Problem 15 - Calculus for Biology and Medicine

Question

The following table contains the number of leaves per basil plant in a sample of size 25 : \(\begin{array}{lllll}19 & 21 & 20 & 13 & 18 \\ 14 & 17 & 14 & 17 & 17 \\ 13 & 15 & 12 & 15 & 17 \\ 15 & 16 & 18 & 17 & 14 \\ 14 & 14 & 13 & 20 & 13\end{array}\) (a) Find the relative frequency distribution. (b) Compute the average value by (i) averaging the values in the table directly and (ii) using the relative frequency distribution obtained in (a).

Step-by-Step Solution

Answer

Relative frequencies are calculated by dividing each frequency by 25. The average is 16.08 from both direct calculation and relative frequencies.

Step 1: Create a Frequency Distribution Table

First, count the number of times each value appears in the data set. The data consists of values from 12 to 21. Create a table with two columns: one for the possible number of leaves (12 through 21) and one for their respective frequencies.

Step 2: Compute Frequencies

Count how many times each number appears in the dataset. For example: - 12 appears once. - 13 appears four times. - 14 appears five times. Continue this for all unique values in the range.

Step 3: Calculate Relative Frequencies

Divide the frequency of each number by the total number of data points (25). This gives the relative frequency of each value. The formula for relative frequency is: \( \text{Relative Frequency} = \frac{\text{Frequency of Value}}{\text{Total Number of Data Points}} \)

Step 4: Calculate Direct Average

Add all the numbers in the dataset and divide by the number of data points to find the average directly:\[ \text{Average} = \frac{19+21+20+...+13}{25} \]

Step 5: Calculate Average from Relative Frequency Distribution

For this, multiply each unique value by its relative frequency and then sum these products to find the average:\[ \text{Average} = \sum ( \text{Value} \times \text{Relative Frequency}) \]

Step 6: Verify Both Averages are Equal

Compare the averages calculated in Step 4 and Step 5 to ensure they are the same, confirming the calculation’s consistency.

Key Concepts

Frequency Distribution Table

In any data analysis, creating a frequency distribution table is a crucial first step. It organizes data in an easy-to-understand format, showing how often each value occurs.
This is particularly useful when you have a large dataset, like the number of leaves on basil plants. In the provided exercise, the range of numbers is 12 to 21, with each representing possible leaves per plant. To construct this table, list possible values in one column and count how many times each appears in your dataset. This count becomes the frequency.
For instance:

• 12 appears once — frequency is 1
• 13 appears four times — frequency is 4

By completing this process for all numbers in the range, you get a complete frequency distribution table. This helps in further steps, like calculating relative frequencies, which are often more insightful in data interpretation.

Average Calculation

There are different methods to calculate the average or mean in a dataset. Both are vital in data analysis for understanding the central tendency. The first method is straightforward, where you sum all data points and divide by the number of points. Given our data:

• Add all leaf counts: 19 + 21 + 20 ...
• Divide by 25 (total number of plants)

This provides a direct average. Another effective approach is using the relative frequency distribution. Here, each unique value is multiplied by its relative frequency. These products are summed to find the average. This method is equally reliable and sometimes preferred for highlighting trends when datasets have repeating values. Ultimately, both methods should give the same average, providing a good reliability check for your calculations.

Data Analysis in Biology

Data analysis is a powerful tool in biology, especially when it comes to understanding patterns in living organisms. Using statistical methods, such as calculating averages or distributions, allows biologists to make meaningful conclusions about their data. For example, analyzing the number of leaves on basil plants can lead to insights into plant health or growth under different conditions. Relative frequency and averages reveal the tendency and variance within a dataset, crucial in biological studies. Patterns in plant leaf data can suggest environmental impacts or genetic differences among plant species.
With this information, biologists can develop hypotheses or shape conservation strategies. In essence, these calculations enable researchers to quantify observations, leading to more robust and scientifically sound conclusions. Regular practice in basic statistical techniques equips students with valuable skills for any field that involves data interpretation.