2.2 Quantitative Data

The graph for quantitative data looks similar to a bar graph, except there are some major differences. First, in a bar graph the categories can be put in any order on the horizontal axis. There is no set order for these data values. You can’t say how the data is distributed based on the shape, since the shape can change just by putting the categories in different orders. With quantitative data, the data are in specific orders, since you are dealing with numbers. With quantitative data, you can talk about a distribution, since the shape only changes a little bit depending on how many categories you set up. This is called a frequency distribution.

This leads to the second difference from bar graphs. In a bar graph, the categories that you made in the frequency table were determined by you. In quantitative data, the categories are numerical categories, and the numbers are determined by how many categories (or what are called classes) you choose. If two people have the same number of categories, then they will have the same frequency distribution. Whereas in qualitative data, there can be many different categories depending on the point of view of the author.

The third difference is that the categories touch with quantitative data, and there will be no gaps in the graph. The reason that bar graphs have gaps is to show that the categories do not continue on, like they do in quantitative data. Since the graph for quantitative data is different from qualitative data, it is given a new name. The name of the graph is a histogram. To create a histogram, you must first create the frequency distribution. The idea of a frequency distribution is to take the interval that the data spans and divide it up into equal subintervals called classes.

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 for all of the classes, then you have a relative frequency distribution.

This gives you percentages of data that fall in each class.

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.

Another useful piece of information is how many data points fall below a particular class boundary. As an example, a teacher may want to know how many students received below an 80%, a doctor may want to know how many adults have cholesterol below 160, or a manager may want to know how many stores gross less than $2000 per day. This is known as a cumulative frequency. If you want to know what percent of the data falls below a certain class boundary, then this would be a cumulative relative frequency. For cumulative frequencies you are finding how many data values fall below the upper class limit.

To create a cumulative frequency distribution, count the number of data points that are below the upper class boundary, starting with the first class and working up to the top class. The last upper class boundary should have all of the data points below it. Also include the number of data points below the lowest class boundary, which is zero.

Again, it is hard to look at the data the way it is. A graph would be useful. The graph for cumulative frequency is called an ogive (o-jive). To create an ogive, first create a scale on both the horizontal and vertical axes that will fit the data. Then plot the points of the class upper class boundary versus the cumulative frequency. Make sure you include the point with the lowest class boundary and the 0 cumulative frequency. Then just connect the dots.

The usefulness of a ogive is to allow the reader to find out how many students pay less than a certain value, and also what amount of monthly rent is paid by a certain number of students. As an example, suppose you want to know how many students pay less than $1500 a month in rent, then you can go up from the $1500 until you hit the graph and then you go over to the cumulative frequency axes to see what value corresponds to this value. It appears that around 20 students pay less than $1500. (See Graph 2.2.4.)

Also, if you want to know the amount that 15 students pay less than, then you start at 15 on the vertical axis and then go over to the graph and down to the horizontal axis where the line intersects the graph. You can see that 15 students pay less than about $1200 a month. (See Graph 2.2.5.)

If you graph the cumulative relative frequency then you can find out what percentage is below a certain number instead of just the number of people below a certain value.

Shapes of the distribution:

When you look at a distribution, look at the basic shape. There are some basic shapes that are seen in histograms. Realize though that some distributions have no shape. The common shapes are symmetric, skewed, and uniform. Another interest is how many peaks a graph may have. This is known as modal.

Symmetric means that you can fold the graph in half down the middle and the two sides will line up. You can think of the two sides as being 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 (backwards from what you would expect). A uniform graph has all the bars the same height.

Modal refers to the number of peaks. Unimodal has one peak and bimodal has two peaks. Usually if a graph has more than two peaks, the modal information is not longer of interest.

Other important features to consider are gaps between bars, a repetitive pattern, how spread out is the data, and where the center of the graph is.

Examples of Graphs:

This graph is roughly symmetric and unimodal:

This graph is symmetric and bimodal:

This graph is skewed to the right:

This graph is skewed to the left and has a gap:

This graph is uniform since all the bars are the same height:

There are occasions where the class limits in the frequency distribution are predetermined. Example $8$ demonstrates this situation.

There are other types of graphs for quantitative data. They will be explored in the next section.

Homework

Exercise $1$

1. The median incomes of males in each state of the United States, including the District of Columbia and Puerto Rico, are given in Table 9 ("Median income of," 2013). Create a frequency distribution, relative frequency distribution, and cumulative frequency distribution using 7 classes.

   Table $9$: Data of Median Income for Males

   $42,951	$52,379	$42,544	$37,488	$49,281	$50,987
   $60,705	$50,411	$66,760	$40,951	$43,902	$45,494
   $41,528	$50,746	$45,183	$43,624	$43,993	$41,612
   $46,313	$43,944	$56,708	$60,264	$50,053	$50,580
   $40,202	$43,146	$41,635	$42,182	$41,803	$53,033
   $60,568	$41,037	$50,388	$41,950	$44,660	$46,176
   $41,420	$45,976	$47,956	$22,529	$48,842	$41,464
   $40,285	$41,309	$43,160	$47,573	$44,057	$52,805
   $53,046	$42,125	$46,214	$51,630

2. The median incomes of females in each state of the United States, including the District of Columbia and Puerto Rico, are given in Table 10 ("Median income of," 2013). Create a frequency distribution, relative frequency distribution, and cumulative frequency distribution using 7 classes.

   Table $10$: Data of Median Income for Females

   $31,862	$40,550	$36,048	$30,752	$41,817	$40,236
   $47,476	$40,500	$60,332	$33,823	$35,438	$37,242
   $31,238	$39,150	$34,023	$33,745	$33,269	$32,684
   $31,844	$34,599	$48,748	$46,185	$36,931	$40,416
   $29,548	$33,865	$31,067	$33,424	$35,484	$41,021
   $47,155	$32,316	$42,113	$33,459	$32,462	$35,746
   $31,274	$36,027	$37,089	$22,117	$41,412	$31,330
   $31,329	$33,184	$35,301	$32,843	$38,177	$40,969
   $40,993	$29,688	$35,890	$34,381

3. The density of people per square kilometer for African countries is in Example $11$ ("Density of people," 2013). Create a frequency distribution, relative frequency distribution, and cumulative frequency distribution using 8 classes.

   Table $11$: Data of Density of People per Square Kilometer

   15	16	81	3	62	367	42	123
   8	9	337	12	29	70	39	83
   26	51	79	6	157	105	42	45
   72	72	37	4	36	134	12	3
   630	563	72	29	3	13	176	341
   415	187	65	194	75	16	41	18
   69	49	103	65	143	2	18	31

4. The Affordable Care Act created a market place for individuals to purchase health care plans. In 2014, the premiums for a 27 year old for the bronze level health insurance are given in Example $12$ ("Health insurance marketplace," 2013). Create a frequency distribution, relative frequency distribution, and cumulative frequency distribution using 5 classes.  

   Table $12$: Data of Health Insurance Premiums

   $114	$119	$121	$125	$132	$139
   $139	$141	$143	$145	$151	$153
   $156	$159	$162	$163	$165	$166
   $170	$170	$176	$177	$181	$185
   $185	$186	$186	$189	$190	$192
   $196	$203	$204	$219	$254	$286

5. Create a histogram and relative frequency histogram for the data in Table 9. Describe the shape and any findings you can from the graph.  
6. Create a histogram and relative frequency histogram for the data in Table 10. Describe the shape and any findings you can from the graph.  
7. Create a histogram and relative frequency histogram for the data in Example $11$. Describe the shape and any findings you can from the graph.  
8. Create a histogram and relative frequency histogram for the data in Example $12$. Describe the shape and any findings you can from the graph.  
9. Create an ogive for the data in Table 9. Describe any findings you can from the graph.  
10. Create an ogive for the data in Table 10. Describe any findings you can from the graph.  
11. Create an ogive for the data in Example $11$. Describe any findings you can from the graph.  
12. Create an ogive for the data in Example $12$. Describe any findings you can from the graph.  
13. Students in a statistics class took their first test. The following are the scores they earned. Create a frequency distribution and histogram for the data using class limits that make sense for grade data. Describe the shape of the distribution.  

    Table $13$: Data of Test 1 Grades

    80	79	89	74	73	67	79
    93	70	70	76	88	83	73
    81	79	80	85	79	80	79
    58	93	94	74

14. Students in a statistics class took their first test. The following are the scores they earned. Create a frequency distribution and histogram for the data using class limits that make sense for grade data. Describe the shape of the distribution. Compare to the graph in question 13.  

    Table $14$: Data of Test 1 Grades

    67	67	76	47	85	70
    87	76	80	72	84	98
    84	64	65	82	81	81
    88	74	87	83

Answer

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