Consider The Following Data:3, 15, 9, 7, 6, 8Step 2 Of 3: Calculate The Value Of The Sample Standard Deviation. Round Your Answer To One Decimal Place.
Understanding Sample Standard Deviation
Sample standard deviation is a measure of the amount of variation or dispersion of a set of values. It is an important concept in statistics and is used to describe the spread of a dataset. In this article, we will walk you through the process of calculating the sample standard deviation using a given dataset.
Step 1: Calculate the Sample Mean
Before we can calculate the sample standard deviation, we need to calculate the sample mean. The sample mean is the average value of the dataset. To calculate the sample mean, we add up all the values in the dataset and divide by the number of values.
Dataset: 3, 15, 9, 7, 6, 8
Step 1.1: Add up all the values in the dataset
3 + 15 = 18 18 + 9 = 27 27 + 7 = 34 34 + 6 = 40 40 + 8 = 48
Step 1.2: Divide the sum by the number of values
There are 6 values in the dataset. To calculate the sample mean, we divide the sum (48) by the number of values (6).
Sample mean = 48 / 6 = 8
Step 2: Calculate the Deviations from the Sample Mean
Now that we have the sample mean, we can calculate the deviations from the sample mean. The deviation is the difference between each value in the dataset and the sample mean.
Step 2.1: Calculate the deviations
| Value | Deviation |
|---|---|
| 3 | 3 - 8 = -5 |
| 15 | 15 - 8 = 7 |
| 9 | 9 - 8 = 1 |
| 7 | 7 - 8 = -1 |
| 6 | 6 - 8 = -2 |
| 8 | 8 - 8 = 0 |
Step 3: Calculate the Squared Deviations
Next, we need to calculate the squared deviations. This is done by squaring each deviation.
Step 3.1: Square the deviations
| Value | Deviation | Squared Deviation |
|---|---|---|
| 3 | -5 | (-5)^2 = 25 |
| 15 | 7 | (7)^2 = 49 |
| 9 | 1 | (1)^2 = 1 |
| 7 | -1 | (-1)^2 = 1 |
| 6 | -2 | (-2)^2 = 4 |
| 8 | 0 | (0)^2 = 0 |
Step 4: Calculate the Sum of the Squared Deviations
Now, we need to calculate the sum of the squared deviations.
Step 4.1: Add up the squared deviations
25 + 49 = 74 74 + 1 = 75 75 + 1 = 76 76 + 4 = 80 80 + 0 = 80
Step 5: Calculate the Sample Variance
The sample variance is the average of the squared deviations. To calculate the sample variance, we divide the sum of the squared deviations by the number of values minus one.
Step 5.1: Divide the sum by the number of values minus one
There are 6 values in the dataset. To calculate the sample variance, we divide the sum (80) by the number of values minus one (5).
Sample variance = 80 / 5 = 16
Step 6: Calculate the Sample Standard Deviation
Finally, we can calculate the sample standard deviation. The sample standard deviation is the square root of the sample variance.
Step 6.1: Take the square root of the sample variance
Sample standard deviation = √16 = 4.0
Conclusion
Q: What is the sample standard deviation?
A: The sample standard deviation is a measure of the amount of variation or dispersion of a set of values. It is an important concept in statistics and is used to describe the spread of a dataset.
Q: How is the sample standard deviation calculated?
A: The sample standard deviation is calculated by following these steps:
- Calculate the sample mean.
- Calculate the deviations from the sample mean.
- Square the deviations.
- Calculate the sum of the squared deviations.
- Calculate the sample variance by dividing the sum of the squared deviations by the number of values minus one.
- Take the square root of the sample variance to get the sample standard deviation.
Q: What is the difference between the sample standard deviation and the population standard deviation?
A: The sample standard deviation is used when the dataset is a sample of the population, while the population standard deviation is used when the dataset is the entire population. The formula for the sample standard deviation is slightly different from the formula for the population standard deviation.
Q: Why is the sample standard deviation important?
A: The sample standard deviation is important because it helps to describe the spread of a dataset. It is used in many statistical analyses, such as hypothesis testing and confidence intervals.
Q: How is the sample standard deviation used in real-life situations?
A: The sample standard deviation is used in many real-life situations, such as:
- Describing the spread of a dataset in a report or presentation.
- Identifying outliers in a dataset.
- Calculating confidence intervals for a population parameter.
- Performing hypothesis testing to determine if there is a significant difference between two or more groups.
Q: What are some common mistakes to avoid when calculating the sample standard deviation?
A: Some common mistakes to avoid when calculating the sample standard deviation include:
- Not using the correct formula for the sample standard deviation.
- Not squaring the deviations correctly.
- Not dividing the sum of the squared deviations by the number of values minus one.
- Not taking the square root of the sample variance.
Q: How can I calculate the sample standard deviation using a calculator or computer software?
A: You can calculate the sample standard deviation using a calculator or computer software by following these steps:
- Enter the dataset into the calculator or software.
- Use the built-in function to calculate the sample standard deviation.
- Follow the prompts to enter the necessary information, such as the number of values and the dataset.
Q: What is the relationship between the sample standard deviation and the sample mean?
A: The sample standard deviation and the sample mean are related in that the sample standard deviation is a measure of the spread of the dataset, while the sample mean is a measure of the central tendency of the dataset. The sample standard deviation can be used to calculate the sample mean, and vice versa.
Q: Can I use the sample standard deviation to compare two or more datasets?
A: Yes, you can use the sample standard deviation to compare two or more datasets. By comparing the sample standard deviations of two or more datasets, you can determine if there is a significant difference in the spread of the datasets.