Problem 6 - Calculus for Biology and Medicine

Question

The following data represent the frequency distribution of the numbers of days that it took a certain ointment to clear up a skin rash: $$ \begin{array}{cc} \hline \text { Number of Days } & \text { Frequency } \\ \hline 1 & 2 \\ 2 & 7 \\ 3 & 9 \\ 4 & 27 \\ 5 & 11 \\ 6 & 5 \\ \hline \end{array} $$ Calculate the sample mean and the sample variance.

Step-by-Step Solution

Answer

The sample mean is 3.87 days and the sample variance is 1.41 days squared.

Step 1: Identify Midpoints and Frequencies

The midpoints for each class (in this case, each "Number of Days" is itself the midpoint) are 1, 2, 3, 4, 5, and 6. The corresponding frequencies are 2, 7, 9, 27, 11, and 5.

Step 2: Determine Total Number of Observations

Calculate the total number of observations by summing all frequencies: \[ ext{Total Observations} = 2 + 7 + 9 + 27 + 11 + 5 = 61 \]

Step 3: Calculate the Mean

Find the mean by calculating the weighted sum of midpoints and dividing by the total number of observations:\[ \text{Mean} = \frac{(1 \cdot 2) + (2 \cdot 7) + (3 \cdot 9) + (4 \cdot 27) + (5 \cdot 11) + (6 \cdot 5)}{61} \] Simplifying, we have:\[ \text{Mean} = \frac{2 + 14 + 27 + 108 + 55 + 30}{61} = \frac{236}{61} = 3.87 \]

Step 4: Calculate Each Midpoint's Deviation from the Mean

Find the deviation of each midpoint from the mean, \(3.87\), and square these deviations:\[ (1 - 3.87)^2, (2 - 3.87)^2, (3 - 3.87)^2, (4 - 3.87)^2, (5 - 3.87)^2, (6 - 3.87)^2 \] These deviations are: 8.23, 3.50, 0.76, 0.02, 1.26, 4.49, respectively.

Step 5: Calculate Weighted Sum of Squared Deviations

Use the squared deviations and frequencies to calculate the weighted sum:\[\sum = 2(8.23) + 7(3.50) + 9(0.76) + 27(0.02) + 11(1.26) + 5(4.49) \]Calculating each:\[16.46 + 24.50 + 6.84 + 0.54 + 13.86 + 22.45 = 84.65\]

Step 6: Calculate the Sample Variance

Finally, divide the sum of squared deviations by the total number of observations minus one:\[ \text{Sample Variance} = \frac{84.65}{61 - 1} = \frac{84.65}{60} = 1.41 \]

Key Concepts

Understanding Frequency Distribution

Frequency distribution is a way to organize data to show how often each value occurs. In our skincare ointment example, the number of days it took to clear the skin rash is listed alongside how many patients experienced each specific duration. Here’s why it’s helpful:

• It provides a quick visual representation of the dataset, making patterns easier to spot.
• It helps in understanding the distribution, such as which outcomes are most or least common.
• It often helps in calculating other statistics, like the mean or variance.

Consider the frequency distribution from the example: - Number of Days: 1, 2, 3, 4, 5, 6 - Frequency: 2, 7, 9, 27, 11, 5 This means that, for instance, 2 patients saw results in 1 day, and 27 patients noted improvement in exactly 4 days. This systematic layout provides a foundation for further statistical analysis like calculating the sample mean and variance.

Calculating the Sample Mean

The sample mean is a crucial concept in statistics, used to find the average of your data set. It gives you a central value that represents your data. To calculate the sample mean:

• Use the formula \( \bar{x} = \frac{\sum{(x_i \times f_i)}}{n} \) where \( x_i \) is the midpoint and \( f_i \) is the frequency.
• Sum up the products of each midpoint and its corresponding frequency.
• Divide this total by the total number of observations (the sum of all frequencies).

In the example, our midpoints are the number of days (1 to 6) and the corresponding frequencies are how often each outcome happened. Calculating:\[\begin{align*}\text{Mean} &= \frac{(1 \times 2) + (2 \times 7) + (3 \times 9) + (4 \times 27) + (5 \times 11) + (6 \times 5)}{61} \&= \frac{236}{61} = 3.87\end{align*}\]This mean of 3.87 days provides a summary measure of the centrality of the dataset. It tells us that, on average, the rash cleared up in about 3.87 days.

Exploring Sample Variance

Sample variance is a measure of how spread out your data points are around the mean. It tells you whether the data points tend to be close to that mean (low variance) or far away from it (high variance).To find the sample variance, follow these steps:

• Calculate each data point's deviation from the mean (difference between each midpoint and the mean).
• Square each of these deviations to make them positive.
• Multiply each squared deviation by its respective frequency to get a weighted value.
• Sum up all these weighted squares.
• Divide by the total number of observations minus one \( (n - 1) \), not by \( n \) because it's a sample variance.

In our case:\[\begin{align*}\text{Sample Variance} &= \frac{2(8.23) + 7(3.50) + 9(0.76) + 27(0.02) + 11(1.26) + 5(4.49)}{60} \&= \frac{84.65}{60} = 1.41\end{align*}\]This result of 1.41 indicates that most data points did not deviate far from the mean of 3.87 days, suggesting a moderate level of consistency in how quickly the ointment worked.