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 data set has a larger standard deviation, we'll first calculate the standard deviation for each set. The standard deviation measures how spread out the numbers in a data set are from the mean.

### Step 1: Calculate the mean (average) of each set.

#### Set A: 9 g, 11 g, 5 g, 7 g, 13 g
\[
\text{Mean of Set A} = \frac{9 + 11 + 5 + 7 + 13}{5} = \frac{45}{5} = 9 \text{ g}
\]

#### Set B: 11 g, 10 g, 19 g, 19 g, 11 g
\[
\text{Mean of Set B} = \frac{11 + 10 + 19 + 19 + 11}{5} = \frac{70}{5} = 14 \text{ g}
\]

### Step 2: Calculate the variance for each set.

Variance is the average of the squared differences from the mean.

#### Variance of Set A:
\[
\text{Variance of Set A} = \frac{(9-9)^2 + (11-9)^2 + (5-9)^2 + (7-9)^2 + (13-9)^2}{5}
\]
\[
= \frac{0^2 + 2^2 + (-4)^2 + (-2)^2 + 4^2}{5}
= \frac{0 + 4 + 16 + 4 + 16}{5} = \frac{40}{5} = 8
\]

#### Variance of Set B:
\[
\text{Variance of Set B} = \frac{(11-14)^2 + (10-14)^2 + (19-14)^2 + (19-14)^2 + (11-14)^2}{5}
\]
\[
= \frac{(-3)^2 + (-4)^2 + 5^2 + 5^2 + (-3)^2}{5}
= \frac{9 + 16 + 25 + 25 + 9}{5} = \frac{84}{5} = 16.8
\]

### Step 3: Calculate the standard deviation for each set.

Standard deviation is the square root of the variance.

#### Standard Deviation of Set A:
\[
\text{Standard Deviation of Set A} = \sqrt{8} \approx 2.83
\]

#### Standard Deviation of Set B:
\[
\text{Standard Deviation of Set B} = \sqrt{16.8} \approx 4.10
\]

### Conclusion
Set B has a larger standard deviation (\(\approx 4.10\)) compared to Set A (\(\approx 2.83\)). This means that the data in Set B is more spread out from the mean than the data in Set A.

Would you like any more details or have any questions about this?

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Here are some related questions you might find interesting:

1. How does the standard deviation relate to the spread of data?
2. What is the difference between variance and standard deviation?
3. How would the standard deviation change if you added a constant to all data points?
4. What are some real-world applications of standard deviation?
5. How does the mean impact the standard deviation?
6. Can you have a negative standard deviation? Why or why not?
7. How do outliers affect the standard deviation?
8. What is the relationship between the standard deviation and the normal distribution?

**Tip:** The standard deviation is sensitive to outliers because it involves squaring the deviations from the mean. Even a single extreme value can significantly increase the standard deviation.

Learn how to compare standard deviations between two data sets, Set A and Set B. Calculate the mean, variance, and standard deviation for each set. Understand the significance of standard deviation in measuring data spread. Suitable for students in Grades 9-12.

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