Find the​ range, variance, and standard deviation for the given sample​ data, if possible. If the measures of variation can be obtained for these​ values, do the results make​ sense?
Biologists conducted experiments to determine whether a deficiency of carbon dioxide in the soil affects the phenotypes of peas. Listed below are the phenotype​ codes, where 1equals
​smooth-yellow, 
2equals
​smooth-green, 
3equals
wrinkled 
​yellow, and 4equals
​wrinkled-green.
1
    
1
    
2
    
2
    
3
    
2
    
4
    
1
    
2
    
2
    
2
    
2
    
1
    
2
    
2
    
3
    
1
    
2
    
2
    
2
 
 

Let's first organize the sample data and then calculate the range, variance, and standard deviation.

The sample data is:

\[ 1, 1, 2, 2, 3, 2, 4, 1, 2, 2, 2, 2, 1, 2, 2, 3, 1, 2, 2, 2 \]

### Step 1: Calculate the Range
The range is the difference between the maximum and minimum values in the data set.

- Minimum value: \( 1 \)
- Maximum value: \( 4 \)

\[
\text{Range} = 4 - 1 = 3
\]

### Step 2: Calculate the Variance
Variance measures the spread of the data. The formula for sample variance is:

\[
s^2 = \frac{1}{n-1} \sum (x_i - \bar{x})^2
\]

Where:
- \( n \) is the sample size
- \( x_i \) are the individual data points
- \( \bar{x} \) is the sample mean

#### Sample Size and Mean:
- Sample size \( n = 20 \)
- Mean \( \bar{x} = \frac{\sum x_i}{n} = \frac{1 + 1 + 2 + \cdots + 2}{20} = \frac{40}{20} = 2 \)

#### Variance Calculation:
Now calculate the squared differences from the mean for each data point and sum them up:

\[
\sum (x_i - \bar{x})^2 = (1-2)^2 + (1-2)^2 + (2-2)^2 + \cdots + (2-2)^2
\]
\[
= 1 + 1 + 0 + 0 + 1 + 0 + 4 + 1 + 0 + 0 + 0 + 0 + 1 + 0 + 0 + 1 + 1 + 0 + 0 + 0 = 11
\]

Now divide by \( n - 1 \):

\[
s^2 = \frac{11}{20 - 1} = \frac{11}{19} \approx 0.579
\]

### Step 3: Calculate the Standard Deviation
The standard deviation is the square root of the variance:

\[
s = \sqrt{s^2} = \sqrt{0.579} \approx 0.761
\]

### Conclusion
- **Range**: 3
- **Variance**: 0.579
- **Standard Deviation**: 0.761

#### Does this make sense?
Yes, the values make sense. The range of 3 shows that the phenotype codes span a difference of three levels. The variance and standard deviation are relatively low, reflecting that the data points are clustered around the mean, indicating limited spread.

Do you want more details or have any questions?  
Here are 5 relative questions:
1. What is the interpretation of variance in a biological context?
2. How would the data change if an additional phenotype code was introduced?
3. What is the significance of the standard deviation in phenotypic experiments?
4. Can range alone provide insights into data spread?
5. How does sample size affect variance and standard deviation?

**Tip**: The standard deviation is always in the same units as the data, making it easier to interpret in context!

Find the​ range, variance, and standard deviation for the given sample​ data, if possible. If the measures of variation can be obtained for these​ values, do the results make​ sense?
Biologists conducted experiments to determine whether a deficiency of carbon dioxide in the soil affects the phenotypes of peas. Listed below are the phenotype​ codes, where 1 equals smooth-yellow, 2 equals smooth-green, 3 equals wrinkled yellow, and 4 equals wrinkled-green.
1, 1, 2, 2, 3, 2, 4, 1, 2, 2, 2, 2, 1, 2, 2, 3, 1, 2, 2, 2

Discover how to calculate the range, variance, and standard deviation for phenotype codes in a biological experiment. Learn the steps for finding these key statistical measures from a sample of 20 data points, including smooth-yellow, smooth-green, wrinkled-yellow, and wrinkled-green phenotypes. Suitable for Grades 10-12.

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