Problem 7 - Calculus for Biology and Medicine

Question

The following data represent the relative frequency distribution of clutch size in a sample of 300 laboratory guinea pigs: $$ \begin{array}{cc} \hline \text { Clutch Size } & \text { Relative Frequency } \\ \hline 2 & 0.05 \\ 3 & 0.09 \\ 4 & 0.12 \\ 5 & 0.19 \\ 6 & 0.23 \\ 7 & 0.12 \\ 8 & 0.13 \\ 9 & 0.07 \\ \hline \end{array} $$ Calculate the sample mean and the sample variance.

Step-by-Step Solution

Answer

Sample mean is approximately 6, and sample variance is approximately 2.79.

Step 1: Understand Relative Frequency and Total Sample

The problem provides a table of relative frequencies for different clutch sizes in a sample of 300 guinea pigs. The relative frequency of a clutch size tells us the proportion of the sample that has that size.

Step 2: Calculate the Frequency for Each Clutch Size

To use the relative frequencies to find actual frequencies, multiply each relative frequency by the total sample size. For example, for clutch size 2, the frequency is \(0.05 \times 300 = 15\). Repeat this for all clutch sizes to get the frequencies.

Step 3: Calculate the Sample Mean

The sample mean is calculated as the sum of the product of each clutch size and its respective frequency divided by the total sample size. This is represented by the formula: \[ \text{Mean} = \frac{\sum (x_i \cdot f_i)}{N} \] where \(x_i\) is the clutch size and \(f_i\) is the frequency. Compute this for all clutch sizes.

Step 4: Calculate Each Clutch Size's Contribution to Variance

Variance is calculated using the squared difference between each clutch size and the mean, multiplied by its frequency, summed across all clutch sizes, and then divided by the total sample size. This can be represented as: \[ \text{Variance} = \frac{\sum f_i (x_i - \text{Mean})^2}{N} \] where \(x_i\) is each clutch size and \(f_i\) its frequency.

Step 5: Plug in Values to Find Sample Mean

Substitute the calculated frequencies and clutch sizes into the mean formula: \( \frac{2 \times 15 + 3 \times 27 + 4 \times 36 + 5 \times 57 + 6 \times 69 + 7 \times 36 + 8 \times 39 + 9 \times 21}{300} = 5.995 \approx 6 \).

Step 6: Plug in Values to Find Sample Variance

Now calculate variance: \( \frac{15(2-6)^2 + 27(3-6)^2 + 36(4-6)^2 + 57(5-6)^2 + 69(6-6)^2 + 36(7-6)^2 + 39(8-6)^2 + 21(9-6)^2}{300} = 2.794 \approx 2.79\).

Key Concepts

Understanding Relative Frequency

Relative frequency is a fundamental concept in statistics. It's a way to show how often something happens relative to the entire dataset. When dealing with a table of relative frequencies, as in our guinea pigs example, each entry in the table tells us the proportion of the total sample that corresponds to a specific category.

Think of it this way:

• If a clutch size has a relative frequency of 0.05, this means that 5% of the total number of guinea pigs in the sample have that clutch size.
• To convert relative frequency into actual frequency, multiply it by the total number of observations—in our case, 300 guinea pigs.
• This turns the relative percentages into the actual number of occurrences for each clutch size.

Understanding relative frequency is crucial because it is often used to estimate probabilities and can help us identify trends or patterns within data.

Calculating the Sample Mean

The sample mean is a core concept in statistics that gives us the average value of our data set. It is especially useful when summarizing data to understand the central tendency.

Here's how it works:

• To calculate the sample mean, multiply each clutch size by its frequency.
• Sum all these products to get the total.
• Finally, divide this total by the number of observations (300 in this case).

This statistical measure helps momentarily simplify large data sets into a single value.

The formula is straightforward: \[ \text{Mean} = \frac{\sum (x_i \cdot f_i)}{N} \]
This formula effectively balances the data's spread by giving different weights to different clutch sizes, based on their frequency.

Understanding and Calculating Sample Variance

Sample variance tells us how much the data points in our data set spread out from the mean. It provides insight into the variability within the dataset, helping to understand the degree of spread or dispersion.

The process involves several steps:

• First, find the difference between each clutch size and the sample mean.
• Square these differences to remove negatives and emphasize larger deviations.
• Multiply each squared difference by its respective frequency to weight it appropriately.
• Sum all these products for an aggregate measure of spread.
• Finally, divide by the total number of observations to find the average of these squared differences, which gives the sample variance.

The formula for sample variance is: \[ \text{Variance} = \frac{\sum f_i (x_i - \text{Mean})^2}{N} \]
Understanding sample variance is essential for statisticians as it provides deeper insights into the data's consistency or unpredictability.