Question

There are many different types of graphical representations (e.g., bar graphs, pie graphs, histograms, dot plots, time series plots, stem-and-leaf-plots, etc.). Provide an example of a real-world application of when a specific type of graphical representation is used to make informed decisions.

Answer 1

Scientists have invented a new dietary supplement that is supposed to increase the weight of a piglet within its first 3 months of growth. Farmer John fed this supplement to his stock of piglets, and at the end of 3 months, he recorded the weights of 50 randomly selected piglets.

The following table is the recorded weights (in pounds) of the 50 selected piglets:

120111651101147211610511911493 113 991181089710795 1137584 12010210484 9712169 10010110711877 1051097889 68 74 10387  67 79 90 1099410696 92 88

Using the above data set and the TI-83, construct a histogram to represent the data.

Solution:

Using the TRACE feature will give you information about the data in each bar of the histogram.

The TRACE feature tells you that in the first bin, which is [60-70), there are 4 values.

The TRACE feature tells you that in the second bin, which is [70-80), there are 6 values.

To advance to the next bin, or bar, of the histogram, use the cursor and move to the right. The information obtained by using the TRACE feature will enable you to create a frequency table and to draw the histogram on paper.

The shape of a histogram can tell you a lot about the distribution of the data, as well as provide you with information about the mean, median, and mode of the data set. The following are some typical histograms, with a caption below each one explaining the distribution of the data, as well as the characteristics of the mean, median, and mode. Distributions can have other shapes besides the ones shown below, but these represent the most common ones that you will see when analyzing data. In each of the graphs below, the distributions are not perfectly shaped, but are shaped enough to identify an overall pattern.

a)

Figure a represents a bell-shaped distribution, which has a single peak and tapers off to both the left and to the right of the peak. The shape appears to be symmetric about the center of the histogram. The single peak indicates that the distribution is unimodal. The highest peak of the histogram represents the location of the mode of the data set. The mode is the data value that occurs the most often in a data set. For a symmetric histogram, the values of the mean, median, and mode are all the same and are all located at the center of the distribution.

b)

Figure b represents a distribution that is approximately uniform and forms a rectangular, flat shape. The frequency of each class is approximately the same.

c)

Figure c represents a right-skewed distribution, which has a peak to the left of the distribution and data values that taper off to the right. This distribution has a single peak and is also unimodal. For a histogram that is skewed to the right, the mean is located to the right on the distribution and is the largest value of the measures of central tendency. The mean has the largest value because it is strongly affected by the outliers on the right tail that pull the mean to the right. The mode is the smallest value, and it is located to the left on the distribution. The mode always occurs at the highest point of the peak. The median is located between the mode and the mean.

d)

Figure d represents a left-skewed distribution, which has a peak to the right of the distribution and data values that taper off to the left. This distribution has a single peak and is also unimodal. For a histogram that is skewed to the left, the mean is located to the left on the distribution and is the smallest value of the measures of central tendency. The mean has the smallest value because it is strongly affected by the outliers on the left tail that pull the mean to the left. The median is located between the mode and the mean.

e)

Figure e has no shape that can be defined. The only defining characteristic about this distribution is that it has 2 peaks of the same height. This means that the distribution is bimodal.

Another type of graph that can be drawn to represent the same set of data as a histogram represents is a frequency polygon. A frequency polygon is a graph constructed by using lines to join the midpoints of each interval, or bin. The heights of the points represent the frequencies. A frequency polygon can be created from the histogram or by calculating the midpoints of the bins from the frequency distribution table. The midpoint of a bin is calculated by adding the upper and lower boundary values of the bin and dividing the sum by 2.

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