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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].
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