Example 5

Consider the two data sets of readings in one of your laboratory activities, Which of the following sets has a larger standard deviation?

Set A. 9 g. 11 g. 5 g. 7 g and 13 g

Set B 11 g. 10 g 19 g. 19 g, and 11 g

To determine which set has a larger standard deviation, we need to calculate the standard deviation for both Set A and Set B. Standard deviation is a measure of the amount of variation or dispersion in a set of values.

### Steps to Calculate Standard Deviation:
1. **Find the mean (average) of each set.**
2. **Calculate the squared differences from the mean for each data point.**
3. **Find the mean of these squared differences (this is called the variance).**
4. **Take the square root of the variance to obtain the standard deviation.**

Let's compute these step by step.

### Set A: 9 g, 11 g, 5 g, 7 g, 13 g
1. **Mean of Set A:**
   \[
   \text{Mean} = \frac{9 + 11 + 5 + 7 + 13}{5} = \frac{45}{5} = 9 \, \text{g}
   \]
   
2. **Squared differences from the mean:**
   \[
   (9-9)^2 = 0^2 = 0
   \]
   \[
   (11-9)^2 = 2^2 = 4
   \]
   \[
   (5-9)^2 = (-4)^2 = 16
   \]
   \[
   (7-9)^2 = (-2)^2 = 4
   \]
   \[
   (13-9)^2 = 4^2 = 16
   \]

3. **Variance of Set A:**
   \[
   \text{Variance} = \frac{0 + 4 + 16 + 4 + 16}{5} = \frac{40}{5} = 8
   \]

4. **Standard Deviation of Set A:**
   \[
   \text{Standard Deviation} = \sqrt{8} \approx 2.83 \, \text{g}
   \]

### Set B: 11 g, 10 g, 19 g, 19 g, 11 g
1. **Mean of Set B:**
   \[
   \text{Mean} = \frac{11 + 10 + 19 + 19 + 11}{5} = \frac{70}{5} = 14 \, \text{g}
   \]
   
2. **Squared differences from the mean:**
   \[
   (11-14)^2 = (-3)^2 = 9
   \]
   \[
   (10-14)^2 = (-4)^2 = 16
   \]
   \[
   (19-14)^2 = 5^2 = 25
   \]
   \[
   (19-14)^2 = 5^2 = 25
   \]
   \[
   (11-14)^2 = (-3)^2 = 9
   \]

3. **Variance of Set B:**
   \[
   \text{Variance} = \frac{9 + 16 + 25 + 25 + 9}{5} = \frac{84}{5} = 16.8
   \]

4. **Standard Deviation of Set B:**
   \[
   \text{Standard Deviation} = \sqrt{16.8} \approx 4.10 \, \text{g}
   \]

### Conclusion:
Set B has a larger standard deviation (approximately 4.10 g) compared to Set A (approximately 2.83 g). This indicates that the values in Set B are more spread out from the mean than the values in Set A.

Would you like more details or have any questions?

### Related Questions:
1. How is variance related to standard deviation?
2. What does a higher standard deviation imply about a data set?
3. How can the standard deviation be used in statistical analysis?
4. What are some real-world applications of standard deviation?
5. How would the standard deviation change if all values in a set are increased by a constant?
6. How do outliers affect the standard deviation?
7. What is the difference between sample standard deviation and population standard deviation?
8. Why is the square root used in calculating standard deviation?

**Tip:** When comparing the variability of two data sets, the set with the larger standard deviation is more dispersed around the mean.

Explore the comparison of standard deviation between Set A and Set B with detailed calculations. Set A includes readings of 9 g, 11 g, 5 g, 7 g, and 13 g, while Set B includes readings of 11 g, 10 g, 19 g, 19 g, and 11 g. Understand how standard deviation measures the dispersion of data points from the mean and how it applies in statistical analysis. Suitable for Grades 10-12.

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