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Chapter 3: Problem 2

Consider a sample with data values of \(10,20,21,17,16,\) and \(12 .\) Compute the mean and median.

Short Answer

Expert verified
The mean is 16, and the median is 16.5.

Step by step solution

01

Understanding the Mean

The mean is the average of all the numbers in a data set. To find it, we add up all the numbers and then divide by the number of values in the data set.
02

Calculating the Mean

Add all the numbers in the data set: \[ 10 + 20 + 21 + 17 + 16 + 12 \] This sum equals 96. Next, divide the sum by the number of data points, which is 6: \[ \frac{96}{6} = 16 \] Thus, the mean is 16.
03

Understanding the Median

The median is the middle value of a data set when the numbers are written in order. If there is an even number of observations, the median is the average of the two middle numbers.
04

Arranging the Data for Median

First, arrange the data in ascending order: \[ 10, 12, 16, 17, 20, 21 \] Since there are six numbers (an even number), the median will be the average of the third and fourth numbers.
05

Calculating the Median

The third and fourth numbers in the ordered data set are 16 and 17. Find their average: \[ \frac{16 + 17}{2} = 16.5 \] Thus, the median is 16.5.

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Key Concepts

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

Mean
The mean is a fundamental concept in statistics and is often what people refer to as the "average." Calculating the mean is straightforward:
  • Add all numbers in the data set.
  • Divide the total by the number of elements in the set.
In our example, we have the numbers 10, 20, 21, 17, 16, and 12. First, add these numbers to get 96. Then divide by the number of data points, which is 6:\[ \frac{96}{6} = 16 \]The mean of this data set is 16. The mean gives us a central value which helps us understand the overall tendency of the numbers. It's like finding the balancing point of a seesaw where all our data values are placed.
Median
The median is the middle value in a list of numbers sorted in order. It helps us find the center of a data distribution, unaffected by extreme values or outliers. Here's how you find it:
  • Order your data from smallest to largest.
  • If the number of data points is odd, the median is the middle number.
  • If it's even, take the average of the two middle numbers.
In our case, we arrange the data: 10, 12, 16, 17, 20, 21. There are six numbers, which is an even count. The median will be the average of the third (16) and fourth (17) numbers:\[ \frac{16 + 17}{2} = 16.5 \]Thus, the median is 16.5. Unlike the mean, the median isn't skewed by very high or low values, making it a robust measure of central tendency.
Data Analysis
Data analysis involves inspecting, cleaning, and modeling data to extract useful information and insights. When we compute the mean and median, we're engaging in a simple form of data analysis aimed at understanding the central tendency and typical values in our data set.
  • By calculating the mean, we determine a typical value that represents the data.
  • Through the median, we gain insight into the center of our data distribution, particularly useful for skewed data.
In everyday applications, these measures can help us:
  • Understand trends in datasets.
  • Compare different data groups.
  • Make informed decisions based on typical data behavior.
Data analysis is crucial for interpreting data and making data-driven decisions. Whether you're handling survey responses, exam scores, or sales figures, understanding these statistics is invaluable.

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Most popular questions from this chapter

The high costs in the California real estate market have caused families who cannot afford to buy bigger homes to consider backyard sheds as an alternative form of housing expansion. Many are using the backyard structures for home offices, art studios, and hobby areas as well as for additional storage. The mean price of a customized wooden, shingled backyard structure is \(\$ 3100\) ( Newsweek , September 29,2003 ). Assume that the standard deviation is \(\$ 1200\). a. What is the \(z\) -score for a backyard structure costing \(\$ 2300 ?\) b. What is the \(z\) -score for a backyard structure costing \(\$ 4900 ?\) c. Interpret the \(z\) -scores in parts (a) and (b). Comment on whether either should be considered an outlier. d. The Newsweek article described a backyard shed-office combination built in Albany, California, for \(\$ 13,000\). Should this structure be considered an outlier? Explain.

The U.S. Census Bureau provides statistics on family life in the United States, including the age at the time of first marriage, current marital status, and size of household (http://www.census.gov, March 20, 2006). The following data show the age at the time of first marriage for a sample of men and a sample of women. \\[ \begin{array}{lcccccccc} \text { Men } & 26 & 23 & 28 & 25 & 27 & 30 & 26 & 35 & 28 \\ & 21 & 24 & 27 & 29 & 30 & 27 & 32 & 27 & 25 \\ \text { Women } & 20 & 28 & 23 & 30 & 24 & 29 & 26 & 25 & \\ & 22 & 22 & 25 & 23 & 27 & 26 & 19 & & \end{array} \\] a. Determine the median age at the time of first marriage for men and women. b. Compute the first and third quartiles for both men and women. c. Twenty-five years ago the median age at the time of first marriage was 25 for men and 22 for women. What insight does this information provide about the decision of when to marry among young people today?

The following times were recorded by the quarter-mile and mile runners of a university track team (times are in minutes). \\[\begin{array}{lrrrrr} \text {Quarter-Mile Times:} & .92 & .98 & 1.04 & .90 & .99 \\ \text {Mile Times:} & 4.52 & 4.35 & 4.60 & 4.70 & 4.50 \end{array}\\] After viewing this sample of running times, one of the coaches commented that the quartermilers turned in the more consistent times. Use the standard deviation and the coefficient of variation to summarize the variability in the data. Does the use of the coefficient of variation indicate that the coach's statement should be qualified?

A department of transportation's study on driving speed and mileage for midsize automobiles resulted in the following data. $$\begin{array}{l|llllllllll} \text { Driving Speed } & 30 & 50 & 40 & 55 & 30 & 25 & 60 & 25 & 50 & 55 \\ \hline \text { Mileage } & 28 & 25 & 25 & 23 & 30 & 32 & 21 & 35 & 26 & 25 \end{array}$$ Compute and interpret the sample correlation coefficient.

Consider a sample with data values of \(27,25,20,15,30,34,28,\) and \(25 .\) Compute the range, interquartile range, variance, and standard deviation.
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