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Chapter 3: Problem 9

How hot does it get in Death Valley? The following data are taken from a study conducted by the National Park System, of which Death Valley is a unit. The ground temperatures \(\left({ }^{\circ} \mathrm{F}\right)\) were taken from May to November in the vicinity of Furnace Creek. \(\begin{array}{lllllll}146 & 152 & 168 & 174 & 180 & 178 & 179 \\ 180 & 178 & 178 & 168 & 165 & 152 & 144\end{array}\) Compute the mean, median, and mode for these ground temperatures.

Short Answer

Expert verified
Mean: 168.71掳F, Median: 171掳F, Mode: 178掳F.

Step by step solution

01

Arrange the Data Set

First, arrange the ground temperature data in ascending order to facilitate computation of the mean, median, and mode. The data set is: \( 144, 146, 152, 152, 165, 168, 168, 174, 178, 178, 178, 179, 180, 180 \).
02

Calculate the Mean

The mean is the average of all the ground temperatures. To find it, sum all the temperatures and divide by the number of data points. Sum: \( 144 + 146 + 152 + 152 + 165 + 168 + 168 + 174 + 178 + 178 + 178 + 179 + 180 + 180 = 2362 \). Number of data points: \( 14 \). Mean: \( \frac{2362}{14} \approx 168.71 \).
03

Determine the Median

The median is the middle value in the ordered data set. For an even number of data points, it is the average of the two middle numbers. The middle values in the data set are the 7th and 8th values: \( 168 \) and \( 174 \). Median: \( \frac{168 + 174}{2} = 171 \).
04

Identify the Mode

The mode is the number that appears most frequently in the set. By inspecting the data, \( 178 \) appears most frequently (three times). Thus, the mode is \( 178 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the Mean
The mean, often referred to as the average, is a fundamental statistic used to summarize a set of data with a single value that represents its central tendency. To determine the mean of a set of numbers, you would follow these steps:
  • Add together all the numbers in your data set to get a total sum.
  • Count how many numbers are in your data set.
  • Divide the total sum by the number of data points to get the mean.
In our exercise regarding the ground temperatures in Death Valley, the mean was calculated by summing all the recorded temperatures: \(144 + 146 + 152 + 152 + 165 + 168 + 168 + 174 + 178 + 178 + 178 + 179 + 180 + 180 = 2362\). With 14 data points, the mean is calculated as:\[\frac{2362}{14} \approx 168.71\]
Therefore, the mean of the ground temperatures is approximately 168.71掳F. The mean gives us a generalized idea of how hot it can get in Death Valley by considering all temperature readings equally.
Exploring the Median
The median offers another way to represent the central tendency of a data set, distinct from the mean. It provides the middle value when your data is arranged in numerical order. Unlike the mean, the median is less affected by extremely high or low values, which can offer a more accurate picture of what is typical in your data.
To find the median:
  • Sort your data from the smallest to the largest value.
  • If your data set has an odd number of observations, the median is the middle number.
  • If your data set has an even number of observations, calculate the median by averaging the two middle numbers.
For our exercise, the data was arranged in ascending order, and with 14 values in total, the two middle numbers are the 7th and 8th values: 168 and 174. The median is calculated as: \[\frac{168 + 174}{2} = 171\]
So, the median ground temperature in Death Valley is 171掳F, offering insight into typical conditions that isn't skewed by any extreme outliers.
Decoding the Mode
The mode is a straightforward statistical concept indicating the number or numbers that appear most frequently in a data set. Unlike the mean and median, a data set can have multiple modes or none at all if each number appears with the same frequency. To find the mode:
  • Organize your data and count how many times each number appears.
  • Identify the number(s) with the highest frequency.
In the Death Valley temperature data, by arranging the numbers in order, it is evident that 178 appears three times: more than any other number in the set. Therefore, the mode is 178掳F.
This method of finding the mode can quickly show the most common temperatures, helping to identify conditions that occur most frequently, which is crucial in understanding the typical environment of a place like Death Valley.

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Most popular questions from this chapter

Some data sets include values so high or so low that they seem to stand apart from the rest of the data. These data are called outliers. Outliers may represent data collection errors, data entry errors, or simply valid but unusual data values. It is important to identify outliers in the data set and examine the outliers carefully to determine if they are in error. One way to detect outliers is to use a box-and-whisker plot. Data values that fall beyond the limits Lower limit: \(Q_{1}-1.5 \times(I Q R)\) Upper limit: \(Q_{3}+1.5 \times(I Q R)\) where \(I Q R\) is the interquartile range, are suspected outliers. In the computer software package Minitab, values beyond these limits are plotted with asterisks \(\left(^{*}\right)\) Students from a statistics class were asked to record their heights in inches. The heights (as recorded) were \(\begin{array}{llllllllllll}65 & 72 & 68 & 64 & 60 & 55 & 73 & 71 & 52 & 63 & 61 & 74 \\ 69 & 67 & 74 & 50 & 4 & 75 & 67 & 62 & 66 & 80 & 64 & 65\end{array}\) (a) Make a box-and-whisker plot of the data. (b) Find the value of the interquartile range \((I Q R)\). (c) Multiply the \(I Q R\) by \(1.5\) and find the lower and upper limits. (d) Are there any data values below the lower limit? above the upper limit? List any suspected outliers. What might be some explanations for the outliers?

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