Get the most out of your questions with the extensive resources available on IDNLearn.com. Join our community to receive timely and reliable responses to your questions from knowledgeable professionals.
Sagot :
Let's go through the problem step-by-step to find the variance and standard deviation for the given data set of bamboo stalk heights: [tex]\( 20, 19, 17, 16, 18, 15, 20, 21 \)[/tex].
### Step 1: Understanding the Formulas
Given formulas are:
- Formula A: [tex]\( s^2 = \frac{(x_1 - \bar{x})^2 + (x_2 - \bar{x})^2 + \ldots + (x_n - \bar{x})^2}{n - 1} \)[/tex]
- Formula B: [tex]\( s = \sqrt{\frac{(x_1 - \bar{x})^2 + (x_2 - \bar{x})^2 + \ldots + (x_n - \bar{x})^2}{n - 1}} \)[/tex]
- Formula C: [tex]\( \sigma^2 = \frac{(x_1 - \mu)^2 + (x_2 - \mu)^2 + \ldots + (x_N - \mu)^2}{N} \)[/tex]
- Formula D: [tex]\( \sigma = \sqrt{\frac{(x_1 - \mu)^2 + (x_2 - \mu)^2 + \ldots + (x_N - \mu)^2}{N}} \)[/tex]
Since "A" is checked for variance and "B" is checked for standard deviation, we will be using:
- [tex]\( A \text{ for variance: } s^2 = \frac{(x_1 - \bar{x})^2 + (x_2 - \bar{x})^2 + \ldots + (x_n - \bar{x})^2}{n - 1} \)[/tex]
- [tex]\( B \text{ for standard deviation: } s = \sqrt{\frac{(x_1 - \bar{x})^2 + (x_2 - \bar{x})^2 + \ldots + (x_n - \bar{x})^2}{n - 1}} \)[/tex]
### Step 2: Calculating the Mean
The mean ([tex]\(\bar{x}\)[/tex]) is given as:
[tex]\[ \bar{x} = 18.25 \][/tex]
### Step 3: Calculating the Variance
Using Formula A for variance [tex]\( s^2 \)[/tex]:
1. Calculate the squared differences from the mean for each data point:
- [tex]\((20 - 18.25)^2 = 3.0625\)[/tex]
- [tex]\((19 - 18.25)^2 = 0.5625\)[/tex]
- [tex]\((17 - 18.25)^2 = 1.5625\)[/tex]
- [tex]\((16 - 18.25)^2 = 5.0625\)[/tex]
- [tex]\((18 - 18.25)^2 = 0.0625\)[/tex]
- [tex]\((15 - 18.25)^2 = 10.5625\)[/tex]
- [tex]\((20 - 18.25)^2 = 3.0625\)[/tex]
- [tex]\((21 - 18.25)^2 = 7.5625\)[/tex]
2. Sum these squared differences:
[tex]\[ 3.0625 + 0.5625 + 1.5625 + 5.0625 + 0.0625 + 10.5625 + 3.0625 + 7.5625 = 31.5 \][/tex]
3. Divide the sum by [tex]\( n - 1 \)[/tex] (where [tex]\( n \)[/tex] is the number of data points, which is 8):
[tex]\[ s^2 = \frac{31.5}{7} = 4.5 \][/tex]
Thus, the variance is:
[tex]\[ \text{Variance} = 4.5 \][/tex]
### Step 4: Calculating the Standard Deviation
Using Formula B for standard deviation [tex]\( s \)[/tex]:
[tex]\[ s = \sqrt{\frac{(20-18.25)^2 + (19-18.25)^2 + \ldots + (21-18.25)^2}{n-1}} \][/tex]
We already have the variance as 4.5, so the standard deviation is:
[tex]\[ s = \sqrt{4.5} \approx 2.1213203435596424 \][/tex]
Thus, the standard deviation is:
[tex]\[ \text{Standard Deviation} \approx 2.1213 \][/tex]
### Summary:
- The variance of the bamboo stalk heights is [tex]\( 4.5 \)[/tex].
- The standard deviation of the bamboo stalk heights is [tex]\( 2.1213 \)[/tex].
### Step 1: Understanding the Formulas
Given formulas are:
- Formula A: [tex]\( s^2 = \frac{(x_1 - \bar{x})^2 + (x_2 - \bar{x})^2 + \ldots + (x_n - \bar{x})^2}{n - 1} \)[/tex]
- Formula B: [tex]\( s = \sqrt{\frac{(x_1 - \bar{x})^2 + (x_2 - \bar{x})^2 + \ldots + (x_n - \bar{x})^2}{n - 1}} \)[/tex]
- Formula C: [tex]\( \sigma^2 = \frac{(x_1 - \mu)^2 + (x_2 - \mu)^2 + \ldots + (x_N - \mu)^2}{N} \)[/tex]
- Formula D: [tex]\( \sigma = \sqrt{\frac{(x_1 - \mu)^2 + (x_2 - \mu)^2 + \ldots + (x_N - \mu)^2}{N}} \)[/tex]
Since "A" is checked for variance and "B" is checked for standard deviation, we will be using:
- [tex]\( A \text{ for variance: } s^2 = \frac{(x_1 - \bar{x})^2 + (x_2 - \bar{x})^2 + \ldots + (x_n - \bar{x})^2}{n - 1} \)[/tex]
- [tex]\( B \text{ for standard deviation: } s = \sqrt{\frac{(x_1 - \bar{x})^2 + (x_2 - \bar{x})^2 + \ldots + (x_n - \bar{x})^2}{n - 1}} \)[/tex]
### Step 2: Calculating the Mean
The mean ([tex]\(\bar{x}\)[/tex]) is given as:
[tex]\[ \bar{x} = 18.25 \][/tex]
### Step 3: Calculating the Variance
Using Formula A for variance [tex]\( s^2 \)[/tex]:
1. Calculate the squared differences from the mean for each data point:
- [tex]\((20 - 18.25)^2 = 3.0625\)[/tex]
- [tex]\((19 - 18.25)^2 = 0.5625\)[/tex]
- [tex]\((17 - 18.25)^2 = 1.5625\)[/tex]
- [tex]\((16 - 18.25)^2 = 5.0625\)[/tex]
- [tex]\((18 - 18.25)^2 = 0.0625\)[/tex]
- [tex]\((15 - 18.25)^2 = 10.5625\)[/tex]
- [tex]\((20 - 18.25)^2 = 3.0625\)[/tex]
- [tex]\((21 - 18.25)^2 = 7.5625\)[/tex]
2. Sum these squared differences:
[tex]\[ 3.0625 + 0.5625 + 1.5625 + 5.0625 + 0.0625 + 10.5625 + 3.0625 + 7.5625 = 31.5 \][/tex]
3. Divide the sum by [tex]\( n - 1 \)[/tex] (where [tex]\( n \)[/tex] is the number of data points, which is 8):
[tex]\[ s^2 = \frac{31.5}{7} = 4.5 \][/tex]
Thus, the variance is:
[tex]\[ \text{Variance} = 4.5 \][/tex]
### Step 4: Calculating the Standard Deviation
Using Formula B for standard deviation [tex]\( s \)[/tex]:
[tex]\[ s = \sqrt{\frac{(20-18.25)^2 + (19-18.25)^2 + \ldots + (21-18.25)^2}{n-1}} \][/tex]
We already have the variance as 4.5, so the standard deviation is:
[tex]\[ s = \sqrt{4.5} \approx 2.1213203435596424 \][/tex]
Thus, the standard deviation is:
[tex]\[ \text{Standard Deviation} \approx 2.1213 \][/tex]
### Summary:
- The variance of the bamboo stalk heights is [tex]\( 4.5 \)[/tex].
- The standard deviation of the bamboo stalk heights is [tex]\( 2.1213 \)[/tex].
Thank you for using this platform to share and learn. Keep asking and answering. We appreciate every contribution you make. For dependable answers, trust IDNLearn.com. Thank you for visiting, and we look forward to assisting you again.
