1. Contingency Table (Cross Tabulation Table)
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- Isaac Dennis Thornton
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1 II. Descriptive Statistics C. Bivariate Data In this section Contingency Table (Cross Tabulation Table) Box and Whisker Plot Line Graph Scatter Plot 1. Contingency Table (Cross Tabulation Table) Bivariate Data is when two observations are made on each subject. Examples Religious Preference and Political Preference Gender and Height Unemployment Rate and Year SAT Score and First Year College GPA Once again the decision of how to display the data will depend on how many variables (in this section always two) there are and the type of the variables. Contingency Table or Cross Tabulation Table used typically to study the responses of two qualitative variables (rows represent categories of first variable and columns represent categories of second variable) Since we are dealing with qualitative data in this case we are limited (as was the case for a single variable) to frequencies or percentages. The most common way to set up a contingency table is with the cells representing frequencies. The table below is for the variables Gender and Major. For example the first cell tells us that there are 5 Male Liberal Arts majors. Gender Major Liberal Arts Business Ad. Technology Row Totals Male Female Column Totals
2 The other way to set up a contingency table is with the cells representing percentages. There are three options in regards to how the percentages are calculated and this should be based on the researcher s interests. It is important when reading the table to pay attention to how the percentages were calculated because that dictates the interpretation of the percentage. The first option is to base the percentages on row totals as is done in the table below. The first cell tells us that 28% (5/18) of Males are Liberal Arts Majors. Percentages Based on Row Totals Major Liberal Arts Business Ad. Technology Row Totals Male 28% 33% 39% % Gender Female % 33% 17% % Column Totals 37% 33% % % The second option is to calculate the percentages based on column totals as is done in the table below. The first cell in this case tells us that 45% (5/11) of Liberal Arts Majors are Male. Percentages Based on Column Totals Major Liberal Arts Business Ad. Technology Row Totals Male 45% 6% 78% 6% Gender Female 55% % 22% % Column Totals % % % % The third and most common option for percentages is to base them on the grand total as is done in the table below. The first cell here tells us that 17% (5/) of the people in the sample are both Male and a Liberal Arts Major. Percentages Based on Grand Totals Major Liberal Arts Business Ad. Technology Row Totals Male 17% % 23% 6% Gender Female % 13% 7% % Column Totals 37% 33% % % 2
3 2. Box and Whisker Plot Box and Whisker Plots also commonly referred to as Boxplots are used for bivariate data when one variable is qualitative and one is quantitative. The example we will start with in this section was introduced when discussing quartiles. There were two variables one being qualitative (student/faculty) and the other being quantitative (money). Below is the data set in stem-and-leaf plots along with the five number summary for each group. Example Students Faculty Low Low Q 1 1 Q 1 15 Q 2 5 Q 2 25 Q 7 3 Q 31 3 High High 73 8 Boxplot of Money vs Group 7 6 Money Faculty Group Students 3
4 In order to understand how the box and whisker plot is created and to better understand the interpretation of the graph it is important to go over the procedure for creating a box and whisker plot. You will not have to create the graph by hand. We will let software actually create the graph, but you do need to understand how to interpret the information presented in the graph. Procedure for creating a box and whisker plot: Draw a scale to include the lowest and highest data value (this will be the y-axis) To the right of the scale draw a box from Q 1 to Include a solid line through the box at the median level If there are any extreme values identify them with an asterisk or dot Draw solid lines, called whiskers, from highest value (if there are extremes, go to the next lowest or highest value) Q 3 Q 1 to the lowest value and from Q 3 to the If you compare the box and whisker plot to the five number summary in the previous example, you should see how it was created. Just look at the faculty category. Start with the box, the bottom lines up with and the top lines up with represents the middle % of the data (IQR). The line that goes through the box lines up with Q This line represents the median. Finally, look at the whiskers which are the lines that extend from the box. If you look at the lower whisker it goes from the bottom of the box to the low value in the data set, Low. The top whisker is a little different, because the High 73 was identified as an extreme value. Therefore, this value is identified by an asterisk and the whisker goes to the next highest value which you can see from the data is 43. Q 1 15 Q The box There are three kinds of questions you should be able to answer from a box-and-whisker plot. You should be able to compare the groups based on a typical value, compare the groups based on variability, and identify the shape of the distribution for each group. Based on the box and whisker plot in the previous example which group tended to have the most money in their pockets, students or faculty? Faculty, we can tell this because of the positioning of the box and the median on the y-axis. The faculty box and median is positioned higher. Based on the box and whisker plot in the previous example which group tended to have more consistent values (less variability)? Students, we can tell this because of the size of the box and the length of the whiskers. The student box is smaller and has shorter whiskers. Now we will consider shape. The following graph illustrates what a box-and-whisker plot will look like for some common shapes of distributions. 4
5 Identifying Shapes from Boxplots The above graphs are smooth curves, but the shape is determined the same way as with a histogram. This is a common way to represent populations and we will look at these graphs later in the course. What I want you to be able to identify from a box-and-whisker plot is if the graph is symmetric or skewed. If it is skewed, you should be able to identify the direction, either right or left. In the above graphs there are two that are symmetric. The Bell-Shaped Distribution and the Uniform (Rectangular) Distribution are both symmetric. The most telling sign of shape based on a box and whisker plot is where the median is in the box. Notice that in a symmetric distribution, the median is in the center of the box. The secondary sign of symmetry is the whiskers being the same length. If a distribution is skewed, then the median will not be in the center. The median being to the right indicates a left skewed distribution. This is because the data to the left is more spread out than the data to the right indicating a longer tail to the left. The secondary characteristic is the whisker to the left is longer than the whisker to the right. This also indicates a longer tail to the left. The right skewed distribution is just the opposite. In order to determine the shape when looking at the box and whisker plots I will give you, most people turn them 9 degrees at least mentally so they look like the ones above. When doing this make sure you turn them clockwise. This will ensure that the y-axis is increasing from left to right. 5
6 Now let us look at another situation. How does tread design affect an automobiles stopping distance? Qualitative variable: Tread Design (A, B, or C) Quantitative variable: Stopping Distance 44 Boxplot of Stopping Distance vs Tread Design 42 Stopping Distance A B Tread Design C You should be able to answer the following questions. If you were to select a tread design just based on stopping distance, which one is the best choice? Tread design B is the best choice because we want a quick stopping distance (low numbers) and the box for B is lower. Also, the median is much lower indicating that the lower % of stopping distances for tread design B is much better than that for the other two tread designs. Which tread design produces the most consistent stopping distances (least variability)? Tread design C has the least variability. We can tell this because the box is smaller than that for the other two designs. Also, the whiskers are short. Notice in this case the stopping distances are consistently bad. We would like to have a choice that is consistently good, but in this situation that is not an option. What is the shape of the distribution for each tread design? All the distributions are right skewed. This is because for each plot, the median is left of center. For each tread design the secondary characteristic of a longer whisker to the right is also satisfied. The important thing with shape is these characteristics. Notice the plots look different but they are all right skewed because they all satisfy the criteria. 6
7 3. Line Graph Line graphs (also called time plots) are typically used for bivariate data when one variable is quantitative and the other variable is time. For example when looking at the price of a stock over time you can quickly see this relationship with a line graph. In this graph the x-axis will represent time and the y-axis will represent the quantitative variable. To discuss the properties of a line graph, a new data set will be introduced. The data follows and represents cell phone subscribers from 199 to. Example Year Cell phone subscribers (in millions of subscribers) Line Graph for Cell Phone Subscribers Count Year The main thing we typically look for in a line graph is the pattern. In the above example it is easy to identify an increasing pattern over time. Technically this variable is increasing exponentially over time. In this class we will cover just the very basics of what to look for in a time series. 7
8 Time Series a record of a variable over time. With this type of data we use a line graph and need to be able to recognize patterns in the data over time. Trend a steady change over time (A graph with trend will be increasing or decreasing.) Seasonal Component in time series this means that the variable tends to be higher at certain points in time and lower at certain points in time (An example of this would be sales in a department store. Every year before Christmas, sales numbers are the highest.) Irregular Cycles an increase or decrease that can be explained, but will not be patterned (If you look at unemployment rates during the Vietnam War, they were very low. We know and can explain why these rates dropped, however, there is not going to be a long term pattern in this case.) Random Fluctuation any fluctuation not accounted for by the above three will be classified as random fluctuation (Pretty much every time series will include some random fluctuation.) You will be expected to identify trend and seasonal variation based on a graph. Irregular cycles will not be identified since you would have to research the variable in order to determine if this component exists. Random fluctuation is in every time series and should not be your focus as it can be assumed that this component is present. Increasing Trend and Random Fluctuation C5 Month JFMAMJJASONDJFMAMJJASONDJFMAMJJASONDJFMAMJJASOND 8
9 Decreasing Trend and Random Fluctuation 6 C6 Month JFMAMJJASONDJFMAMJJASONDJFMAMJJASONDJFMAMJJASOND Seasonal Variation and Random Fluctuation C7 Month JFMAMJJASONDJFMAMJJASONDJFMAMJJASONDJFMAMJJASOND Increasing Trend, Seasonal Variation and Random Fluctuation C8 6 Month JFMAMJJASONDJFMAMJJASONDJFMAMJJASONDJFMAMJJASOND 9
10 Decreasing Trend, Seasonal Variation and Random Fluctuation C Month JFMAMJJASONDJFMAMJJASONDJFMAMJJASONDJFMAMJJASOND Only Random Fluctuation; No Trend and No Seasonal Variation C1 5 Month JFMAMJJASONDJFMAMJJASONDJFMAMJJASONDJFMAMJJASOND
11 4. Scatter Plot Scatter plot or Scatter diagram displays the relationship between two quantitative variables. X axis (independent variable or explanatory variable) Y axis (dependent variable or response variable) A scatter plot is set up in a similar way to how you graph in an algebra class. The big difference is that there is not a perfect pattern so you cannot connect the points. We are simply looking for patterns in the data points themselves. The variables in a scatterplot are the same as when dealing with functions in algebra. You must always think about which variable is your X and which is your Y based on which variable is dependent on the other. The difference between what is done in algebra and in this class comes from the definition of a function; each X will have exactly one Y. That is not the case when dealing with data. Commonly you will give two individuals the same X and get different values for Y. Identifying correlation graphically Correlation an association or relationship between two variables Example Suppose you have two variables, dosage of drug and reduction in blood pressure. Which one should be your X and which one should by your Y? In this case reduction in blood pressure is the Y variable because the reduction in blood pressure would be dependent on the dosage of the drug. Therefore, the dosage of the drug is X, your independent variable. Below is some sample data for this situation. X = Dosage of Drug Y = Reduction in Blood Pressure In order to graph this data, you simply plot the points on the xy-plane. We will not actually graph these by hand. Instead we will use software to create scatterplots. Remember that we are looking for a relationship or pattern in the data. The graph follows. 11
12 Dosage of Drug and Reduction in Blood Pressure 6 Y X Notice in the above graph there is a pattern in the data. As the dosage of the drug increases, the reduction in blood pressure also increases. We will now look at the specific patterns you need to be able to identify. Perfect positive linear correlation We will focus on linear patterns in this class. For the graph below, all the points are exactly on a line with a positive slope. We only say perfect if all the points are exactly on a given function. Positive is used because the line has a positive slope. It is important you use these exact terms. C1 C2 12
13 Perfect negative linear correlation The only difference between this graph and the previous one is the negative slope. C1 C2 Positive linear correlation The graph below is a much more reasonable situation. In practice, nothing is ever perfect. In this case there is no exact function that fits all of the points. However, if you draw a line with a positive slope, all of the points are close to the line. This is typically what we look for with this type of data. C1 C2 13
14 Negative linear correlation Once again, the only difference between this graph and the previous one is that in this case we have a negative slope. C1 C2 Non-linear correlation Since our focus is on linear relationships, you do not have to identify other types of correlation. If we have a pattern that is not linear, then you should identify it as non-linear correlation. Technically the relationship below is quadratic, but you do not have to identify the specific type. C2 C1 14
15 No correlation If there is not a pattern or relationship seen in the graph then there is no correlation as is seen in the graph below. C2 C1 15
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