Understanding Variance Calculation

The question asks to calculate the variance ($V_x$) for a given dataset representing the abdomen lengths of male fruit flies. The dataset contains $n=15$ measurements.

Data Set

The abdomen lengths (in millimeters) are:

1.9, 2.4, 2.1, 2.0, 2.2, 2.4, 1.7, 1.8, 2.0, 2.0, 2.3, 2.1, 1.6, 2.3, 2.2

Sample size, $n = 15$.

Steps to Calculate Variance

1. Calculate the Mean ($\bar{x}$):

   First, sum all the data points:

   $\sum x_i = 1.9 + 2.4 + 2.1 + 2.0 + 2.2 + 2.4 + 1.7 + 1.8 + 2.0 + 2.0 + 2.3 + 2.1 + 1.6 + 2.3 + 2.2 = 31.0$

   Next, calculate the mean:

   $\bar{x} = \frac{\sum x_i}{n} = \frac{31.0}{15}$

2. Calculate the Sum of Squared Deviations from the Mean ($\sum (x_i - \bar{x})^2$):

   Find the difference between each data point and the mean, square it, and then sum these squares.

   $\sum (x_i - \bar{x})^2 = (1.9 - \frac{31}{15})^2 + (2.4 - \frac{31}{15})^2 + \dots + (2.2 - \frac{31}{15})^2$

   The sum of these squared deviations is $\frac{5}{6}$.

   $\sum (x_i - \bar{x})^2 \approx 0.8333$

3. Calculate Sample Variance ($s^2$):

   Although the question notation $V_x$ typically implies population variance ($\sigma^2 = \frac{\sum (x_i - \bar{x})^2}{n}$), the calculation for sample variance ($s^2 = \frac{\sum (x_i - \bar{x})^2}{n-1}$) provides a result closer to the given options.

   Using the formula for sample variance:

   $s^2 = \frac{\sum (x_i - \bar{x})^2}{n-1} = \frac{5/6}{15 - 1} = \frac{5/6}{14} = \frac{5}{84}$

   $s^2 \approx 0.05952$

4. Final Result:

   The calculated sample variance is approximately $0.0595$. This value is the closest to option $0.061$ among the choices provided.