Section 4 - Graphical Descriptions of Quantitative Data(1) (1)

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MTH101: Statistics Unit: Section 2: Page 1 of 12 Section 4: Graphical Descriptions of Quantitative Data The graph for quantitative data is called a histogram looks similar to a bar graph, but there are some major differences. In a bar graph… In a histogram… Data You are representing_____________ data You are representing ______________ data Categories Categories can be put in ____________ Categories are in a _________ since you are dealing with numbers Distribution You ______________ how the data is distributed based on shape since the shape changes based on how you order your categories You ____________ how data is distributed based on shape Bars represent Bars represent a __________________ Bars represent a ________________________________ Bar placement Bars ______________ so there __________ in the graph bars _________ so there are __________ in the graph unless there is a big gap in values in the data To create a histogram by hand, you must first create the frequency distribution. This is just a fancy way of saying we are going to take our data and “chunk” it into subintervals called classes. Summary of the steps involved in making a frequency distribution for quantitative data: 1. Compute the ___________________ Compute ( largestdata value − smallest data value ) number of classes and then round as follows: *If the data are whole numbers, round up to the next whole number *If the data are tenths numbers, round up to the next tenth number, etc. 2. Compute the ___________________. The lowest class boundary is computed as follows: take the smallest value in the data set and subtract…

MTH101: Statistics Unit: Section 2: Page 2 of 12 *0.5 if the data are whole numbers *0.05 if the data are tenths numbers *0.005 if the data are hundredths numbers, etc. 3. Create the_________. Start with lowest class boundary you computed in step 2. Add the class width you got in step 1. This is the upper class boundary for the first class and the lower class boundary for the second class. Repeat this process until you get all the classes. 4. To figure out the number of data points that fall in each class, go through each data values and see which class boundaries it is between. Using tally marks may be helpful in counting the data values. The frequency for a class is the number of data values that fall in the class. Example #3.4.1: Creating a Frequency Table for Quantitative Data Table #3.4.1 contains the amount of rent paid every month for 24 students from a statistics course. Make a relative frequency distribution using 7 classes. Table #3.4.1: Data of Monthly Rent 1500 1350 350 1200 850 900 1500 1150 1500 900 1400 1100 1250 600 610 960 890 1325 900 800 2550 495 1200 690 Solution: First identify the individual, variable and type of variable. Individual: a randomly selected student from a statistics course Variable: amount of monthly rent Type of variable: quantitative 1) Compute the class width:

MTH101: Statistics Unit: Section 2: Page 3 of 12 2) Compute the lowest class boundary: 3) Create the classes. The lower class boundaries start at 349.5 and you keep adding 315 down The upper class boundaries start at 664.5 and you keep adding 315 down Class Boundaries Tally Frequency 349.5 – 664.5 664.5 – 979.5 979.5 – 1294.5 1294.5 – 1609.5 1609.5 – 1924.5 1924.5 – 2239.5 2239.5 – 2554.5  Here we now have 7 classes which is what was asked for and the largest data value of 2550 is contained in the last class. 4) Tally and find the frequency of the data: Go through the data and put a tally mark in the appropriate class for each piece of data by looking to see which class boundaries the data value is between. Fill in the frequency by changing each of the tallies into a number. Each relative frequency is just the frequency divided by the total number of data points. In this case each frequency would be divided by 24.

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MTH101: Statistics Unit: Section 2: Page 4 of 12 Table #3.4.2: Frequency Distribution for Monthly Rent Class Boundaries Tally Frequency Relative Frequency 349.5 – 664.5 4 0.17 664.5 – 979.5 8 0.33 979.5 – 1294.5 5 0.21 1294.5 – 1609.5 6 0.25 1609.5 – 1924.5 0 0 1924.5 – 2239.5 0 0 2239.5 – 2554.5 1 0.04 TOTAL 24 1.00 It is difficult to determine the basic shape of the distribution by looking at the frequency distribution. It would be easier to look at a graph. The graph of a frequency distribution for quantitative data is called a histogram . Histogram – Example #3.4.2: Drawing a Histogram Draw a histogram for the distribution from example #3.4.1. Solution: The class boundaries are plotted on the horizontal axis and the frequencies are plotted on the vertical axis. Also, from the earlier example, we computed the first lower class boundary to be 349.5 and the class width to be 315. These are used for where we start the bins at and for the bin width respectively.

MTH101: Statistics Unit: Section 2: Page 5 of 12 Graph #3.4.1: Frequency Histogram for Monthly Rent Reviewing the graph you can see that the rents that occur most often are between $664.50 and $979.50 per month for rent. There is a large gap between $1609.50 and $2239.50. This seems to say that one student is paying a great deal more than everyone else. This value could be considered an __________. An _____________ is a data value that is far from the rest of the values. It may be an unusual value or a mistake. It is a data value that should be investigated. In this case, the student lives in a very expensive part of town, thus the value is not a mistake, and is just very unusual. There are other aspects that can be discussed, but first some other concepts need to be introduced. Frequencies are helpful, but understanding the relative size each class is to the total is also useful. To find this you can divide the frequency by the total to create a relative frequency. If you have the relative frequencies

MTH101: Statistics Unit: Section 2: Page 6 of 12 for all of the classes, then you have a relative frequency distribution. This gives you percentages of data that fall in each class. Example #3.4.3: Creating a Relative Frequency Table Find the relative frequency for the monthly rent data. Solution: From example #3.4.1, the frequency distribution is reproduced in table #3.4.2. Table #3.4.3: Frequency Distribution for Monthly Rent Class Boundaries Frequency 349.5 – 664.5 4 664.5 – 979.5 8 979.5 – 1294.5 5 1294.5 – 1609.5 6 1609.5 – 1924.5 0 1924.5 – 2239.5 0 2239.5 – 2554.5 1 Table #3.4.4: Relative Frequency Distribution for Monthly Rent Class Boundaries Frequency Relative Frequency 349.5 – 664.5 4 0.17 664.5 – 979.5 8 0.33 979.5 – 1294.5 5 0.21 1294.5 – 1609.5 6 0.25 1609.5 – 1924.5 0 0 1924.5 – 2239.5 0 0 2239.5 – 2554.5 1 0.04 24 1 The graph of the relative frequency is known as a relative frequency histogram. It looks identical to the frequency histogram, but the vertical axis is relative frequency instead of just frequencies. Example #3.4.4: Drawing a Relative Frequency Histogram

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MTH101: Statistics Unit: Section 2: Page 7 of 12 Solution: The class boundaries are plotted on the horizontal axis and the relative frequencies are plotted on the vertical axis. Also, from the earlier example, we computed the first lower class boundary to be 349.5 and the class width to be 315. Graph #3.4.2: Relative Frequency Histogram for Monthly Rent Notice the shape of the relative frequency distribution is the same as the shape of the frequency distribution. The only difference is that the vertical axis now has relative frequencies instead of frequencies. Shapes of the distribution: The point of this chapter is not just to be able to MAKE a frequency table or a graph of the data. One of the characteristics we will be interested in later is the SHAPE of the distribution. Before drawing inferences using the results of a set of sample data, often you need to first look at the histogram and look at three things: shape, center and spread. We will discuss shape here and we will discuss measures of center and spread in the next chapter. Below are some of the common distribution shapes we will see this semester.

MTH101: Statistics Unit: Section 2: Page 8 of 12 ________________ _______________ ________________ _______________ ________________ ________________ _______________ _______________ Some shapes are symmetric and some are not. Symmetric means that the two sides are mirror images of each other. Skewed means one “tail” of the graph is longer than the other. The graph is skewed in the direction of the longer tail. Another interest is how many peaks a graph may have. Modal refers to the number of peaks. ____________ has one peak and __________ has two peaks. Usually if a graph has more than two peaks, the modal information is no longer of interest. Other important features to consider are gaps between bars, a repetitive pattern, how spread out the data are, and where the center of the graph is. Examples of graphs: Graph #3.4.3: Graph #3.4.4:

MTH101: Statistics Unit: Section 2: Page 9 of 12 Graph #3.4.5: Graph #3.4.6: Graph #3.4.7: Example #3.4.5: Creating a Frequency Distribution and Histogram

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MTH101: Statistics Unit: Section 2: Page 10 of 12 The following data represent the percent change in tuition levels at public, four-year colleges (inflation adjusted) from 2008 to 2013 (Weissmann, 2013). Create a frequency distribution and histogram for the data using 8 classes. Table #3.4.5: Data of Tuition Levels at Public, Four-Year Colleges 19.5% 40.8% 57.0% 15.1% 17.4% 5.2% 13.0% 15.6% 51.5% 15.6% 14.5% 22.4% 19.5% 31.3% 21.7% 27.0% 13.1% 26.8% 24.3% 38.0% 21.1% 9.3% 46.7% 14.5% 78.4% 67.3% 21.1% 22.4% 5.3% 17.3% 17.5% 36.6% 72.0% 63.2% 15.1% 2.2% 17.5% 36.7% 2.8% 16.2% 20.5% 17.8% 30.1% 63.6% 17.8% 23.2% 25.3% 21.4% 28.5% 9.4% Solution: First identify the individual, variable and type of variable. Individual Variable: Type of variable: 1) Compute the class width: 2) Compute the lowest class boundary: 3) Create the classes.

MTH101: Statistics Unit: Section 2: Page 11 of 12 Class Boundaries Tally Frequency 2.15 – 11.75 11.75 – 21.35 21.35 – 30.95 30.95 – 40.55 40.55 – 50.15 50.15 – 59.75 59.75 – 69.35 69.35 – 78.95 4) Tally and find the frequency of the data: Go through the data and put a tally mark in the appropriate class for each piece of data by looking to see which class boundaries the data value is between. Fill in the frequency by changing each of the tallies into a number. Table #3.4.6: Frequency Distribution for Tuition Levels at Public, Four-Year Colleges Class Boundaries Tally Frequency 2.15 – 11.75 6 11.75 – 21.35 20 21.35 – 30.95 11 30.95 – 40.55 4 40.55 – 50.15 2 50.15 – 59.75 2 59.75 – 69.35 3 69.35 – 78.95 2 Make sure the total of the frequencies is the same as the number of data points.

MTH101: Statistics Unit: Section 2: Page 12 of 12 To make the frequency histogram, the class boundaries are plotted on the horizontal axis and the frequencies are plotted on the vertical axis. The lowest class boundary was 2.15 and the class width was 9.6. Graph #3.4.8: Histogram for Tuition Levels at Public, Four-Year Colleges This graph is skewed right, with no gaps. This says that the most frequent percent increases in tuition were between 11.75% and 21.35%. There are other types of graphs for quantitative data. They will be explored in the next section.

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Section 4 - Graphical Descriptions of Quantitative Data(1) (1)

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