Problem 2 - Calculus for Biology and Medicine

Question

The following data represent the number of seeds per flower head in a sample of nine flowering plants: $$ 27,39,42,18,21,33,45,37,21 $$ Find the median, the sample mean, and the sample variance.

Step-by-Step Solution

Answer

Median is 33, sample mean is 31.444, and sample variance is 125.861.

Step 1: Organize the Data

Sort the given data set in ascending order: \[ 18, 21, 21, 27, 33, 37, 39, 42, 45 \]

Step 2: Find the Median

Since there are 9 numbers, which is odd, the median is the middle number. The median is the 5th number in the ordered data set. Therefore, the median is 33.

Step 3: Calculate the Sample Mean

The sample mean \( \bar{x} \) is calculated using the formula: \[ \bar{x} = \frac{\sum_{i=1}^{n} x_i}{n} \] where \( n = 9 \) and \( x_i \) represents each value in the data set. Sum of the data values is \( 18 + 21 + 21 + 27 + 33 + 37 + 39 + 42 + 45 = 283 \). Thus, the sample mean is \( \frac{283}{9} = 31.444 \).

Step 4: Find Deviations from the Mean

Calculate the deviation of each data point from the mean: \( 18 - 31.444, 21 - 31.444, 21 - 31.444, 27 - 31.444, 33 - 31.444, 37 - 31.444, 39 - 31.444, 42 - 31.444, 45 - 31.444 \), which results in: \(-13.444, -10.444, -10.444, -4.444, 1.556, 5.556, 7.556, 10.556, 13.556\).

Step 5: Calculate the Variance

Use the deviations to compute the sample variance \( s^2 \) with the formula: \[ s^2 = \frac{\sum (x_i - \bar{x})^2}{n - 1} \]. Compute the squared deviations and their sum: \( (-13.444)^2 + (-10.444)^2 + (-10.444)^2 + (-4.444)^2 + 1.556^2 + 5.556^2 + 7.556^2 + 10.556^2 + 13.556^2 = 1006.888 \). Therefore, the sample variance is \( \frac{1006.888}{8} = 125.861 \).

Key Concepts

Median

The median is a measure of central tendency that indicates the middle value of a data set when it is ordered in ascending or descending order. To find the median of a data set, follow these simple steps:

• Sort the data set in order from least to greatest.
• If the number of data points is odd, the median is the middle number.
• If the number of data points is even, the median is the average of the two middle numbers.

In our given problem, we had a list of nine numbers:
18, 21, 21, 27, 33, 37, 39, 42, 45.
Since nine is an odd number, the median is the fifth number after sorting, which is 33.
This means that half of the numbers are smaller than 33 and half are larger, making it an excellent representation of the dataset's central value.

Sample Mean

The sample mean is another central tendency measure, representing the average value of a sample. Calculating the sample mean involves the following steps:

• Add up all the numbers in the data set to obtain the sum.
• Divide the sum by the number of data points.

For the given data set of seed counts:
18, 21, 21, 27, 33, 37, 39, 42, 45,
the sum is 283, and there are 9 data points. To find the mean:\[\bar{x} = \frac{283}{9} = 31.444\]Hence, the mean number of seeds per flower head in this sample is approximately 31.444.
This mean gives us an idea of the average seed count, providing a useful summary of the overall dataset.

Sample Variance

The sample variance is a measure of how dispersed or spread out the numbers in a data set are relative to the mean. It's calculated using the following steps:

• Subtract the mean from each data value to find the deviation of each data point.
• Square each deviation.
• Sum all the squared deviations.
• Divide this sum by one less than the total number of values in the data set (i.e., \(n - 1\)), where \(n\) is the sample size.

Applying these steps to the given data, with a mean of 31.444 and 9 data points:
The sum of the squared deviations is 1006.888. The sample variance is computed as:\[s^2 = \frac{1006.888}{8} = 125.861\]This variance indicates the average squared deviation from the mean, giving insight into the extent to which individual data points differ from the average in the sample.
A higher variance means more spread in the data, while a lower variance indicates that the data points tend to be closer to the mean.