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.

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

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