we will discuss the standard deviation formula and provide solved examples to help you understand how to calculate it.
Standard Deviation Formula
The formula for calculating the standard deviation is as follows:
σ = √(Σ (Xi – μ)^2 / N)
Where:
- σ is the standard deviation
- Σ is the sum of
- Xi is the individual value in the data set
- μ is the mean or average of the data set
- N is the total number of values in the data set
Let’s break down this formula into its components.
Step 1: Calculate the mean or average value of the data set.
The first step in calculating the standard deviation is to find the mean or average value of the data set. This is done by adding up all the values in the data set and then dividing the sum by the total number of values in the data set.
μ = (X1 + X2 + … + Xn) / n
Where:
- μ is the mean or average value
- X1, X2, … Xn are the individual values in the data set
- n is the total number of values in the data set
Step 2: Calculate the deviation of each value from the mean.
The next step is to calculate the deviation of each value in the data set from the mean. This is done by subtracting the mean from each individual value in the data set.
Xi – μ
Step 3: Square the deviations.
The next step is to square the deviations obtained in step 2. This is done to eliminate negative values and to give more weight to the larger deviations.
(Xi – μ)^2
Step 4: Calculate the sum of the squared deviations.
The next step is to add up all the squared deviations calculated in step 3.
Σ (Xi – μ)^2
Step 5: Calculate the variance.
The next step is to calculate the variance by dividing the sum of squared deviations obtained in step 4 by the total number of values in the data set.
Variance = Σ (Xi – μ)^2 / n
Step 6: Calculate the standard deviation.
The final step is to calculate the standard deviation by taking the square root of the variance obtained in step 5.
σ = √(Σ (Xi – μ)^2 / N)
Now that we have broken down the formula for calculating standard deviation, let’s look at some solved examples.
Solved Examples
Example 1:
Suppose we have a data set of 10 students’ scores on a math test: 65, 70, 75, 80, 85, 90, 95, 100, 105, and 110. Calculate the standard deviation of the scores.
Step 1: Calculate the mean or average value.
μ = (65 + 70 + 75 + 80 + 85 + 90 + 95 + 100 + 105 + 110) / 10
μ = 85
Step 2: Calculate the deviation of each value from the mean.
65 – 85 = -20
70 – 85 = -15
75 – 85 = -10
80 – 85 = -5
85 – 85 = 0
90 – 85 = 5
95 – 85 = 10
100 – 85 = 15
105 – 85 = 20
Step 3: Square the deviations.
(-20)^2 = 400
(-15)^2 = 225
(-10)^2 = 100
(-5)^2 = 25
0^2 = 0
5^2 = 25
10^2 = 100
15^2 = 225
20^2 = 400
Step 4: Calculate the sum of squared deviations.
Σ (Xi – μ)^2 = 400 + 225 + 100 + 25 + 0 + 25 + 100 + 225 + 400
Σ (Xi – μ)^2 = 1500
Step 5: Calculate the variance.
Variance = Σ (Xi – μ)^2 / n
Variance = 1500 / 10
Variance = 150
Step 6: Calculate the standard deviation.
σ = √(Σ (Xi – μ)^2 / N)
σ = √(150)
σ = 12.25
Therefore, the standard deviation of the scores is 12.25.
Example 2:
Suppose we have a data set of 8 employees’ salaries: $30,000, $35,000, $40,000, $45,000, $50,000, $55,000, $60,000, and $65,000. Calculate the standard deviation of the salaries.
Step 1: Calculate the mean or average value.
μ = ($30,000 + $35,000 + $40,000 + $45,000 + $50,000 + $55,000 + $60,000 + $65,000) / 8
μ = $47,500
Step 2: Calculate the deviation of each value from the mean.
$30,000 – $47,500 = -$17,500
$35,000 – $47,500 = -$12,500
$40,000 – $47,500 = -$7,500
$45,000 – $47,500 = -$2,500
$50,000 – $47,500 = $2,500
$55,000 – $47,500 = $7,500
$60,000 – $47,500 = $12,500
$65,000 – $47,500 = $17,500
Step 3: Square the deviations.
(-$17,500)^2 = $306,250,000
(-$12,500)^2 = $156,250,000
(-$7,500)^2 = $56,250,000
(-$2,500)^2 = $6,250,000
($2,500)^2 = $6,250,000
($7,500)^2 = $56,250,000
($12,500)^2 = $156,250,000
($17,500)^2 = $306,250,000
Step 4: Calculate the sum of squared deviations.
Σ (Xi – μ)^2 = $1,045,000,000
Step 5: Calculate the variance.
Variance = Σ (Xi – μ)^2 / n
Variance = $1,045,000,000 / 8
Variance = $130,625,000
Step 6: Calculate the standard deviation.
σ = √(Σ (Xi – μ)^2 / N)
σ = √($130,625,000)
σ = $11,425.78
Therefore, the standard deviation of the salaries is $11,425.78.
Standard deviation is a useful statistical tool that helps in analyzing and understanding the variability of data in a data set. It is a measure of how spread out the data is from the mean or average value. The higher the standard deviation, the more spread out the data is, while a lower standard deviation indicates that the data is clustered closer to the mean.
In this article, we have discussed the formula to calculate standard deviation along with two examples to illustrate the calculation process. It is important to note that standard deviation should not be used as the sole measure of variability in a data set, and other measures such as variance, range, and interquartile range should also be considered.
Standard deviation is used in various fields such as finance, science, engineering, and social sciences. In finance, it is used to analyze the risk and volatility of investments. In science and engineering, it is used to analyze the precision and accuracy of measurements. In social sciences, it is used to analyze the variability and dispersion of data in surveys and polls.
Understanding the concept and calculation of standard deviation is essential in statistical analysis and data interpretation. It helps in making informed decisions based on data and identifying trends and patterns in a data set. By calculating and interpreting standard deviation, we can gain insights into the variability and spread of data, which is crucial in many fields of study and research.
FAQs for Standard Deviation Formula
Q: What is standard deviation?
A: Standard deviation is a measure of the variability or spread of a set of data from the mean or average value.
Q: Why is standard deviation important?
A: Standard deviation is important because it provides a quantitative measure of how much the data in a set varies from the mean or average value. This can help in identifying patterns and trends in the data, and in making informed decisions based on the data.
Q: What is the formula for calculating standard deviation?
A: The formula for calculating standard deviation is: σ = √(Σ (Xi – μ)^2 / N), where σ is the standard deviation, Σ (Xi – μ)^2 is the sum of squared deviations, N is the total number of data points, and μ is the mean or average value.
Q: How is standard deviation used in finance?
A: In finance, standard deviation is used to analyze the risk and volatility of investments. It measures the amount of fluctuation or variability in the returns of an investment, and helps investors to make informed decisions about risk management and portfolio diversification.
Q: What are some other measures of variability that can be used alongside standard deviation?
A: Other measures of variability that can be used alongside standard deviation include variance, range, and interquartile range. These measures can provide additional insights into the spread and distribution of data in a data set.
Understanding standard deviation and how to calculate it is important in statistical analysis and data interpretation. It provides a measure of the variability or spread of a set of data from the mean or average value, and can be used in various fields such as finance, science, engineering, and social sciences. While standard deviation should not be used as the sole measure of variability, it is an important tool in identifying patterns and trends in the data, and in making informed decisions based on the data. By calculating and interpreting standard deviation, we can gain insights into the variability and spread of data, which is crucial in many fields of study and research.
TalentEdge offers various online certification courses for working professionals from top institutes, some of the courses are:
- Digital Marketing Courses
- Finance Courses
- Data Science Courses
- Supply Chain Management Courses
- Human Resource Management Courses
- Risk Management Courses
- IT & Technology Courses
- Business Management Courses
- Brand Management Courses
- Project Management Courses
- General Management Courses
- Entrepreneurship Courses
- Healthcare Courses
