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Chapter 3: Problem 1

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

Short Answer

Expert verified
The mean is 15 and the median is 16.

Step by step solution

01

Calculate the Mean

The mean is the average value of a dataset and it's calculated by adding together all the values in the data set and then dividing by the number of data values. For this dataset \[ 10, 20, 12, 17, 16 \]add the values: \[ 10 + 20 + 12 + 17 + 16 = 75 \]Then divide by the number of values, which is 5:\[ \text{Mean} = \frac{75}{5} = 15 \]
02

Finding the Median

The median is the middle value in a sorted list of numbers. First, arrange the data in increasing order:\[ 10, 12, 16, 17, 20 \]As there are 5 data values, which is an odd number, the median is the third number in this ordered list, which is 16.

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Key Concepts

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

Mean Calculation
When you look at a set of numbers and need to find a typical value, the mean offers a concise summary. The mean, also known as the average, gives you an idea of what the central value of a data set is. To calculate the mean, follow these simple steps:

  • Add all the numbers in the dataset together to get a single total value.
  • Count how many numbers are in your dataset.
  • Divide the total value by the number of numbers you have.
For example, in a dataset composed of the values 10, 20, 12, 17, 16, after adding them all up, you get 75. Since there are 5 numbers in total, divide 75 by 5 to get 15. Hence, the mean of this dataset is 15.

This calculation smooths out individual extremes and articulates a central tendency. It's a fundamental tool in data analysis that helps transform raw data into interpretable information.
Median Calculation
The median is another key metric used in data analysis to identify the center of a dataset. Unlike the mean, the median is less affected by outliers, and thus can sometimes provide a better sense of a typical value.

Finding the median involves organizing your numbers from smallest to largest:

  • First, sort your dataset.
  • For an odd number of values, the median is the number that falls exactly in the middle of your list.
  • If your dataset has an even number of values, find the two middle numbers, add them, and divide by two to get the median.

In the dataset containing 10, 20, 12, 17, 16, the numbers rearranged in ascending order are 10, 12, 16, 17, 20. Because there are 5 numbers, the median is the middle number, which is 16. The median is a valuable statistic, especially when there are outliers that might skew the mean. It accurately represents the central point of a dataset in many situations.
Data Analysis
Data analysis involves processing data to extract useful insights. Understanding both the mean and median is crucial for data analysis, as they reveal different aspects of the data's distribution.

When analyzing data:
  • Use the mean to find the average value, which helps in understanding the overall trend of the dataset.
  • Use the median to comprehend the central tendency without being affected by extremely high or low values.
  • Compare the mean and median to identify possible skewness in the data. If these values differ significantly, it may indicate an asymmetric distribution.
Understanding how to calculate and interpret the mean and median allows for a more comprehensive analysis of data, making you better equipped to draw insightful conclusions. Data analysis is a powerful skill utilized across various fields and industries, helping to make data-driven decisions and foster informed strategies.

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Most popular questions from this chapter

A panel of economists provided forecasts of the U.S. economy for the first six months of 2007 (The Wall Street Journal, January 2,2007 ). The percent changes in the gross domestic product (GDP) forecasted by 30 economists are as follows. $$\begin{array}{cccccccccc} 2.6 & 3.1 & 2.3 & 2.7 & 3.4 & 0.9 & 2.6 & 2.8 & 2.0 & 2.4 \\ 2.7 & 2.7 & 2.7 & 2.9 & 3.1 & 2.8 & 1.7 & 2.3 & 2.8 & 3.5 \\ 0.4 & 2.5 & 2.2 & 1.9 & 1.8 & 1.1 & 2.0 & 2.1 & 2.5 & 0.5 \end{array}$$ a. What is the minimum forecast for the percent change in the GDP? What is the maximum? b. Compute the mean, median, and mode. c. \(\quad\) Compute the first and third quartiles. d. Did the economists provide an optimistic or pessimistic outlook for the U.S. economy? Discuss.

A data set has a first quartile of 42 and a third quartile of \(50 .\) Compute the lower and upper limits for the corresponding box plot. Should a data value of 65 be considered an outlier?

The Los Angeles Times regularly reports the air quality index for various areas of Southern California. A sample of air quality index values for Pomona provided the following data: \(28,42,58,48,45,55,60,49,\) and 50 a. Compute the range and interquartile range. b. Compute the sample variance and sample standard deviation. c. \(\quad\) A sample of air quality index readings for Anaheim provided a sample mean of 48.5 a sample variance of \(136,\) and a sample standard deviation of \(11.66 .\) What comparisons can you make between the air quality in Pomona and that in Anaheim on the basis of these descriptive statistics?

The grade point average for college students is based on a weighted mean computation. For most colleges, the grades are given the following data values: \(A(4), B(3), C\) \((2), D(1),\) and \(F(0) .\) After 60 credit hours of course work, a student at State University earned 9 credit hours of \(A, 15\) credit hours of \(B, 33\) credit hours of \(C,\) and 3 credit hours of \(\mathrm{D}\) a. Compute the student's grade point average. b. Students at State University must maintain a 2.5 grade point average for their first 60 credit hours of course work in order to be admitted to the business college. Will this student be admitted?

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