Problem 44 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

The relative humidity, in percent, in the morning for the months of January through December in Boston follows: $$\begin{array}{llllll}68 & 67 & 69 & 69 & 71 & 73 \\ 74 & 76 & 79 & 77 & 74 & 70\end{array}$$ Find the average and the median of these humidity readings.

Step-by-Step Solution

Answer

The average humidity in Boston for the months of January through December is approximately 71.42%, and the median humidity is 70.5%.

Step 1: Calculate the sum of all humidity readings.

Add all the given humidity readings to find the total: \(68 + 67 + 69 + 69 + 71 + 73 + 74 + 76 + 79 + 77 + 74 + 70 = 857\)

Step 2: Calculate the average.

Divide the sum by the total number of readings (in this case, 12 months) to find the average humidity: \(\text{Average} = \frac{857}{12} \approx 71.42\)

Step 3: Arrange the data in an ordered list.

Order the humidity readings from least to greatest: \(67, 68, 69, 69, 70, 71, 73, 74, 74, 76, 77, 79\)

Step 4: Find the median.

There are two ways to find the median based on the number of readings: 1. If there are an odd number of readings, the median is the middle value. 2. If there are an even number of readings (like in this case - 12 readings), find the average of the middle two values. So, we will find the average of the middle two values in our ordered list: \((70 + 71)/2 = 70.5\) The average humidity in Boston is approximately 71.42%, and the median humidity is 70.5%.

Key Concepts

Average Calculation

In applied mathematics, calculating the average gives you a central idea of a data set. When we talk about the average, we usually mean the arithmetic mean. It is a measure that represents the center of a dataset.

To find the average of humidity readings, follow these steps:

• Add together all the readings to get the sum, which in this case is 857.
• Count how many readings are there; here, we have 12 months.
• Divide the total sum by the number of readings to get the average.

These steps together provide the equation for the average:\[\text{Average} = \frac{\text{Total Sum of Readings}}{\text{Number of Readings}} = \frac{857}{12} \approx 71.42\]
The average gives us an idea of what a typical humidity reading might be in Boston over the year, smoothing out the high and low variations.

Median Calculation

The median is another type of average that gives us the central value of a dataset. It’s particularly useful in statistical analysis when the dataset contains outliers that may skew the average.

To find the median of the provided humidity readings:

• Arrange the data in ascending order.
• If the number of data points is odd, the median is the middle number.
• If it is even, as in this case with 12 readings, the median is determined by averaging the two middle numbers.

For our example, once the readings are ordered:\[67, 68, 69, 69, 70, 71, 73, 74, 74, 76, 77, 79\]The two middle numbers are 70 and 71, so the median is:\[\text{Median} = \frac{70 + 71}{2} = 70.5\]
The median here gives us an idea of what the typical humidity reading is when the data is centered, unaffected by extreme highs or lows.

Statistics in Education

Statistics is a significant part of applied mathematics, helping students understand how to interpret data in real-world contexts. In the context of education, it equips learners with skills to collect, analyze, and interpret numerical data.

Understanding statistics through concepts like average and median helps students:

• Make informed decisions based on data.
• Analyze surveys or research results.
• Understand trends and patterns.

Statistics in education often uses everyday examples, such as weather data or exam scores, to illustrate key points.
By practicing with statistical measures like average and median, students gain a critical understanding of data representation, fostering analytical thinking skills and preparation for future academic or professional endeavors. These concepts demystify numbers, transforming them into stories that describe everyday phenomena, like humidity changes in Boston throughout the year.