A study of the relationship between age and various visual functions (such as acuity and depth perception) reported the following observations on the area of scleral lamina (mm2) from human optic nerve heads:

2.76   

2.69   

2.77   

3.84   

2.33   

2.65   

3.94   

4.26   

3.93   

4.43   

3.43   

4.51   

2.50   

3.56   

2.77   

3.62   

3.04   

(a) Calculate Σ*x*i and Σ*x*i2.

Σ*x*i

=  mm2

Σ*x*i2

=  mm4


(b) Use the values calculated in part (a) to compute the sample variance *s*2 and then the sample standard deviation *s*. (Round your answers to four decimal places.)

*s*2

=  mm4

*s*

=  mm2

To compute the sample variance \( s^2 \) and the sample standard deviation \( s \), we first need to find the following:

1. \( \Sigma x_i \): The sum of all the observations.
2. \( \Sigma x_i^2 \): The sum of the squares of all the observations.

Given dataTo compute the sample variance \( s^2 \) and the sample standard deviation \( s \), we first need to find the following:

1. \( \Sigma x_i \): The sum of all the observations.
2. \( \Sigma x_i^2 \): The sum of the squares of all the observations.

Given 

\[
\text{Data (mm}^2\text{)}: 2.76, 2.69, 2.77, 3.84, 2.33, 2.65, 3.94, 4.26, 3.93, 4.43, 3.43, 4.51, 2.50, 3.56, 2.77, 3.62, 3.04
\]

### Step 1: Compute \( \Sigma x_i \) (Sum of all observations)
We need to sum all the given data points.

### Step 2: Compute \( \Sigma x_i^2 \) (Sum of the squares of all observations)
We need to square each data point and then sum those squares.

### Step 3: Calculate the Sample Variance \( s^2 \)
The formula for the sample variance is:
\[
s^2 = \frac{1}{n-1} \left( \Sigma x_i^2 - \frac{(\Sigma x_i)^2}{n} \right)
\]

### Step 4: Calculate the Sample Standard Deviation \( s \)
The sample standard deviation is simply the square root of the sample variance:
\[
s = \sqrt{s^2}
\]

Now, let's calculate these values.### Results:

- **Σ*x*i** (Sum of all observations):
  \[
  \Sigma x_i = 57.03 \, \text{mm}^2
  \]

- **Σ*x*i^2** (Sum of the squares of all observations):
  \[
  \Sigma x_i^2 = 199.5121 \, \text{mm}^4
  \]

- **Sample Variance** \( s^2 \):
  \[
  s^2 = 0.5121 \, \text{mm}^4 \, \text{(rounded to four decimal places)}
  \]

- **Sample Standard Deviation** \( s \):
  \[
  s = 0.7156 \, \text{mm}^2 \, \text{(rounded to four decimal places)}
  \]

Do you need further explanation on any of these calculations? Here are 5 related questions to explore:

1. How would the results change if a new observation were added?
2. What is the interpretation of the sample standard deviation in the context of this study?
3. How does variance help in understanding data dispersion compared to the mean?
4. Can you calculate the coefficient of variation using the mean and standard deviation?
5. What impact does sample size have on the reliability of variance estimates?

**Tip:** Always report both variance and standard deviation, as they provide complementary information about data dispersion.

Learn how to calculate sample variance and standard deviation from a set of data points in statistics. This problem involves computing sums of squares and understanding measures of dispersion in data. Suitable for college-level students and researchers.

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