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Chapter 3: Problem 13

Consider the numbers 2 3 4 5 5 (a) Compute the mode, median, and mean. (b) If the numbers represent codes for the colors of T-shirts ordered from a catalog, which average(s) would make sense? (c) If the numbers represent 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
Mode: 5, Median: 4, Mean: 3.8; Median and mode for T-shirt colors; mean, median, and mode for mileages; median and mode for survey responses.

Step by step solution

01

Compute the Mode

The mode is the number that appears most frequently in a data set. In the set \( \{2, 3, 4, 5, 5\} \), the number \( 5 \) appears twice, while all other numbers appear only once. Therefore, the mode is \(5\).
02

Compute the Median

To find the median, first, arrange the numbers in ascending order: \( 2, 3, 4, 5, 5 \). Since there is an odd number of data points, the median is the middle number. This means the median is \(4\).
03

Compute the Mean

The mean is the average of all data points. Add the numbers: \( 2 + 3 + 4 + 5 + 5 = 19 \). Then divide by the number of data points (5): \( \frac{19}{5} = 3.8 \). Thus, the mean is \(3.8\).
04

Determine Suitable Averages for T-Shirt Colors

For T-shirt colors, both mode and median can make sense because these measures do not require numerical interpretation. The mode shows the most frequently ordered color, while the median indicates the central tendency in a ranked sense.
05

Determine Suitable Averages for Trail Mileages

For mileages, the mean, median, and mode all make sense. The mean provides the average distance, the median gives the middle distance, and the mode can indicate the most common trail length.
06

Determine Suitable Averages for Survey Responses

For survey responses, the median and mode are appropriate as they summarize ordinal data without assuming equality between scale points. The mode shows the most common sentiment, while the median conveys a central tendency. The mean is less appropriate as it implies an interval scale.

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Key Concepts

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

Mode
In statistics, the mode is the number that appears most frequently in a data set. It's particularly useful because it tells us what is most common or popular within a dataset. To find the mode, you simply need to identify which number(s) occur most often. For example, in the set \( \{2, 3, 4, 5, 5\} \), we notice that the number \(5\) appears twice, whereas all other numbers appear only once. Therefore, the mode of this set is \(5\). This makes the mode a valuable measure for categorical data, like T-shirt sizes, where the most common size is of interest. It can also help in identifying popular items or forms in different datasets, thus providing insights into consumer or survey response patterns.
Median
The median represents the middle value of a data set when the numbers are arranged in either ascending or descending order. For a dataset like \( \{2, 3, 4, 5, 5\} \), you first arrange the numbers in order, which they already are. Since the dataset has an odd number of points (5 in this case), the median is simply the middle number. Thus, the median here is \(4\).

  • If there had been an even number of values, the median would be the average of the two middle numbers.
  • The median is particularly useful because it is not affected by outliers or extreme values, making it a robust measure of central tendency.
  • In contexts like survey responses, where data is ordinal, the median provides a meaningful measure of central tendency without assuming equal spacing between points.
Mean
The mean, also known as the average, is a measure of central tendency calculated by summing all the values in a dataset and then dividing by the number of values. For the data set \( \{2, 3, 4, 5, 5\} \), you add them up to get \(2 + 3 + 4 + 5 + 5 = 19\), and then divide by the total number of values, which is 5. This gives a mean of \(3.8\).

  • The mean is very useful for numerical data where all values contribute equally, like trail mileages.
  • One key aspect of the mean is that it takes into account every data point, providing a comprehensive measure of central tendency.
  • However, the mean can be distorted by extreme values or outliers, which might not always reflect a true center for datasets with wide variability.
  • This makes it less suitable for ordinal data, where the mean might suggest a level of precision not valid for categories like survey responses.

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Most popular questions from this chapter

The Hill of Tara in Ireland is a place of great archaeological importance. This region has been occupied by people for more than 4000 years. Geomagnetic surveys detect subsurface anomalies in the earth's magnetic field. These surveys have led to many significant archaeological discoveries. After collecting data, the next step is to begin a statistical study. The following data measure magnetic susceptibility (centimeter-gram-second \(\times 10^{-6}\) ) on two of the main grids of the Hill of Tara (Reference: Tara: An Archaeological Survey by Conor Newman, Royal Irish Academy, Dublin). Grid \(\mathbf{E}: x\) variable $$\begin{array}{ccccccc} 13.20 & 5.60 & 19.80 & 15.05 & 21.40 & 17.25 & 27.45 \\ 16.95 & 23.90 & 32.40 & 40.75 & 5.10 & 17.75 & 28.35 \end{array}$$ Grid H: \(y\) variable $$\begin{array}{lllllll} 11.85 & 15.25 & 21.30 & 17.30 & 27.50 & 10.35 & 14.90 \\ 48.70 & 25.40 & 25.95 & 57.60 & 34.35 & 38.80 & 41.00 \\ 31.25 & & & & & \end{array}$$ (a) Compute \(\Sigma x, \Sigma x^{2}, \Sigma y,\) and \(\Sigma y^{2}\). (b) Use the results of part (a) to compute the sample mean, variance, and standard deviation for \(x\) and for \(y\). (c) Compute a \(75 \%\) Chebyshev interval around the mean for \(x\) values and also for \(y\) values. Use the intervals to compare the magnetic susceptibility on the two grids. Higher numbers indicate higher magnetic susceptibility. However, extreme values, high or low, could mean an anomaly and possible archaeological treasure. (d) Compute the sample coefficient of variation for each grid. Use the \(C V\) s to compare the two grids. If \(s\) represents variability in the signal (magnetic susceptibility) and \(\bar{x}\) represents the expected level of the signal, then \(s / \bar{x}\) can be thought of as a measure of the variability per unit of expected signal. Remember, a considerable variability in the signal (above or below average) might indicate buried artifacts. Why, in this case, would a large \(C V\) be better, or at least more exciting? Explain.

In this problem, we explore the effect on the standard deviation of adding the same constant to each data value in a data set. Consider the data set 5,9,10,11,15. (a) Use the defining formula, the computation formula, or a calculator to compute \(s\) (b) Add 5 to each data value to get the new data set 10,14,15,16,20 Compute \(s\) (c) Compare the results of parts (a) and (b). In general, how do you think the standard deviation of a data set changes if the same constant is added to each data value?

Basic Computation: Weighted Average Find the weighted average of a data set where 10 has a weight of \(5 ; 20\) has a weight of \(3 ; 30\) has a weight of 2

What symbol is used for the arithmetic mean when it is a sample statistic? What symbol is used when the arithmetic mean is a population parameter?

One indicator of an outlier is that an observation is more than 2.5 standard deviations from the mean. Consider the data value \(80 .\) (a) If a data set has mean 70 and standard deviation \(5,\) is 80 a suspect outlier? (b) If a data set has mean 70 and standard deviation \(3,\) is 80 a suspect outlier?
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