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Chapter 3: Problem 10

The following data set belongs to a sample: \(\begin{array}{llllll}14 & 18 & -10 & 8 & 8 & -16\end{array}\) Calculate the mean, median, and mode.

Short Answer

Expert verified
The mean of the data set is approximately 3.67, the median is 8, and the mode is 8

Step by step solution

01

Calculate the Mean

To calculate the mean, add up all the numbers in the data set and then divide by the count of numbers. In this case, \(14 + 18 - 10 + 8 + 8 - 16 = 22\), and there are six numbers, so the mean is \(22/6 = 3.67\)
02

Calculate the Median

To find the median, you need to arrange the numbers in order and find the middle value. Since there are six numbers, the median is the average of the third and fourth numbers. If arranged in ascending order, the series become \(-16,-10,8,8,14,18\). Hence, the median is \((8+8)/2 = 8\)
03

Calculate the Mode

The mode is the number that appears most often. In this case, the number 8 appears twice so the mode is 8

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Key Concepts

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

Mean Calculation
The mean, often referred to as the average, is a central value that provides a general idea of the data set. To calculate the mean, you should follow these simple steps:
  • Add all the numbers in the data set together. This sum represents the total of all observed values.
  • Count how many numbers there are in the data set, which will be used to divide the total sum.
  • Finally, divide the sum by the count of numbers. This result will be the mean.
In our example, the data set is { 14, 18, -10, 8, 8, -16 }, and their sum is 22. Since there are six numbers in total, we divide 22 by 6 to find the mean, resulting in a mean value of approximately 3.67. The mean provides a good estimation of the typical value in the data set.
Median Calculation
Finding the median can be very informative, as it identifies the middle value of a data set when it is ordered. Here's how to calculate it:
  • First, sort the numbers in ascending order. This step is crucial for locating the center value accurately.
  • If the count of numbers is odd, the median is simply the middle number.
  • If the count is even, the median is the average of the two middle numbers.
In the given set { 14, 18, -10, 8, 8, -16 }, arranging the numbers in order gives us {-16, -10, 8, 8, 14, 18}. Since there are six numbers, the median is calculated as the average of the third and fourth numbers: (8+8)/2, equaling 8. The median is particularly useful for understanding the distribution of a data set, especially when it has outliers.
Mode Calculation
The mode is an important measure within descriptive statistics, highlighting the most frequently occurring number in a data set. Here is how to determine it:
  • Examine the data set to find out which number(s) appear the most times. A data set may have one, more than one, or no mode at all.
  • If one number appears more often than all others, that number is the mode.
  • In cases where multiple numbers appear with the same highest frequency, each of these numbers is a mode (multimodal).
In our example { 14, 18, -10, 8, 8, -16 }, the number 8 appears twice, more than any other number. Therefore, the mode of this data set is 8. The mode is especially useful in understanding trends within the data.

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Most popular questions from this chapter

Although the standard workweek is 40 hours a week, many people work a lot more than 40 hours a week. The following data give the numbers of hours worked last week by 50 people. \(\begin{array}{llllllllll}40.5 & 41.3 & 41.4 & 41.5 & 42.0 & 42.2 & 42.4 & 42.4 & 42.6 & 43.3 \\ 43.7 & 43.9 & 45.0 & 45.0 & 45.2 & 45.8 & 45.9 & 46.2 & 47.2 & 47.5 \\ 47.8 & 48.2 & 48.3 & 48.8 & 49.0 & 49.2 & 49.9 & 50.1 & 50.6 & 50.6 \\ 50.8 & 51.5 & 51.5 & 52.3 & 52.3 & 52.6 & 52.7 & 52.7 & 53.4 & 53.9 \\\ 54.4 & 54.8 & 55.0 & 55.4 & 55.4 & 55.4 & 56.2 & 56.3 & 57.8 & 58.7\end{array}\) a. The sample mean and sample standard deviation for this data set are \(49.012\) and \(5.080\), respectively. Using Chebyshev's theorem, calculate the intervals that contain at least \(75 \%, 88.89 \%\) and \(93.75 \%\) of the data. b. Determine the actual percentages of the given data values that fall in each of the intervals that you calculated in part a. Also calculate the percentage of the data values that fall within one standard deviation of the mean. c. Do you think the lower endpoints provided by Chebyshev's theorem in part a are useful for this problem? Explain your answer. d. Suppose that the individual with the first number \((54.4)\) in the fifth row of the data is a workaholic who actually worked \(84.4\) hours last week and not \(54.4\) hours. With this change now \(\bar{x}=49.61\) and \(s=7.10\). Recalculate the intervals for part a and the actual percentages for part b. Did your percentages change a lot or a little? e. How many standard deviations above the mean would you have to go to capture all 50 data values? What is the lower bound for the percentage of the data that should fall in the interval, according to the Chebyshev theorem.

The following data give the numbers of computer keyboards assembled at the Twentieth Century Electronics Company for a sample of 25 days. \(\begin{array}{llllllllll}45 & 52 & 48 & 41 & 56 & 46 & 44 & 42 & 48 & 53 \\\ 51 & 53 & 51 & 48 & 46 & 43 & 52 & 50 & 54 & 47 \\ 44 & 47 & 50 & 49 & 52 & & & & & \end{array}\) Prepare a box-and-whisker plot. Comment on the skewness of these data.

Using the population formulas, calculate the mean, variance, and standard deviation for the following grouped data. \begin{tabular}{l|ccccc} \hline\(x\) & \(2-4\) & \(5-7\) & \(8-10\) & \(11-13\) & \(14-16\) \\ \hline\(f\) & 5 & 9 & 14 & 7 & 5 \\ \hline \end{tabular}

Briefly describe how the three quartiles are calculated for a data set. Illustrate by calculating the three quartiles for two examples, the first with an odd number of observations and the second with an even number of observations.

The mean life of a certain brand of auto batteries is 44 months with a standard deviation of 3 months. Assume that the lives of all auto batteries of this brand have a bell-shaped distribution. Using the empirical rule, find the percentage of auto batteries of this brand that have a life of if. 41 to 47 months b. 38 to 50 months c. 35 to 53 months
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