Question

For Excercises 87 and \(88,\) use the following table, which contains the high and low temperatures in degrees Fahrenheit each day from March 14 to March 20 in Nome, Alaska. $$\begin{array}{c|c|c|c|c|c|c|c} & 3 / 14 & 3 / 15 & 3 / 16 & 3 / 17 & 3 / 18 & 3 / 19 & 3 / 20 \\ \hline \text { High } & -6 & 0 & 19 & 25 & 26 & 29 & 26 \\ \hline \text { Low } & -30 & -25 & -11 & -1 & 13 & 17 & 9 \\ \hline \end{array}$$ Find the mean, median, and mode high temperature.

Short answer

Answer: The mean high temperature is \(17^{\circ}\text{F}\), the median high temperature is \(19^{\circ}\text{F}\), and the mode high temperature is \(26^{\circ}\text{F}\).

Step-by-step solution

Arrange the temperatures in ascending order

To find the median and mode, we first need to arrange the given high temperatures for the week in ascending order. Here's the ordered list: \(-6, 0, 19, 25, 26, 29, 26\)

Calculate the mean

To find the mean (average) temperature, we'll add up all the temperatures and then divide the sum by the number of days (7). Thus: Mean \(= \dfrac{-6 + 0 + 19 + 25 + 26 + 29 + 26}{7} = \dfrac{119}{7} \approx 17\)

Find the median

Now that the temperatures are arranged in ascending order, we can easily find the median (middle value). Since there are 7 days, we can take the 4th value as the median: Median \(= 19\)

Find the mode

Finally, we need to find the mode (the value that occurs most frequently). In the list of ordered temperatures, we notice that "26" appears twice, which is more frequent than any other temperature: Mode \(= 26\) The mean high temperature is \(17^{\circ}\text{F}\), the median high temperature is \(19^{\circ}\text{F}\), and the mode high temperature is \(26^{\circ}\text{F}\).

Key concepts

Statistical Measures

Statistical measures are crucial for summarizing and understanding data in a more manageable form. In the context of temperature data, three commonly used statistical measures are the mean, median, and mode.

The mean, often referred to as the average, is the sum of all data points divided by the number of points. It gives us a general idea of the central value of the data. To calculate the mean temperature, you simply add up all the high temperatures for a given period and divide by the number of observations.

The median represents the middle value when data points are arranged in ascending or descending order. If there’s an odd number of data points, the median is the middle one. If there’s an even number, it’s the average of the two middle values. For temperature, this tells us the mid-point of temperature ranges within a dataset, offering insight into central tendencies that might not be affected by extreme values, unlike the mean.

The mode is the most frequently occurring value within a dataset. In terms of temperature, the mode is the temperature that appeared most often during a specific timeframe. It can be particularly informative when the data has one or more values that occur significantly more often than others.

Data Analysis

Data analysis involves examining, cleaning, transforming, and modeling data to discover useful information, draw conclusions, and support decision-making. Analyzing temperature data can be quite complex, depending on the context, since temperature is influenced by many factors and can vary widely from day to day.

In the exercise at hand, the analysis process started by arranging the temperatures and then calculating the statistical measures. During analysis, these measures help to extract insights from the data. For instance, the mean temperature could suggest that a week was generally warm or cold, the median could reflect a typical day's temperature devoid of outliers, and the mode could indicate the most stable temperature pattern experienced.

Temperature Data

Temperature data are readings that show how hot or cold the environment is at different times. Such data are usually recorded in degrees and it's vital to use a consistent system, like Fahrenheit or Celsius, for the entire dataset. In our example, the temperatures are in Fahrenheit (°F).

Understanding temperature data requires considering the context in which the temperatures were measured. For example, daily high and low temperatures in Nome, Alaska reflect not only the weather patterns but also the seasonal change, geography, and climate of the region. Statistical measures applied to this data can provide us with insights about typical weather conditions or reveal an unusual climate behavior during the period in question.

Arranging Data

Before you can find the median or mode, it is essential to arrange the data in order, usually ascending or descending. This step helps to visualize and understand how the data points are distributed.

In the exercise about temperature data, arranging the high temperatures in ascending order made it possible to immediately identify the median temperature as the middle value. For finding the mode, this ordering also aided in quickly spotting which value appeared most frequently. This technique not only simplifies the process of finding statistical measures but also helps in spotting patterns, trends, and outliers within the dataset. Arranging data lays the foundation for a structured and insightful analysis.