Welcome to Westonci.ca, your go-to destination for finding answers to all your questions. Join our expert community today! Experience the ease of finding quick and accurate answers to your questions from professionals on our platform. Get immediate and reliable solutions to your questions from a community of experienced professionals on our platform.
Sagot :
To determine which formula is used to calculate the standard deviation of sample data, let's analyze each given option:
1. [tex]\( s = \sqrt{\frac{(x_1 - \bar{x})^2 + (x_2 - \bar{x})^2 + \ldots + (x_n - \bar{x})^2}{n - 1}} \)[/tex]
2. [tex]\( a^2 = \frac{(x_1 - \mu)^2 + (x_2 - \mu)^2 + \ldots + (x_N - \mu)^2}{N} \)[/tex]
3. [tex]\( \sigma = \sqrt{\frac{(x_1 - \mu)^2 + (x_2 - \mu)^2 + \ldots + (x_N - \mu)^2}{N}} \)[/tex]
4. [tex]\( s = \frac{(x_1 - \bar{x})^2 + (x_2 - \bar{x})^2 + \ldots + (x_n - \bar{x})^2}{n - 1} \)[/tex]
### Definitions & Symbols:
- [tex]\( s \)[/tex] is the standard deviation of a sample.
- [tex]\( \sigma \)[/tex] is the standard deviation of a population.
- [tex]\( \bar{x} \)[/tex] is the sample mean.
- [tex]\( \mu \)[/tex] is the population mean.
- [tex]\( n \)[/tex] is the sample size.
- [tex]\( N \)[/tex] is the population size.
### Explanation of Each Option:
1. Option 1:
[tex]\[ s = \sqrt{\frac{(x_1 - \bar{x})^2 + (x_2 - \bar{x})^2 + \ldots + (x_n - \bar{x})^2}{n - 1}} \][/tex]
This is the formula for the standard deviation of a sample. The divisor is [tex]\( n - 1 \)[/tex] which is known as Bessel's correction and it is used to correct the bias in the estimation of the population variance and standard deviation when sample data is used.
2. Option 2:
[tex]\[ a^2 = \frac{(x_1 - \mu)^2 + (x_2 - \mu)^2 + \ldots + (x_N - \mu)^2}{N} \][/tex]
This formula computes the population variance ([tex]\( \sigma^2 \)[/tex]), not the standard deviation, and uses the population mean ([tex]\( \mu \)[/tex]) and population size ([tex]\( N \)[/tex]).
3. Option 3:
[tex]\[ \sigma = \sqrt{\frac{(x_1 - \mu)^2 + (x_2 - \mu)^2 + \ldots + (x_N - \mu)^2}{N}} \][/tex]
This is the formula for the population standard deviation. It uses the population mean ([tex]\( \mu \)[/tex]) and population size ([tex]\( N \)[/tex]).
4. Option 4:
[tex]\[ s = \frac{(x_1 - \bar{x})^2 + (x_2 - \bar{x})^2 + \ldots + (x_n - \bar{x})^2}{n - 1} \][/tex]
This is not a standard deviation formula, but rather an expression for the sample variance without taking the square root.
### Conclusion:
The correct formula to calculate the standard deviation of sample data is:
[tex]\[ s = \sqrt{\frac{(x_1 - \bar{x})^2 + (x_2 - \bar{x})^2 + \ldots + (x_n - \bar{x})^2}{n - 1}} \][/tex]
Therefore, the correct option is:
[tex]\[ \boxed{1} \][/tex]
1. [tex]\( s = \sqrt{\frac{(x_1 - \bar{x})^2 + (x_2 - \bar{x})^2 + \ldots + (x_n - \bar{x})^2}{n - 1}} \)[/tex]
2. [tex]\( a^2 = \frac{(x_1 - \mu)^2 + (x_2 - \mu)^2 + \ldots + (x_N - \mu)^2}{N} \)[/tex]
3. [tex]\( \sigma = \sqrt{\frac{(x_1 - \mu)^2 + (x_2 - \mu)^2 + \ldots + (x_N - \mu)^2}{N}} \)[/tex]
4. [tex]\( s = \frac{(x_1 - \bar{x})^2 + (x_2 - \bar{x})^2 + \ldots + (x_n - \bar{x})^2}{n - 1} \)[/tex]
### Definitions & Symbols:
- [tex]\( s \)[/tex] is the standard deviation of a sample.
- [tex]\( \sigma \)[/tex] is the standard deviation of a population.
- [tex]\( \bar{x} \)[/tex] is the sample mean.
- [tex]\( \mu \)[/tex] is the population mean.
- [tex]\( n \)[/tex] is the sample size.
- [tex]\( N \)[/tex] is the population size.
### Explanation of Each Option:
1. Option 1:
[tex]\[ s = \sqrt{\frac{(x_1 - \bar{x})^2 + (x_2 - \bar{x})^2 + \ldots + (x_n - \bar{x})^2}{n - 1}} \][/tex]
This is the formula for the standard deviation of a sample. The divisor is [tex]\( n - 1 \)[/tex] which is known as Bessel's correction and it is used to correct the bias in the estimation of the population variance and standard deviation when sample data is used.
2. Option 2:
[tex]\[ a^2 = \frac{(x_1 - \mu)^2 + (x_2 - \mu)^2 + \ldots + (x_N - \mu)^2}{N} \][/tex]
This formula computes the population variance ([tex]\( \sigma^2 \)[/tex]), not the standard deviation, and uses the population mean ([tex]\( \mu \)[/tex]) and population size ([tex]\( N \)[/tex]).
3. Option 3:
[tex]\[ \sigma = \sqrt{\frac{(x_1 - \mu)^2 + (x_2 - \mu)^2 + \ldots + (x_N - \mu)^2}{N}} \][/tex]
This is the formula for the population standard deviation. It uses the population mean ([tex]\( \mu \)[/tex]) and population size ([tex]\( N \)[/tex]).
4. Option 4:
[tex]\[ s = \frac{(x_1 - \bar{x})^2 + (x_2 - \bar{x})^2 + \ldots + (x_n - \bar{x})^2}{n - 1} \][/tex]
This is not a standard deviation formula, but rather an expression for the sample variance without taking the square root.
### Conclusion:
The correct formula to calculate the standard deviation of sample data is:
[tex]\[ s = \sqrt{\frac{(x_1 - \bar{x})^2 + (x_2 - \bar{x})^2 + \ldots + (x_n - \bar{x})^2}{n - 1}} \][/tex]
Therefore, the correct option is:
[tex]\[ \boxed{1} \][/tex]
Thanks for stopping by. We are committed to providing the best answers for all your questions. See you again soon. We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Thank you for visiting Westonci.ca. Stay informed by coming back for more detailed answers.