Problem 1 - Calculus for Biology and Medicine

Question

The following data represent the number of aphids per plant found in a sample of 10 plants: $$ 17,13,21,47,3,6,12,25,0,18 $$ Find the median, the sample mean, and the sample variance.

Step-by-Step Solution

Answer

Median is 15, mean is 16.2, and variance is approximately 179.91.

Step 1: Organize the Data

First, list the data in ascending order for easier calculation of the median. The data in order is: 0, 3, 6, 12, 13, 17, 18, 21, 25, 47.

Step 2: Find the Median

Since there are 10 data points (an even number), the median is the average of the 5th and 6th values in the ordered list. Calculate the median as follows: \[ \text{Median} = \frac{13 + 17}{2} = 15 \]

Step 3: Calculate the Sample Mean

Add all the values together and divide by the number of values (10). Calculating this, we have: \[ \text{Mean} = \frac{0 + 3 + 6 + 12 + 13 + 17 + 18 + 21 + 25 + 47}{10} = \frac{162}{10} = 16.2 \]

Step 4: Calculate the Sample Variance

First, find the value of each data point minus the mean (16.2), square those results, then find the average of those squares. 1. Calculate deviations: - (0 - 16.2)^2 = 262.44 - (3 - 16.2)^2 = 174.24 - (6 - 16.2)^2 = 104.04 - (12 - 16.2)^2 = 17.64 - (13 - 16.2)^2 = 10.24 - (17 - 16.2)^2 = 0.64 - (18 - 16.2)^2 = 3.24 - (21 - 16.2)^2 = 23.04 - (25 - 16.2)^2 = 77.44 - (47 - 16.2)^2 = 947.24 2. Sum of squared deviations: 262.44 + 174.24 + 104.04 + 17.64 + 10.24 + 0.64 + 3.24 + 23.04 + 77.44 + 947.24 = 1619.20 3. Divide by the number of data points minus one \[ \text{Variance} = \frac{1619.20}{10 - 1} = \frac{1619.20}{9} \approx 179.91 \]

Key Concepts

Understanding the Median

The median is a measure of central tendency that is often used to find the middle value in a set of data. It is especially useful when dealing with skewed distributions as it isn't affected by extreme values.
The process of finding the median involves ordering the data from the smallest to the largest value. Once the data is ordered, identify the middle value. If the number of data points is odd, the median is the middle number. However, if the number of values is even, as in our exercise, the median is the average of the two middle numbers.
For example, with the aphid data set of 10 ordered values: 0, 3, 6, 12, 13, 17, 18, 21, 25, 47, the median is calculated by finding the average of the 5th and 6th numbers (13 and 17), thus:

• Median = (13 + 17) / 2 = 15

This means that 50% of the aphids per plant in the data set were less than or equal to 15, and 50% were greater than or equal to 15.

Calculating the Sample Mean

The sample mean is the average of all data points in a sample and is calculated by adding all the observations together, then dividing by the number of observations. It's a useful measure to determine the "center" of the data set.
For our given data set, the formula to compute the sample mean is:

• Sample Mean = (sum of all data values) / (number of data values)

Applying this to the aphid data (0, 3, 6, 12, 13, 17, 18, 21, 25, 47), we find the sum is 162. Dividing this sum by 10 (the number of aphid counts), we have:

• Mean = 162 / 10 = 16.2

Thus, on average, each plant has about 16.2 aphids. The sample mean gives an overall sense of the average number of aphids per plant across the sample.

Explaining Sample Variance

Sample variance measures the spread or variability of the data points around the mean. It shows how much the values differ from the mean, providing insights into the data's consistency.
Calculating the sample variance involves the following steps:

• Find the deviation of each data point from the mean.
• Square each deviation to make them positive.
• Sum all these squared deviations.
• Divide the sum by the number of observations minus one (n-1), where n is the total number of data points.

For the given data set, with a mean of 16.2, each deviation squared is summed to get 1619.20. We then divide by 9 (10 - 1) as there are 10 data points:

• Variance = 1619.20 / 9 ≈ 179.91

This variance suggests that the number of aphids on the plants is widely spread from the mean, indicating variability in the aphid counts per plant.