14 Descriptive Statistics. 14.1 Graphical Descriptions of Data 14.2 Variables 14.3 Numerical Summaries 14.4 Measures of Spread. Data Set.
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14.1 Graphical Descriptions of Data
14.3 Numerical Summaries
14.4 Measures of Spread
A data set is a collection of data values. Statisticians often refer to the individualdata values in a data set as data points. For the sake of simplicity, we will workwith data sets in which each data point consists of a single number, but in morecomplicated settings, a single data point can consist of many numbers.
As usual, we will use the letter N to represent the size of the data set. In real-life applications, data sets can range in size from reasonably small (a dozen or sodata points) to very large (hundreds of millions of data points), and the larger thedata set is, the more we need a good way to describe and summarize it.
The day after the midterm exam in his Stat 101 class, Dr.Blackbeard hasposted the results online. The data set consists of N = 75data points(the number of students who took the test). Each data point (listed in the secondcolumn) is a score between 0 and 25 (Dr. Blackbeard gives no partial credit).Notice that the numbers listed in the first column are not data points–they arenumerical IDs used as substitutes for names to protect the students’ rights ofprivacy.
Like students everywhere, the students in the Stat 101 class have one question foremost on their mind when they look at the results: How did I do? Eachstudent can answer this question directly from the table. It’s the next questionthat is statistically much more interesting. How did the class as a whole do? Toanswer this last question, we will have to find a way to package the resultsinto a compact, organized, and intelligible whole.
The first step in summarizing the information in Table 14-1 is to organize thescores in a frequency table such as Table 14-2. In this table, the number beloweach score gives the frequency of the score–that is, the number of students gettingthat particular score.
We can readily see from Table 14-2 that there was onestudent with a score of 1, one with a score of 6, two with a score of 7, six with ascore of 8, and so on. Notice that the scores with a frequency of zero are not listedin the table.
We can doeven better. Figure 14-1 (next slide) shows the same information in a much more visual waycalled a bar graph, with the test scores listed in increasing order on a horizontalaxis and the frequency of each test score displayed by the heightof the columnabove that test score. Notice that in the bar graph, even the test scores with afrequency of zero show up–there simply is no column above these scores.
Bar graphs are easy to read, and they are a nice way to present a good general picture of the data. With a bar graph, for example, it is easy to detectoutliers–extreme data points that do not fit into the overall pattern of thedata. In this example there are two obvious outliers–the score of 24 (head andshoulders above the rest of the class) and the score of 1 (lagging way behindthe pack).
Sometimes it is more convenient to express the bar graph in terms ofrelative frequencies –that is, the frequencies given in terms of percentages ofthe total population. Figure 14-2 shows a relative frequency bar graph for theStat 101 data set. Notice that we indicated on the graph that we are dealingwith percentages rather than total counts and that the size of the data set is N = 75.
This allows anyone whowishes to do so to compute the actual frequencies. For example,Fig. 14-2 indicates that 12% of the75 students scored a 12 on theexam, so the actual frequency isgiven by 75 0.12 = 9students.
The change from actual frequencies to percentages (or vice versa)does not change the shape of thegraph–it is basically a change ofscale.
Frequency charts that use iconsor pictures instead of bars to showthe frequencies are commonly referred to as pictograms. The point ofa pictogram is that a graph is oftenused not only to inform but also toimpress and persuade, and, in suchcases, a well-chosen icon or picturecan be a more effective tool thanjust a bar.
Here’s a pictogram displaying the same data as in figure 14-2.
This figure is a pictogram showing the growth in yearly sales of theXYZ Corporation between 2001 and 2006. It’s a good picture to
show at ashareholders meeting,but the picture is actually quite misleading.
This figure shows apictogram for exactlythe same data with amuch more accurateand sobering picture ofhow well the XYZ
Corporation had beendoing.
The difference between the two pictograms can be attributed to a coupleof standard tricks of the trade: (1) stretching the scale of the vertical axis and(2) “cheating” on the choice of starting value on the vertical axis. As an educatedconsumer, you should always be on the lookout for these tricks. In graphicaldescriptions of data, a fine line separates objectivity from propaganda.