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Chapter 3: Problem 5

Consider the numbers \(\begin{array}{lllll}2 & 3 & 4 & 5 & 5\end{array}\) (a) Compute the mode, median, and mean. (b) If the numbers represented codes for the colors of \(\mathrm{T}\) -shirts ordered from a catalog, which average(s) would make sense? (c) If the numbers represented one-way mileages for trails to different lakes, which average(s) would make sense? (d) Suppose the numbers represent survey responses from 1 to 5, with \(1=\) disagree strongly, \(2=\) disagree, \(3=\) agree, \(4=\) agree strongly, and \(5=\) agree very strongly. Which averages make sense?

Short Answer

Expert verified
(a) Mode: 5, Median: 4, Mean: 3.8. (b) Mode. (c) Mean and Median. (d) All averages make sense.

Step by step solution

01

Finding the Mode

The mode is the number that appears the most frequently in a data set. In the given list of numbers \(2, 3, 4, 5, 5\), the number \(5\) appears twice, more than any other number. Therefore, the mode is \(5\).
02

Calculating the Median

To find the median, first arrange the numbers in ascending order (if not already). For the list \(2, 3, 4, 5, 5\), the numbers are already sorted. The median is the middle number, so since there are five numbers, the median is the third number: \(4\).
03

Calculating the Mean

The mean is the average of the numbers. To find it, sum all the numbers: \(2 + 3 + 4 + 5 + 5 = 19\). Then, divide by the count of numbers, which is \(5\). So, the mean is \(\frac{19}{5} = 3.8\).
04

Determining Suitable Averages for Colors of T-shirts

When dealing with categorical data like codes for colors, the mode is typically the most informative average as it shows the most frequently ordered color. The mean and median do not make sense for categorical data.
05

Determining Suitable Averages for Trail Mileages

For mileage data, both the mean and median are appropriate as they each provide useful information about the central tendency of the distances. The mode is less relevant unless you need the most common mileage.
06

Determining Suitable Averages for Survey Responses

For ordinal data like survey responses, all three averages can be used but with varying appropriateness. The mode identifies the most common opinion, the median provides a central tendency without influence from outliers, and the mean offers a measure of overall tendency but may be less meaningful due to distortion by categories.

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Key Concepts

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

Understanding Mode
The mode is a simple yet powerful measure of central tendency in descriptive statistics. It refers to the number that appears most frequently in a data set. To calculate the mode, simply count how often each number appears. In the exercise numbers \[2, 3, 4, 5, 5\], the number **5** occurs twice, making it the mode. The mode is particularly useful with categorical data such as color codes because it identifies the most common item, in this case, the most frequently ordered T-shirt color.

The useful aspects of mode include:
  • Ability to highlight the most popular choice in a dataset.
  • It can work with non-numeric data like colors or sizes.
  • It is not affected by extremely high or low values.
Exploring the Median
The median is the middle value in a list of numbers when listed in ascending order. Finding the median is simple: sort the numbers, and pick the middle one. When there is an odd number of observations, the median is a single number, as in the case of the exercise dataset \[2, 3, 4, 5, 5\], where the median is **4**.

The median is valuable because:
  • It represents the point that divides the dataset in half.
  • Unlike the mean, it is not sensitive to extreme values or outliers.
  • It provides a clear perspective on the distribution of values, which is useful for non-numeric ordinal data like survey responses.
For survey responses, the median helps identify a central tendency without giving undue weight to extreme opinions, making it preferable for ranking or ordinal data.
Calculating the Mean
The mean, often referred to as the average, is a measure that sums all values and divides by the number of values. For the given numbers \[2, 3, 4, 5, 5\], you would add them (2 + 3 + 4 + 5 + 5 = 19) and divide by 5, the count of numbers, resulting in a mean of \[3.8\].

The mean offers several benefits:
  • It provides a quick summary of the dataset's overall trend.
  • Useful for interval and ratio data, such as trail mileages.
  • Affected by each value in the data set, giving equal importance to all numbers.
However, because the mean is sensitive to extremely high or low values, it might misrepresent the "typical" value in skewed distributions. For example, when assessing trail mileages, the mean and median together can offer a more comprehensive view of the distances traveled.

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