Solved Problems: Standard Deviation Calculations for Algebra 1

📚 Understanding Standard Deviation

Standard deviation is a measure of how spread out numbers are in a data set. It tells you the average distance of each data point from the mean of the set. A low standard deviation indicates that the data points tend to be close to the mean, while a high standard deviation indicates that the data points are spread out over a wider range.

📜 History and Background

The concept of standard deviation was introduced by Karl Pearson in the late 19th century as part of his work on statistical analysis. It has since become a fundamental tool in various fields, including mathematics, statistics, finance, and science.

📌 Key Principles

• 🔢 Calculate the Mean: Find the average of all the numbers in the data set.
• ➖ Find the Deviations: Subtract the mean from each number in the data set.
• 🧮 Square the Deviations: Square each of the deviations obtained in the previous step.
• ➕ Sum the Squared Deviations: Add up all the squared deviations.
• ➗ Divide by (n-1): Divide the sum of the squared deviations by $n-1$, where $n$ is the number of data points (this gives you the variance). This is called Bessel's correction and provides an unbiased estimator of the population variance when working with a sample.
• ✅ Take the Square Root: Take the square root of the variance to get the standard deviation.

➗ Formula for Standard Deviation

The formula for standard deviation ($s$) is:

$s = \sqrt{\frac{\sum_{i=1}^{n}(x_i - \bar{x})^2}{n-1}}$

Where:

• 📈 $x_i$ represents each individual data point.
• 📊 $\bar{x}$ represents the mean of the data set.
• 𝑛 represents the number of data points in the sample.

📝 Solved Problems

Problem 1:

Find the standard deviation of the following data set: 4, 8, 6, 5, 3.

1. 💡 Calculate the Mean: $\frac{4+8+6+5+3}{5} = \frac{26}{5} = 5.2$
2. ➖ Find the Deviations: 4-5.2 = -1.2, 8-5.2 = 2.8, 6-5.2 = 0.8, 5-5.2 = -0.2, 3-5.2 = -2.2
3. 🧮 Square the Deviations: (-1.2)^2 = 1.44, (2.8)^2 = 7.84, (0.8)^2 = 0.64, (-0.2)^2 = 0.04, (-2.2)^2 = 4.84
4. ➕ Sum the Squared Deviations: 1.44 + 7.84 + 0.64 + 0.04 + 4.84 = 14.8
5. ➗ Divide by (n-1): $\frac{14.8}{5-1} = \frac{14.8}{4} = 3.7$
6. ✅ Take the Square Root: $\sqrt{3.7} \approx 1.92$

Therefore, the standard deviation is approximately 1.92.

Problem 2:

Calculate the standard deviation for the dataset: 10, 12, 15, 18, 20.

1. 💡 Calculate the Mean: $\frac{10+12+15+18+20}{5} = \frac{75}{5} = 15$
2. ➖ Find the Deviations: 10-15 = -5, 12-15 = -3, 15-15 = 0, 18-15 = 3, 20-15 = 5
3. 🧮 Square the Deviations: (-5)^2 = 25, (-3)^2 = 9, (0)^2 = 0, (3)^2 = 9, (5)^2 = 25
4. ➕ Sum the Squared Deviations: 25 + 9 + 0 + 9 + 25 = 68
5. ➗ Divide by (n-1): $\frac{68}{5-1} = \frac{68}{4} = 17$
6. ✅ Take the Square Root: $\sqrt{17} \approx 4.12$

The standard deviation is approximately 4.12.

Problem 3:

Determine the standard deviation of: 2, 4, 6, 8, 10.

1. 💡 Calculate the Mean: $\frac{2+4+6+8+10}{5} = \frac{30}{5} = 6$
2. ➖ Find the Deviations: 2-6 = -4, 4-6 = -2, 6-6 = 0, 8-6 = 2, 10-6 = 4
3. 🧮 Square the Deviations: (-4)^2 = 16, (-2)^2 = 4, (0)^2 = 0, (2)^2 = 4, (4)^2 = 16
4. ➕ Sum the Squared Deviations: 16 + 4 + 0 + 4 + 16 = 40
5. ➗ Divide by (n-1): $\frac{40}{5-1} = \frac{40}{4} = 10$
6. ✅ Take the Square Root: $\sqrt{10} \approx 3.16$

The standard deviation is approximately 3.16.

Problem 4:

Find the standard deviation for: 1, 2, 3, 4, 5.

1. 💡 Calculate the Mean: $\frac{1+2+3+4+5}{5} = \frac{15}{5} = 3$
2. ➖ Find the Deviations: 1-3 = -2, 2-3 = -1, 3-3 = 0, 4-3 = 1, 5-3 = 2
3. 🧮 Square the Deviations: (-2)^2 = 4, (-1)^2 = 1, (0)^2 = 0, (1)^2 = 1, (2)^2 = 4
4. ➕ Sum the Squared Deviations: 4 + 1 + 0 + 1 + 4 = 10
5. ➗ Divide by (n-1): $\frac{10}{5-1} = \frac{10}{4} = 2.5$
6. ✅ Take the Square Root: $\sqrt{2.5} \approx 1.58$

The standard deviation is approximately 1.58.

Problem 5:

What is the standard deviation of the data set: 7, 9, 11, 13, 15?

1. 💡 Calculate the Mean: $\frac{7+9+11+13+15}{5} = \frac{55}{5} = 11$
2. ➖ Find the Deviations: 7-11 = -4, 9-11 = -2, 11-11 = 0, 13-11 = 2, 15-11 = 4
3. 🧮 Square the Deviations: (-4)^2 = 16, (-2)^2 = 4, (0)^2 = 0, (2)^2 = 4, (4)^2 = 16
4. ➕ Sum the Squared Deviations: 16 + 4 + 0 + 4 + 16 = 40
5. ➗ Divide by (n-1): $\frac{40}{5-1} = \frac{40}{4} = 10$
6. ✅ Take the Square Root: $\sqrt{10} \approx 3.16$

The standard deviation is approximately 3.16.

Problem 6:

Calculate the standard deviation of: 25, 30, 35, 40, 45.

1. 💡 Calculate the Mean: $\frac{25+30+35+40+45}{5} = \frac{175}{5} = 35$
2. ➖ Find the Deviations: 25-35 = -10, 30-35 = -5, 35-35 = 0, 40-35 = 5, 45-35 = 10
3. 🧮 Square the Deviations: (-10)^2 = 100, (-5)^2 = 25, (0)^2 = 0, (5)^2 = 25, (10)^2 = 100
4. ➕ Sum the Squared Deviations: 100 + 25 + 0 + 25 + 100 = 250
5. ➗ Divide by (n-1): $\frac{250}{5-1} = \frac{250}{4} = 62.5$
6. ✅ Take the Square Root: $\sqrt{62.5} \approx 7.91$

The standard deviation is approximately 7.91.

Problem 7:

Find the standard deviation of the following set: 100, 102, 104, 106, 108.

1. 💡 Calculate the Mean: $\frac{100+102+104+106+108}{5} = \frac{520}{5} = 104$
2. ➖ Find the Deviations: 100-104 = -4, 102-104 = -2, 104-104 = 0, 106-104 = 2, 108-104 = 4
3. 🧮 Square the Deviations: (-4)^2 = 16, (-2)^2 = 4, (0)^2 = 0, (2)^2 = 4, (4)^2 = 16
4. ➕ Sum the Squared Deviations: 16 + 4 + 0 + 4 + 16 = 40
5. ➗ Divide by (n-1): $\frac{40}{5-1} = \frac{40}{4} = 10$
6. ✅ Take the Square Root: $\sqrt{10} \approx 3.16$

The standard deviation is approximately 3.16.

🌍 Real-world Applications

• 🌡️ Science: Analyzing experimental data to determine the reliability of results.
• 🏦 Finance: Assessing the risk associated with investments.
• 📊 Quality Control: Monitoring the consistency of manufacturing processes.
• 🩺 Healthcare: Evaluating the effectiveness of medical treatments.

🎉 Conclusion

Understanding standard deviation is crucial for analyzing data and making informed decisions. By following the steps and practicing with examples, you can master this essential concept in Algebra 1!