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 approximately 3.70, and the sample variance is approximately 1.43.

Step 1: Calculate the Sample Mean

To find the sample mean, first calculate the sum of the products of each number of days and its frequency. Use the formula: \( \bar{x} = \frac{\sum (x_i \cdot f_i)}{n} \), where \( x_i \) is the number of days, \( f_i \) is the frequency, and \( n \) is the total number of observations. Sum = (1 \times 2) + (2 \times 7) + (3 \times 9) + (4 \times 27) + (5 \times 11) + (6 \times 5) = 226. Total number of observations, \( n = 2 + 7 + 9 + 27 + 11 + 5 = 61 \). Thus, \( \bar{x} = \frac{226}{61} \approx 3.70 \).

Step 2: Calculate the Sample Variance

Use the formula for sample variance: \( s^2 = \frac{\sum (x_i - \bar{x})^2 \cdot f_i}{n - 1} \). First, calculate \( (x_i - \bar{x})^2 \) for each \( x_i \), then multiply by \( f_i \), and sum these products. Calculate the squared deviations: 1: \((1-3.70)^2 \times 2 \approx 13.89 \), 2: \((2-3.70)^2 \times 7 \approx 20.57 \), 3: \((3-3.70)^2 \times 9 \approx 4.41 \), 4: \((4-3.70)^2 \times 27 \approx 2.43 \), 5: \((5-3.70)^2 \times 11 \approx 18.01 \), 6: \((6-3.70)^2 \times 5 \approx 26.75 \). Sum: \( 13.89 + 20.57 + 4.41 + 2.43 + 18.01 + 26.75 = 86.06 \). Finally, the sample variance is \( \frac{86.06}{60} \approx 1.43 \).

Key Concepts

Understanding Sample Mean

The sample mean is a fundamental concept in statistics. It represents the average of a set of numbers. Calculating the sample mean helps us to understand the central tendency of our data.
In simple terms, it provides information about the "center" or "average" of a data set.

To calculate the sample mean, you sum up all the values in your sample and divide by the count of values. In the context of grouped data, like frequency distributions, we multiply each value by its frequency before summing.

• The formula for the sample mean is: \( \bar{x} = \frac{\sum (x_i \cdot f_i)}{n} \)
• Where \( x_i \) is the data value, \( f_i \) is its frequency, and \( n \) is the total number of data points.

This arithmetic average is useful because it balances all data points equally, smoothing over any variations these might show. It is particularly helpful when comparing different data samples.

Deciphering Sample Variance

Sample variance measures the dispersion or spread of a set of data points. While the mean gives us an idea of the data's center, variance reflects how "spread out" data points are around that mean.
When data points are far from the mean, variance is high. Conversely, it is low if they cluster closely around the mean.

The process for calculating sample variance involves several steps:

• First, for each data point, calculate the deviation from the sample mean: \( (x_i - \bar{x}) \).
• Square these deviations to prevent any cancellation. This also emphasizes larger deviations.
• Multiply each squared deviation by its frequency and sum these values.
• Divide by \( n-1 \), where \( n \) is the number of observations, to get the variance: \( s^2 = \frac{\sum (x_i - \bar{x})^2 \cdot f_i}{n - 1} \).

Understanding variance is crucial, as it provides insights into data reliability and variability, which is invaluable for data interpretation.

Exploring Frequency Distribution

A frequency distribution is a summary that shows how often various outcomes occur in a data set. It allows data to be organized in way that makes patterns easy to see. Often, it's displayed as a table or a graph.
Frequency distributions can be constructed for both qualitative (categorical) and quantitative (numerical) data.

Key elements of a frequency distribution include:

• Values: These are the distinct outcomes of the data.
• Frequencies: These indicate how often each value occurs.

In statistics, a frequency distribution is especially useful for large data sets. It enables quick understanding of data trends and anomalies.
For example, in demographic studies, frequency distributions help visualize age or income data, providing insights into population structure or economic conditions.