Problem 5 - Calculus for Biology and Medicine

Question

The following data represent the frequency distribution of seed numbers per flower head in a flowering plant: $$ \begin{array}{cc} \hline \text { Seed Number } & \text { Frequency } \\ \hline 9 & 37 \\ 10 & 48 \\ 11 & 53 \\ 12 & 49 \\ 13 & 61 \\ 14 & 42 \\ 15 & 31 \\ \hline \end{array} $$ Calculate the sample mean and the sample variance.

Step-by-Step Solution

Answer

Sample mean is 11.93 and sample variance is 2.45.

Step 1: Calculate the Total Number of Flower Heads

To find the sample mean and variance, first we need the total number of flower heads. This is the sum of all frequencies. Sum up the frequencies: \( 37 + 48 + 53 + 49 + 61 + 42 + 31 = 321 \). So, the total number of flower heads is 321.

Step 2: Calculate the Sample Mean

The sample mean is calculated using the formula \( \bar{x} = \frac{\sum (x_i \cdot f_i)}{N} \), where \( x_i \) is the seed number, \( f_i \) is the frequency, and \( N \) is the total number. Calculate \( 9 \cdot 37 = 333 \), \( 10 \cdot 48 = 480 \), \( 11 \cdot 53 = 583 \), \( 12 \cdot 49 = 588 \), \( 13 \cdot 61 = 793 \), \( 14 \cdot 42 = 588 \), \( 15 \cdot 31 = 465 \). Sum these products: \( 333 + 480 + 583 + 588 + 793 + 588 + 465 = 3830 \). Divide by total frequency: \( \bar{x} = \frac{3830}{321} \approx 11.93 \).

Step 3: Calculate the Squared Differences from the Mean

For each seed number, calculate \((x_i - \bar{x})^2 \cdot f_i\). Using \( \bar{x} = 11.93 \):\[ \begin{align*} (9 - 11.93)^2 \cdot 37 & = 318.92, \ (10 - 11.93)^2 \cdot 48 & = 169.55, \ (11 - 11.93)^2 \cdot 53 & = 47.01, \ (12 - 11.93)^2 \cdot 49 & = 0.32, \ (13 - 11.93)^2 \cdot 61 & = 64.92, \ (14 - 11.93)^2 \cdot 42 & = 91.48, \ (15 - 11.93)^2 \cdot 31 & = 92.73 \end{align*} \] Sum the squared differences: \( 318.92 + 169.55 + 47.01 + 0.32 + 64.92 + 91.48 + 92.73 = 784.93 \).

Step 4: Calculate the Sample Variance

The sample variance \( s^2 \) is calculated by dividing the sum of squared differences by the total number of observations minus one: \( s^2 = \frac{784.93}{321 - 1} = \frac{784.93}{320} = 2.45 \).

Step 5: Review

The calculations show that the sample mean is approximately 11.93 and the sample variance is 2.45 based on the given frequency distribution.

Key Concepts

Frequency Distribution

Frequency distribution is a useful way to organize and summarize large sets of data. It shows how often each value in a set of data occurs, making it easier to visualize patterns or trends.
For example, consider a flowering plant with various numbers of seeds per flower head as shown in the exercise.

• Seed Number 9 occurs 37 times.
• Seed Number 10 occurs 48 times, and so on.

By listing seed numbers alongside their frequencies, the table gives us a quick view of the distribution of seed numbers across these flower heads.
This allows statisticians to extract meaningful insights about the data, such as determining which seed numbers appear most frequently or what the overall spread and average might look like.
Frequency distribution is a critical step in descriptive statistics before moving on to more complex calculations like mean and variance.

Sample Mean

The sample mean provides a measure of the center of a dataset. It is calculated by dividing the sum of all data values by the total frequency.
In the example provided, we must first find the weighted sum of seed numbers by multiplying each seed number by its frequency, then summing these products:
\( 333 + 480 + 583 + 588 + 793 + 588 + 465 = 3830 \).

• This gives the total combined value of all seeds in all flower heads.
• We then divide this by the total number of observations, 321 flower heads, to find the sample mean.

The sample mean here is \( \frac{3830}{321} \approx 11.93 \), indicating the average seed number per flower head in the sample.
A useful measure, the sample mean is a primary descriptor in statistical analysis and provides vital insights into the data's central tendency.

Sample Variance

Sample variance measures how much the data points in a set differ from the mean, or in other words, it shows data variability.
To find the variance of seed numbers, we first determine how far each seed number deviates from the mean, square these deviations to ensure they are positive, and then multiply each by its respective frequency.
Subsequently, sum these values:
\( 318.92 + 169.55 + 47.01 + 0.32 + 64.92 + 91.48 + 92.73 = 784.93 \).
Divide by the number of observations minus one (\( 321 - 1 = 320 \)) to find the variance:
\( s^2 = \frac{784.93}{320} \approx 2.45 \).
A higher variance would indicate a wider spread of seed numbers, whereas a lower variance suggests the seed numbers are close to the mean.

This measure aids in understanding data dispersion beyond simply looking at the mean, providing a fuller picture of data collection characteristics.