Mean, Variance and Standard Deviation

Mean, Variance and Standard Deviation are fundamental concepts in statistics and engineering mathematics, essential for analyzing and interpreting data. These measures provide insights into data's central tendency, dispersion, and spread, which are crucial for making informed decisions in various engineering fields.

Here, we will discuss the definitions, formulas, and applications of mean, variance, and standard deviation in engineering, along with solved examples.

Mean, variance, and standard deviation are all fundamental concepts in statistics, and they help to describe the distribution and spread of data.

Mean

The mean also known as the average, is a measure of the central tendency of a dataset. It is calculated by summing up all the values in the dataset and dividing them by the number of values. It is denoted by the symbol μ.

Mean Formula

For a dataset with n values x₁, x₂, x₃, ......., x_n the mean μ is given by:

μ = \frac{1}{n} \sum_{i=1}^{n}x_i

Example: Find the mean (average) of the following dataset: {4, 8, 6, 5, 3, 7}

μ = 4 + 8 + 6 + 5 + 3 + 7 / 6
= 33/6
= 5.5

Variance

Variance measures the dispersion of a dataset, indicating how much the values differ from the mean. It is the average of the squared differences from the mean.

Variance Formula

For a dataset with n values x₁, x₂, x₃, ......., x_n the mean σ² is given by:

σ² = \frac{1}{n} \sum_{i=1}^{n} (x_i - \mu)^2

Example: Find the variance of the following dataset {4, 8, 6, 5, 3, 7} with mean = 5.5.

σ² = (4 - 5.5)² + (8 - 5.5)² + (6−5.5)² + (5−5.5)² + (3−5.5)² + (7- 5.5)² / 6
σ² = 17.5/6 = 2.92

Standard Deviation

Standard deviation is the square root of the variance, providing a measure of the spread of the dataset in the same units as the data.

Standard Deviation Formula

For a dataset with n values x₁, x₂, x₃, ......., x_n the mean σ is given by:

σ = \sqrt σ2 = \sqrt {\frac{1}{n} \sum_{i=1}^{n} (x_i - \mu)^2}

Example: Find the standard deviation of the following dataset {4, 8, 6, 5, 3, 7}, with variance σ² = 2.92.

To find the standard deviation of the dataset {4, 8, 6, 5, 3, 7} with a given variance of σ² = 2.92, we use the following formula:
σ = √σ²​

Given that the variance σ² = 2.92, we can calculate the standard deviation σ:
σ = 2.92 = 1.71

So, the standard deviation is 1.71.

Relationship between Mean, Variance, and Standard Deviation

The mean is the average of all numbers in a dataset and shows the center of the data. Once we have the mean, we can find the variance, which tells us how spread out the numbers are from the mean.

To calculate variance, we look at how far each number is from the mean, square those differences, and then find their average. Since variance is in squared units, we take the square root of it to get the standard deviation, which tells us the spread in the same units as the original data. So, the mean helps us find the variance, and the variance helps us find the standard deviation.

Example: The dataset below represents the scores of 5 students in a quiz: {5, 7, 9, 11, 13}

1. Calculate the mean of the dataset.
2. Use the mean to calculate the variance.
3. Find the standard deviation from the variance.

Solution:

Step 1: Calculate the Mean
Mean = 5 + 7 + 9 + 11 + 13 /5
= 45 / 5 = 9.

Step 2: Calculate the Variance

Subtract the mean from each number, square the result, and find the average:
(5 - 9)² + (7 - 9)² + (9 - 9)² + (11 - 9)² + (13 - 9)
= 16 + 4 + 0 + 4 + 16
= 40
Then, divide by 5: 40 / 5 = 8.

Step 3: Calculate the Standard Deviation

• Take the square root of the variance: √8 = 2.83.
• So, the standard deviation is 2.83.

Solved Question on Mean, Variance, and Standard Deviation

Question 1. Consider the data set: 4, 8, 6, 5, 3, 9.

Solution:

Step 1: Calculate the Mean

• Add up all the numbers: 4 + 8 + 6 + 5 + 3 + 9 = 35.
• Divide by the number of values (6): 35 / 6 = 5.83.
• So, the mean is 5.83.

Step 2: Calculate the Variance

Subtract the mean from each number, square the result, and find the average:

• (4 - 5.83)² + (8 - 5.83)² + (6 - 5.83)² + (5 - 5.83)² + (3 - 5.83)² + (9 - 5.83)²
• = 3.35 + 4.68 + 0.03 + 0.69 + 8.00 + 10.03 = 26.78.
• Then, divide by 6: 26.78 / 6 = 4.80.
• So, the variance is 4.80.

Step 3: Calculate the Standard Deviation

• Take the square root of the variance: √4.80 = 2.19.
• So, the standard deviation is 2.19.

Question 2. Consider the dataset: 2, 4, 6, 8, 10.

Solution:

Step 1: Calculate the Mean

• Add up all the numbers: 2 + 4 + 6 + 8 + 10 = 30.
• Divide by the number of values (5): 30 / 5 = 6.
• So, the mean is 6.

Step 2: Calculate the Variance

• Subtract the mean from each number, square the result, and find the average:(2−6)2+(4−6)2+(6−6)2+(8−6)2+(10−6)2=(−4)2+(−2)2+02+22+42=16+4+0+4+16=40.
• Then, divide by the number of values (5): 40 / 5 = 8.
• So, the variance is 8.

Step 3: Calculate the Standard Deviation

• Take the square root of the variance: √8 ≈ 2.83.
• So, the standard deviation is approximately 2.83.

Question 3. Consider the dataset: 3, 7, 7, 19, 24.

Solution:

Step 1: Calculate the Mean

• Add up all the numbers: 3 + 7 + 7 + 19 + 24 = 60.
• Divide by the number of values (5): 60 / 5 = 12.
• So, the mean is 12.

Step 2: Calculate the Variance

• Subtract the mean from each number, square the result, and find the average:(3−12)2+(7−12)2+(7−12)2+(19−12)2+(24−12)2=(−9)2+(−5)2+(−5)2+72+122=81+25+25+49+144=324.
• Then, divide by the number of values (5): 324 / 5 = 64.8.
• So, the variance is 64.8.

Step 3: Calculate the Standard Deviation

• Take the square root of the variance: √64.8 ≈ 8.05.
• So, the standard deviation is approximately 8.05.

Question 4. Consider the dataset: 5, 10, 15, 20, 25.

Step 1: Calculate the Mean

• Add up all the numbers: 5 + 10 + 15 + 20 + 25 = 75.
• Divide by the number of values (5): 75 / 5 = 15.
• So, the mean is 15.

Step 2: Calculate the Variance

• Subtract the mean from each number, square the result, and find the average:(5−15)2+(10−15)2+(15−15)2+(20−15)2+(25−15)2=(−10)2+(−5)2+02+52+102=100+25+0+25+100=250.
• Then, divide by the number of values (5): 250 / 5 = 50.
• So, the variance is 50.

Step 3: Calculate the Standard Deviation

• Take the square root of the variance: √50 ≈ 7.07.
• So, the standard deviation is approximately 7.07.

Question 5. Consider the dataset: 11, 13, 15, 17, 19.

Solution:

Step 1: Calculate the Mean

• Add up all the numbers: 11 + 13 + 15 + 17 + 19 = 75.
• Divide by the number of values (5): 75 / 5 = 15.
• So, the mean is 15.

Step 2: Calculate the Variance

• Subtract the mean from each number, square the result, and find the average:(11−15)2+(13−15)2+(15−15)2+(17−15)2+(19−15)2=(−4)2+(−2)2+02+22+42=16+4+0+4+16=40.
• Then, divide by the number of values (5): 40 / 5 = 8.
• So, the variance is 8.

Step 3: Calculate the Standard Deviation

• Take the square root of the variance: √8 ≈ 2.83.
• So, the standard deviation is approximately 2.83.

Unsolved Question on Mean, Variance, and Standard Deviation

Question 1: Find the mean, variance, and standard deviation for the dataset: 10, 15, 20, 25, 30.

Question 2: Calculate the mean, variance, and standard deviation for: 5, 10, 15, 20, 25.

Question 3: Determine the mean, variance, and standard deviation for: 4, 6, 8, 10, 12, 14.

Question 4: Find the mean, variance, and standard deviation for the dataset: 1, 4, 9, 16, 25.

Question 5: Calculate the mean, variance, and standard deviation for: 3, 6, 9, 12, 15.

Question 6: Determine the mean, variance, and standard deviation for: 8, 16, 24, 32, 40.

Question 4: Find the mean, variance, and standard deviation for the dataset: 2, 4, 6, 8, 10, 12.

Question 8: Calculate the mean, variance, and standard deviation for: 11, 22, 33, 44, 55.

Question 9: Determine the mean, variance, and standard deviation for: 7, 14, 21, 28, 35.

Question 10: Find the mean, variance, and standard deviation for the dataset: 13, 26, 39, 52, 65.

Answer Key: Mean, Variance, and Standard deviation

Ans 1: 20, 50, 7.07
Ans 2: 15, 50, 7.07
Ans 3: 9, 14, 3.74
Ans 4: 11, 92.8, 9.63
Ans 5: 9, 18, 4.24
Ans 6: 24, 128, 11.31
Ans 7: 7, 14, 3.74
Ans 8: 33, 308, 17.55
Ans 9: 21, 98, 9.90
Ans 10: 39, 338, 18.38

Related Articles:

• Variance and Standard Deviation
• Real Life Applications of Standard Deviation
• Difference Between Variance and Standard Deviation
• Probability Theory
• Normal Distribution

Applications in Engineering

• In manufacturing and quality control, these measures help monitor and maintain product quality by analyzing variations in production processes.
• In signal processing, mean, variance, and standard deviation are used to analyze noise and signal strength, helping to improve the accuracy of signal transmission and reception.
• In reliability engineering, these measures are used to predict the lifespan and failure rates of components and systems, aiding in the design of more reliable products.
• In financial engineering, mean, variance, and standard deviation are used to analyze investment risks and returns, helping to make better investment decisions.
• In civil engineering, these statistical measures are used to analyze data from material tests, environmental studies, and structural performance, ensuring safety and compliance with standards.

Summary

Mean, variance, and standard deviation are key statistical measures that provide insights into the central tendency, dispersion, and spread of a dataset. These concepts are crucial in various fields, including engineering, finance, and data analysis, helping to understand and interpret data effectively.