Problem 8 - Calculus for Biology and Medicine

Question

The following data represent the relative frequency distribution of clutch size in a sample of 42 mallards: $$ \begin{array}{cc} \hline \text { Clutch Size } & \text { Relative Frequency } \\ \hline 6 & 0.10 \\ 7 & 0.24 \\ 8 & 0.29 \\ 9 & 0.21 \\ 10 & 0.16 \\ \hline \end{array} $$ Calculate the sample mean and the sample variance.

Step-by-Step Solution

Answer

The sample mean is 8.09, and the sample variance is 1.4819.

Step 1: Calculate Frequencies from Relative Frequencies

First, we need to calculate the actual frequency for each clutch size. Since the total sample size is 42, multiply each relative frequency by 42. - Frequency for clutch size 6: \(0.10 \times 42 = 4.2\) - Frequency for clutch size 7: \(0.24 \times 42 = 10.08\) - Frequency for clutch size 8: \(0.29 \times 42 = 12.18\) - Frequency for clutch size 9: \(0.21 \times 42 = 8.82\) - Frequency for clutch size 10: \(0.16 \times 42 = 6.72\)However, since frequencies must be whole numbers, these values indicate the expected number based on relative frequency. Still, for the purpose of mean and variance calculations, we'll multiply directly with sample size.

Step 2: Calculate the Sample Mean

The sample mean \(\bar{x}\) is calculated as the sum of each clutch size multiplied by its corresponding frequency divided by the total number of observations.\[\bar{x} = \frac{(6 \times 0.10) + (7 \times 0.24) + (8 \times 0.29) + (9 \times 0.21) + (10 \times 0.16)}{1}\]\[\bar{x} = 6(0.10) + 7(0.24) + 8(0.29) + 9(0.21) + 10(0.16)\]\[\bar{x} = 0.6 + 1.68 + 2.32 + 1.89 + 1.6 = 8.09\]So the sample mean is 8.09.

Step 3: Calculate the Sample Variance

Sample variance is calculated using:\[s^2 = \sum{(x_i - \bar{x})^2 \cdot f_i}\]Where \(x_i\) is the clutch size, \(\bar{x}\) is the sample mean, and \(f_i\) is the relative frequency for each clutch size.Calculate each \((x_i - \bar{x})^2 \):- For clutch size 6: \((6 - 8.09)^2 = 4.3681\)- For clutch size 7: \((7 - 8.09)^2 = 1.1881\)- For clutch size 8: \((8 - 8.09)^2 = 0.0081\)- For clutch size 9: \((9 - 8.09)^2 = 0.8281\)- For clutch size 10: \((10 - 8.09)^2 = 3.6481\)Now, multiply each by its frequency:- 4.3681 \(\times\) 0.10 = 0.43681- 1.1881 \(\times\) 0.24 = 0.285144- 0.0081 \(\times\) 0.29 = 0.002349- 0.8281 \(\times\) 0.21 = 0.173901- 3.6481 \(\times\) 0.16 = 0.583696Add these values together:\[s^2 = 0.43681 + 0.285144 + 0.002349 + 0.173901 + 0.583696 = 1.4819 \]This is the sample variance.

Key Concepts

Understanding the Sample Mean

The sample mean is a critical concept in statistics as it provides the average value of a data set. It is calculated by multiplying each data value by its relative frequency, summing these products, and dividing the total by the sum of the relative frequencies.
The formula for the sample mean \( \bar{x} \) is:

• \( \bar{x} = \frac{\sum (x_i \times f_i)}{ \sum f_i} \)

Where:

• \( x_i \) represents each data value (e.g., clutch sizes in our exercise: 6, 7, 8, 9, 10)
• \( f_i \) is the relative frequency of each data value (e.g., 0.10 for clutch size 6).

In our exercise, the sample mean was calculated by following these steps:

• Multiply each clutch size by its respective relative frequency
• Add all the products: 0.6 (from clutch size 6) + 1.68 (from clutch size 7) + 2.32 (from clutch size 8) + 1.89 (from clutch size 9) + 1.6 (from clutch size 10)
• The sum of these products gives us a value of 8.09, which is the sample mean.

The sample mean provides a simple yet powerful summary of the data set and helps in understanding the average trend within the data.

Delving into Sample Variance

Sample variance is a measure that describes how much the values in a data set differ from the mean of the data set. It essentially tells you how spread out the values are around the sample mean.
The formula for sample variance \( s^2 \) is:

• \( s^2 = \sum{(x_i - \bar{x})^2 \cdot f_i} \)

Where:

• \( x_i \) is the individual data value
• \( \bar{x} \) is the sample mean
• \( f_i \) is the relative frequency for each data value.

To calculate sample variance, you follow these steps:

• First, find the difference between each data value and the sample mean (e.g., for clutch size 6, it's \( 6 - 8.09 \))
• Square the result to eliminate negative differences
• Multiply each squared difference by its corresponding relative frequency
• Add all these products together to get the total variance

In our particular example:

• After squaring each difference and multiplying by its relative frequency, the sum of these products gives us the total sample variance of 1.4819.

Sample variance is essential as it provides detailed insight into the variability of the data set, complementing the information provided by the sample mean.

The Importance of Relative Frequency

Relative frequency is a way to express the number of times a particular value occurs relative to the total number of observations.
This allows us to compare how often different values appear even if the sample sizes are different.
The calculation is straightforward:

• Relative frequency = \( \frac{\text{frequency of value}}{\text{total number of observations}} \)

In our exercise, relative frequency was given directly (e.g., 0.10 for clutch size 6), meaning that:

• 10% of the mallards had a clutch size of 6.

The benefits of using relative frequency include:

• It gives a clearer picture of distribution compared to just using absolute counts.
• Allows easy comparison between different data sets.

In statistical analyses, relative frequency enables the calculation of other key measures like sample mean and sample variance. These two, in turn, help us to understand the central tendency and spread of data.