Use The Standard Deviation To Compare The Variability Of The Datasets. Round To The Nearest Hundredth.Dataset A: 4, 5, 6, 12, 13 Dataset B: 4, 5, 7, 9, 10(1 Point)A. The Standard Deviation Of Dataset A Is 3.74. The Standard Deviation Of Dataset B Is
Introduction
When working with datasets, it's essential to understand the variability or dispersion of the data points. This can be achieved by calculating the standard deviation, which is a measure of the amount of variation or dispersion of a set of values. In this article, we will explore how to use the standard deviation to compare the variability of two different datasets.
Calculating Standard Deviation
The standard deviation is calculated using the following formula:
σ = √[(Σ(xi - μ)²) / (n - 1)]
where:
- σ is the standard deviation
- xi is each individual data point
- μ is the mean of the dataset
- n is the number of data points
- Σ denotes the sum of the values
Dataset A
Let's start by calculating the standard deviation of Dataset A: 4, 5, 6, 12, 13.
Step 1: Calculate the mean
To calculate the mean, we add up all the data points and divide by the number of data points.
μ = (4 + 5 + 6 + 12 + 13) / 5 μ = 40 / 5 μ = 8
Step 2: Calculate the deviations from the mean
Next, we calculate the deviations from the mean for each data point.
| Data Point | Deviation from Mean |
|---|---|
| 4 | 4 - 8 = -4 |
| 5 | 5 - 8 = -3 |
| 6 | 6 - 8 = -2 |
| 12 | 12 - 8 = 4 |
| 13 | 13 - 8 = 5 |
Step 3: Calculate the squared deviations
Now, we square each deviation from the mean.
| Data Point | Squared Deviation |
|---|---|
| 4 | (-4)² = 16 |
| 5 | (-3)² = 9 |
| 6 | (-2)² = 4 |
| 12 | 4² = 16 |
| 13 | 5² = 25 |
Step 4: Calculate the sum of the squared deviations
Next, we add up the squared deviations.
Σ(xi - μ)² = 16 + 9 + 4 + 16 + 25 Σ(xi - μ)² = 70
Step 5: Calculate the standard deviation
Finally, we calculate the standard deviation using the formula.
σ = √[(Σ(xi - μ)²) / (n - 1)] σ = √[70 / (5 - 1)] σ = √[70 / 4] σ = √17.5 σ = 4.18
Dataset B
Now, let's calculate the standard deviation of Dataset B: 4, 5, 7, 9, 10.
Step 1: Calculate the mean
To calculate the mean, we add up all the data points and divide by the number of data points.
μ = (4 + 5 + 7 + 9 + 10) / 5 μ = 35 / 5 μ = 7
Step 2: Calculate the deviations from the mean
Next, we calculate the deviations from the mean for each data point.
| Data Point | Deviation from Mean |
|---|---|
| 4 | 4 - 7 = -3 |
| 5 | 5 - 7 = -2 |
| 7 | 7 - 7 = 0 |
| 9 | 9 - 7 = 2 |
| 10 | 10 - 7 = 3 |
Step 3: Calculate the squared deviations
Now, we square each deviation from the mean.
| Data Point | Squared Deviation |
|---|---|
| 4 | (-3)² = 9 |
| 5 | (-2)² = 4 |
| 7 | 0² = 0 |
| 9 | 2² = 4 |
| 10 | 3² = 9 |
Step 4: Calculate the sum of the squared deviations
Next, we add up the squared deviations.
Σ(xi - μ)² = 9 + 4 + 0 + 4 + 9 Σ(xi - μ)² = 26
Step 5: Calculate the standard deviation
Finally, we calculate the standard deviation using the formula.
σ = √[(Σ(xi - μ)²) / (n - 1)] σ = √[26 / (5 - 1)] σ = √[26 / 4] σ = √6.5 σ = 2.55
Comparing the Variability of the Datasets
Now that we have calculated the standard deviations of both datasets, we can compare their variability.
Dataset A has a standard deviation of 4.18, while Dataset B has a standard deviation of 2.55. This means that Dataset A has a higher variability than Dataset B.
Conclusion
In this article, we have learned how to use the standard deviation to compare the variability of two different datasets. We have calculated the standard deviations of Dataset A and Dataset B and compared their variability. The results show that Dataset A has a higher variability than Dataset B.
Rounding to the Nearest Hundredth
Finally, we round the standard deviations to the nearest hundredth.
Dataset A: 4.18 → 4.18 Dataset B: 2.55 → 2.55
Answer
Q: What is standard deviation?
A: Standard deviation is a measure of the amount of variation or dispersion of a set of values. It represents how spread out the values are from the mean.
Q: How is standard deviation calculated?
A: Standard deviation is calculated using the following formula:
σ = √[(Σ(xi - μ)²) / (n - 1)]
where:
- σ is the standard deviation
- xi is each individual data point
- μ is the mean of the dataset
- n is the number of data points
- Σ denotes the sum of the values
Q: What is the difference between standard deviation and variance?
A: Standard deviation is the square root of variance. Variance is the average of the squared differences from the mean, while standard deviation is the square root of that value.
Q: Why is standard deviation important?
A: Standard deviation is important because it helps to understand the variability of a dataset. It can be used to compare the spread of different datasets and to identify outliers.
Q: How do I interpret the standard deviation?
A: The standard deviation can be interpreted as follows:
- A small standard deviation indicates that the data points are close to the mean.
- A large standard deviation indicates that the data points are spread out from the mean.
Q: Can standard deviation be negative?
A: No, standard deviation cannot be negative. The square root of a negative number is not a real number, so standard deviation is always a positive value.
Q: Can standard deviation be zero?
A: Yes, standard deviation can be zero. This occurs when all the data points are equal to the mean.
Q: How do I calculate the standard deviation of a sample?
A: To calculate the standard deviation of a sample, you use the following formula:
s = √[(Σ(xi - x̄)²) / (n - 1)]
where:
- s is the sample standard deviation
- xi is each individual data point
- x̄ is the sample mean
- n is the number of data points
- Σ denotes the sum of the values
Q: How do I calculate the standard deviation of a population?
A: To calculate the standard deviation of a population, you use the following formula:
σ = √[(Σ(xi - μ)²) / N]
where:
- σ is the population standard deviation
- xi is each individual data point
- μ is the population mean
- N is the number of data points
- Σ denotes the sum of the values
Q: What is the difference between sample standard deviation and population standard deviation?
A: Sample standard deviation is used when you are working with a sample of data, while population standard deviation is used when you are working with the entire population of data.
Q: Can I use standard deviation to compare the variability of different datasets?
A: Yes, you can use standard deviation to compare the variability of different datasets. By comparing the standard deviations of two or more datasets, you can determine which dataset has the most variability.
Q: How do I round the standard deviation to the nearest hundredth?
A: To round the standard deviation to the nearest hundredth, you simply round the calculated value to two decimal places.
Conclusion
In this article, we have answered some frequently asked questions about standard deviation. We have covered topics such as how to calculate standard deviation, how to interpret standard deviation, and how to compare the variability of different datasets using standard deviation.