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Chapter 2: Problem 66

Which is the least, the mean, the mode, and the median of the data set? 56; 56; 56; 58; 59; 60; 62; 64; 64; 65; 67

Short Answer

Expert verified
Mean: 60.64, Mode: 56, Median: 60.

Step by step solution

01

Sort the Data Set

The given data set is already sorted in ascending order: 56, 56, 56, 58, 59, 60, 62, 64, 64, 65, 67.
02

Find the Mean

To find the mean of the data set, add all the numbers together and divide by the total number of data points. \[\text{Mean} = \frac{56 + 56 + 56 + 58 + 59 + 60 + 62 + 64 + 64 + 65 + 67}{11} = \frac{667}{11} = 60.64\]
03

Identify the Mode

The mode is the number that appears most frequently in the data set. Looking through the numbers, 56 appears three times, which is more frequent than any other number, so the mode is 56.
04

Find the Median

The median is the middle number when the data set is ordered. Since there are 11 numbers (which is odd), the median is the sixth number in this ordered set, which is 60.

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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 called the average, is one of the most common ways to represent a data set with a single value. To compute the mean of a given data set, follow these steps:

  • Step 1: Sum all the numbers in the data set. For our example, this means adding up all the values: 56 + 56 + 56 + 58 + 59 + 60 + 62 + 64 + 64 + 65 + 67, which equals 667.
  • Step 2: Count how many numbers are in the data set. In our case, there are 11 numbers.
  • Step 3: Divide the total sum from Step 1 by the count from Step 2 to find the mean. So, \[\text{Mean} = \frac{667}{11} = 60.64 \approx 60.6\]
Remember, the mean gives you an idea of what the average or central value in your data is, but it may not always represent the data perfectly, especially if there are outliers.
Mode Identification
The mode is an important descriptive statistic that tells us about the most common value(s) within a given data set. To find the mode:

  • Look for the number that appears most frequently in your data. In our sorted set of numbers, 56 is the most frequent since it appears three times, whereas other numbers like 58, 59, and others appear once or twice.

  • If two or more numbers appear with the same highest frequency, all such numbers are modes, and the data set is termed multimodal.

  • If no number repeats, then the data set is considered amodal because it doesn't have a mode.

    • The mode is helpful in understanding the most common or repetitive values in a data set, providing insight into distribution patterns.
    Median Determination
    The median provides a measure of the center of the data set and is a valuable statistic, especially when dealing with skewed distributions. To find the median:

    • First, ensure your data is sorted. Since ours already is: 56, 56, 56, 58, 59, 60, 62, 64, 64, 65, 67.
    • Count the number of data points, which is 11 in this case.
    • If the number of data points is odd, the median is the middle number. This is the 6th number here, which is 60.\[\text{Median} = 60\]
    • If the number of data points was even, the median would be calculated as the average of the two middle numbers.
    The median gives a better measure of central tendency when there are outliers in the data that might skew the mean.

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    Most popular questions from this chapter

    Use the following information to answer the next nine exercises: The population parameters below describe the full-time equivalent number of students (FTES) each year at Lake Tahoe Community College from 1976鈥�1977 through 2004鈥�2005. \(\bullet \mu=1000 \mathrm{FTES}\) \(\bullet\) median \(=1,014 \mathrm{FTES}\) \(\bullet \quad \sigma=474 \mathrm{FTES}\) \(\cdot\) first quartile \(=528.5\) FTES \(\cdot\) third quartile \(=1,447.5\) FTES \(\cdot n=29\) years How many standard deviations away from the mean is the median?

    Use the two frequency tables to compare the life expectancy of men and women from 20 randomly selected countries. Include an overlayed frequency polygon and discuss the shapes of the distributions, the center, the spread, and any outliers. What can we conclude about the life expectancy of women compared to men? $$\begin{array}{|c|c|}\hline \text { Life Expectancy at Birth - women } & {\text { Frequency }} \\ \hline 49-55 & {3} \\ \hline 56-62 & {3} \\ \hline 63-62 & {1} \\ \hline 70-76 & {3} \\ \hline 77-83 & {8} \\ \hline 84-90 & {2} \\ \hline\end{array}$$ Table 2.47 $$\begin{array}{|c|c|}\hline \text { Life Expectancy at Birth - Men } & {\text { Frequency }} \\ \hline 49-55 & {3} \\ \hline 56-62 & {3} \\ \hline 63-62 & {1} \\ \hline 70-6 & {1} \\ \hline 77-83 & {7} \\ \hline 84-90 & {5} \\\ \hline\end{array}$$ Table 2.48

    The median age of the U.S. population in 1980 was 30.0 years. In 1991, the median age was 33.1 years. a. What does it mean for the median age to rise? b. Give two reasons why the median age could rise. c. For the median age to rise, is the actual number of children less in 1991 than it was in 1980? Why or why not?

    Use the following information to answer the next three exercises: The following data show the lengths of boats moored in a marina. The data are ordered from smallest to largest: 16; 17; 19; 20; 20; 21; 23; 24; 25; 25; 25; 26; 26; 27; 27; 27; 28; 29; 30; 32; 33; 33; 34; 35; 37; 39; 40 Identify the median.

    $$\begin{array}{|l|l|l|}\hline \text { Baseball Player } & {\text { Batting Average }} & {\text { Team Batting Average }} & {\text { Team Standard Deviation }} \\ \hline \text { Fredo } & {0.158} & {0.166} & {0.012} \\\ \hline \text { Karl } & {0.177} & {0.189} & {0.015} \\ \hline\end{array}$$ Use Table 2.57 to find the value that is three standard deviations: a. above the mean b. below the mean
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