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Data Analysis On the ISEE, Data Analysis will require that you are able to analyze charts and graphs, as well as understand the basics of statistical ideas such as mean, median, range, mode, and standard deviation. All of these basic concepts are covered in more detail in their The two most common types of charts and graphs that you will encounter on the ISEE are Bar graphs Line graphs Bar Graphs Bar graphs are the bread and butter of any data analysis task. They are most often used to show numbers of people who fit into some sort of category. Essentially, a bar graph is a way to count how many people fit into a category and see it represented visually. In the bar graph above, we see that the number of students who read a certain number of books is measured. On the ISEE, you could be asked what the mean, median, mode, or range of this data is. Let’s tackle the idea of mode first, since this is the easiest to see from a bar graph. Mode The mode of any data set is the value that occurs most often. In our bar graph, it is simply the highest bar as pictured below. Thus, the mode of our data is 5! This means that the highest number of students read 5 books for their summer reading. Range Dealing with range is the next easiest task for a bar graph. The range is just the highest value of the data set minus the lowest value. Remember that the y-axis gives you a count of how many people did something and the x-axis gives you the actual values of the data. This means that the range of the data for the summer reading bar graph is simply 6 - 1 = 5, as pictured below. Both the mean and the median present more complexity than either the mode or the range. Mean The mean of any data set is simply the average of all of the values in that data set. It would be very easy to calculate if we were just given the numbers representing the data, but it is much less clear what to do with a bar graph that represents data. Let’s interpret the bar graph as an actual data set. This shows that there are 5 students who read 1 book, 4 students who read 2 books, 2 students who read 3 books, 1 student who read 4 books, 6 students who read 5 books, and 2 students who read 6 books. This translates to the data set {1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 4, 5, 5, 5, 5, 5, 5, 6, 6}. Notice how each set of 1’s, 2’s, etc. is almost a horizontal bar now. We can simply add up all of these values and divide by the total number of students! The total number of students is 5 + 4 + 2 + 1 + 6 + 2 = 20. The total sum of the number of books they read is 1 + 1 + 1 + 1 + 1 + 2 + 2 + 2 + 2 + 3 + 3 + 4 + 5 + 5 + 5 + 5 + 5 + 5 + 6 + 6 = 65. The mean is 65/20 = 3.25. That method of calculating the mean is valid, but it is incredibly long. A much better idea is to multiply each x-value by each corresponding y-value to speed up the process. This gives that the total number of books read is 1(5) + 2(4) + 3(2) + 4(1) + 5(6) + 6(2) = 5 + 8 + 6 + 4 + 30 + 12 = 65. Note that each color circle corresponds to the number of students who read that number of books. This is where we get the equation above the graph from! Also, notice that both 3 and 6 share the fact that 2 students read that number of books, so there is a double circle around the 2 on the y-axis. This was a much faster way to calculate the total number of books that were read and this technique shows up incredibly often on standardized tests. Be sure to remember this and know how to use it. Calculating the total number of students is done just like before, we add up the y-value of every bar. This gives that the total number of students is 5 + 4 + 2 + 1 + 6 + 2 = 20. Our mean is 65/20 = 3.25, just like the longer method. Median The median is the trickiest value to calculate for a bar graph and most high school seniors will make the same mistakes that ISEE students do. The mistake that occurs for all students is that they simply take the median of the x-values. For this bar graph, that means that the students would write out the list {1, 2, 3, 4, 5, 6} and then cross off the highest and lowest values until they reach the middle, since that’s how the median is calculated. However, this results in the list {1, 2, 3, 4, 5, 6}. Since there are two middle values, the students average these two and find that the median is 7/2 = 3.5. Sometimes, this method works, but it is absolutely not the correct way to solve for the median! Remember that our data set has 20 students and therefore has 20 individual values. This means that the actual data set is {1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 4, 5, 5, 5, 5, 5, 5, 6, 6} just like we determined before. Following the same procedure here, we find that crossing off values to get to the middle gives us the data set {1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 4, 5, 5, 5, 5, 5, 5, 6, 6}. Notice that the two middle values are now 3! Averaging these gives us that 3 is the median value. Simply following your intuition and finding the median of the x-axis is not enough. This is incorrect math that can sometimes work, but not for any of the right reasons. If you don’t want to write out the whole data set, you can actually cancel off individual values on the bar graph itself. This would look like the following bar graph. This is the exact same strategy as crossing off the first 1 and the last 6 and then moving your way to the center, but you don’t have to write out the whole data set. Line Graphs Line graphs represent how a variable is changing over time. This means that time is always on the x-axis and whatever variable is being measured is on the y-axis. On the ISEE, you are not expected to perform many calculations on line graphs, besides just looking at the data. There is no analysis of mean, median, or mode. It is possible that you may be asked about the range of values in a line graph, but it is more likely that you will be asked to identify a line graph that corresponds with a specific trend. Some of the most common variables to measure with a line graph are the temperature of something over time or the distance from somewhere over time. For instance, if I have a hot apple pie that was baked at 350 degrees Fahrenheit for one hour, what would the graph of the temperature of the pie look like as it cools on the kitchen counter? Obviously, this graph should show the temperature going down, but the exact relationship is a little bit trickier than just that. This line graph shows that the temperature of the pie decreases rapidly, but does not decrease below a certain level. The temperature of the pie cannot decrease infinitely, as it would just freeze! This means that the room temperature limits how much the pie can cool. Analyzing this data, we see that initially the temperature of the pie drops quickly, but begins to cool more slowly the longer that it is left out to cool. Generally speaking, you will only be required to describe trends in the data for line graphs and will not have to perform calculations for mean, median, mode, and range. A sample question using the graph above could be: During which of the following 10-minute intervals did the temperature of the pie decrease the most? A. 0-10 minutes B. 10-20 minutes C. 20-30 minutes D. 30-40 minutes Looking at the graph, the most negative slope occurs between 10-20 minutes, where the pie cools off 300 - 200 = 100 degrees Fahrenheit. This is the largest decrease, so 10-20 minutes is the correct answer. Practice Questions 1. The graph shows the distance that Susan was from the start of a hiking trail as a function of time during a hike. At one point in the hike, Susan stopped for several minutes and admired a waterfall. How far was Susan from the start of the trail when she stopped and admired the waterfall? A. 1 mile B. 1.25 miles C. 1.5 miles D. 1.6 miles 2. The graph below shows the distance that Susan was from the start of a hiking trail as a function of time during the hike. During which of the following 15-minute intervals was Susan hiking the fastest? A. 30-45 minutes B. 60-75 minutes C. 75-90 minutes D. 90-105 minutes 3. A scientist collected data for analysis. The data is presented in the bar graph below. What is the positive difference between the range and the mean of the data? A. 3 B. 5 C. 8 D. 9 4. A scientist collected data for analysis. The data is presented in the bar graph below. What is the average of the median and the mode? A. 1 B. 5 C. 5.5 D. 10 5. A teacher polled her class to see how many siblings each student had. The students who have 0 siblings were excluded from the bar graph. The data is summarized in the bar graph below. The class has a total of 25 students. What is the median number of siblings for a student in this class to have? A. 0 B. 1 C. 2 D. 3 Answers to Practice Questions: D D A B B
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Understanding diagrams and analyzing data is a critical part of the ISEE exam. You will have to understand how tables and graphs convey information and be able answer questions about them. The type of chart, table, or graph may change, but your method should be the same: Read the title and labels to understand what each component represents Look to see if an explanatory key offers more information about the data. Study the graph to see what the visualization is meant to show. This lesson will cover a few common types of data visualization that you may come across on the exam. Tables Tables are the most common kind of diagram you will find, and can be used to convey data in multiple ways. A table may compare the amount or intensity of multiple variables, or it may compare how those variables change over time. You will see this in the following examples. In this problem, you are tasked with analyzing the table to assess the popularity of each ice cream flavor. You can see from the column headings that the table shows each of the flavors, and the number of students who voted for that flavor. Given that the top three flavors will be served at the party, you should look for the three flavors with the highest number of votes. You can see from the table that these are cookies and cream, chocolate, and vanilla; therefore, the correct answer is D. To solve this problem, you must figure out what piece of data you need, and then find how it is represented in the table. What Shelly wants to know is whether Houston is 21 degrees hotter than Chicago in the month of April. Looking at the row representing April, we see that Houston’s average temperature is 62 degrees, while Chicago is 39 degrees. Since 62-39 is 23, this surpasses Shelly’s requirement—she will move to Houston in April and not Chicago because Chicago is too cold. Bar Graphs Bar graphs are another common kind of graph. They help you visualize data by comparing the height of each bar in the diagram. In the bar graph above, you can see four bars representing the number of miles each person ran in a week. These bars show: Sam ran 22 miles, Kathy ran 30 miles, Kelly ran 15 miles, and George ran 12 miles. Now, you can figure out which statement is true. B is false, since Kathy did not run fewer miles than Sam (she ran 30 and he ran 22). C is also false, because Kelly and George combined to run 27 miles while Sam ran 22 miles. You can now assess that D is false, the average is not over 25; the total number of miles is 79, and 79 divided by 4 is less than 25. A is true, because the range, 18, is more than the number of miles Kelly ran, 15. Pie Charts Pie charts are another way of visualizing data. Just like a real pie, you can think of each portion as a slice of pie. The bigger the slice, the bigger the percentage. Based on the diagram, you can see that approximately half of all students chose Math as their favorite subject. Then, looking at the remaining half, English looks to be about two thirds of that half. Therefore, History was chosen by a third of students within the second half of the pie chart. One third of one half is equal to one sixth. Therefore, the answer is B! Practice Problems Answers to Practice Problems B Question 2 Movie 29 A
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