To solve this problem, we need to identify the correct formulas for calculating the variance and standard deviation based on the given sample data.
### Step 1: Define the Data
We are given the daily heights of bamboo stalks:
[tex]\[ 20, 19, 17, 16, 18, 15, 20, 21 \][/tex]
### Step 2: Calculate the Mean ([tex]\(\bar{x}\)[/tex])
The mean ([tex]\(\bar{x}\)[/tex]) is calculated as:
[tex]\[
\bar{x} = \frac{\sum_{i=1}^n x_i}{n}
\][/tex]
where [tex]\( x_i \)[/tex] are the individual heights and [tex]\( n \)[/tex] is the number of observations.
For the given data:
[tex]\[
\bar{x} = \frac{20 + 19 + 17 + 16 + 18 + 15 + 20 + 21}{8} = \frac{146}{8} = 18.25
\][/tex]
### Step 3: Calculate the Sum of Squared Deviations
We calculate the squared deviation of each observation from the mean:
[tex]\[
(x - \bar{x})^2
\][/tex]
For each height:
[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]
Summing these squared deviations:
[tex]\[
3.0625 + 0.5625 + 1.5625 + 5.0625 + 0.0625 + 10.5625 + 3.0625 + 7.5625 = 31.5
\][/tex]
### Step 4: Calculate the Variance
Using 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]
For the data:
[tex]\[
s^2 = \frac{31.5}{8 - 1} = \frac{31.5}{7} = 4.5
\][/tex]
### Step 5: Calculate the Standard Deviation
Using 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]
For the data:
[tex]\[
s = \sqrt{4.5} \approx 2.1213
\][/tex]
### Conclusion
The variance of the sample is given by 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]
The standard deviation of the sample is given by 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]
So, the formulas we should use are:
- For variance: Formula A [tex]\( \boxed{A} \)[/tex]
- For standard deviation: Formula B [tex]\( \boxed{B} \)[/tex]
