Satyagopal Mandal
Department of Mathematics
University of Kansas
Office: 624 Snow Hall  Phone: 785-864-5180
  • e-mail: mandal@math.ukans.edu
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    Chapter 14: Graphing and Summarizing Data

    We will have to deal with two types of data, namely, the sample data and the population data. Most often, for us data will be a long list of numbers. Our job here is to make inferences or predictions about the population data on the basis of the sample data.

    The branch of statistics that deals with organization, representation and summarizing (huge set of) data in understandable ways is called descriptive statistics. The other part that deals with making inferences about the population is called the inferential statistics. In this chapter we study descriptive statistics.

     

     

    Descriptive Statistics

     

    There are two ways of summarizing or representing data, namely, the Graphical (or pictorial) representation and the numerical representation. Before we proceed, let us fix some definitions. Each individual piece of data is called a data value or data point or data member. A list of data values is called a data set (which could be the whole sample data or the whole population data).

    Definition: A variable is a characteristic of the population members that varies with each member of the population.

     

    Example: Suppose we are studying the KU student population.

    1. If GPA is the "characteristic" that we are studying, then X= the GPA of students is a variable. So, given a student, X has a value. Like:


      X(Donald Smith) = 3.25 X(Sam Donaldson)=3.11
      X(Karen Currie)=3.89 X(King Who)=2.13

      and so on.
    2. On the other hand if Gender is the "characteristic" that we are studying then Y= Gender of students is a variable. Like:
      Y(Donald Smith) = Male Y(Sam Donaldson) =Male
      Y(Karen Currie) =Female Y(King Who)=Male

      and so on.
    3. If weight (in pound) is the "characteristic" that we are studying then Z= weight of students is a variable. Like:
      Z(Donald Smith) = 190 Z(Sam Donaldson) =166
      Z(Karen Currie) =130 Z(King Who)=187

      and so on.
    4. If the number of credit hours course completed is the "characteristic" that we are studying then T= the number of credit hours course completed by students is a variable. Like:
      T(Donald Smith) = 33 T(Sam Donaldson) = 102
      T(Karen Currie) = 72 T(King Who)=89

      and so on.

      Similarly, height, annual income, annual expenditure of KU students are variable.

    Types of Variables:

    1. A variable that takes numerical values is called a quantitative (or numerical) variable. There are two types of quantitative variables as follows.
      1. A quantitative variable is called a discrete variable if there is a "gap" between the values that the variable can assume. In particular, if a variable can assume only a finitely many values then it is a discrete variable. So, T= the number of credit hours completed is a discrete variable.
      2. If the difference between the values that a quantitative variable can assume can be arbitrarily small then the variable is called a continuous variable. So, weight, height above are continuous variables.
    2. A variable that assumes non-numerical values is called a qualitative variable. So, sex, blood group, hair color are examples of quantitative variables.
    Remark. We also define discrete, continuous and qualitative data in a similar way. That means, the data is called discrete, continuous or qualitative if the corresponding variable so.

    Graphical Representation

    The Pictorial representation of data is found everyday in newspapers and TV news. The most common among them are pie charts, bar graphs and histograms. We are going to discuss these three ways of pictorial representation of data.

    Frequency Table and Bar Graph of discrete data

    We will explain this with the help of the following example. (You may like to have a look at example 1, page 454, in the reference textbook. I will pick a different problem.)

    Example (see page 475 of the reference textbook): Following are scores in a test that consisted of 10 questions worth 10 points each.

    Data Set 1:Test Scores of 24 students:
    50 80 70 50 70 60 70 70
    80 70 60 100 60 80 60 80
    70 80 100 60 10 60 50 60

    Definition 1: The frequency of a data member is the number of times the data member is present in the data.

    For example, the data member 10 was present only once in the data, so the frequency of 10 is one. The data member 60 is present 7 times, so the frequency of 60 is seven. On the other hand, 90 was not present in the data, so the frequency of 90 is zero.

    Definition 2: The frequency table of a data set is a table that gives values of the variable and the corresponding frequencies. (If a value has zero frequency then we may or may not mention it in the table.) The following is the frequency table of the above data set.

    The Frequency Table of Data Set 1:
    Scores 10 50 60 70 80 90 100
    Frequency 1 3 7 6 5 0 2

    The following is the bar graph of the above data:


    Picture 6

    Definition 3: The bar graph is a pictorial representation of the data set constructed as follows:

    1. The variable values are listed in an increasing order on the horizontal axis.
    2. The frequency of each value is displayed by erecting a column, above the value, of height equal to the frequency of the value.
    Definition 4: The relative frequency bar graph is similar to bar graphs. It is constructed by erecting columns, above the data values, of height equal to the relative frequency of the value. Here relative frequency (more accurately percentage frequency) is given by

    Relative frequency = 100(frequency)¤ (Data Size).

    Remarks:

    1. The bar graphs are also known as bar charts. Many software programs (e.g. Excel) will can be used to construct bar graphs.
    2. There should be a gap between the bars in the graph.

    Use of Calculators:

    It is not easy to construct a frequency table of a data set, unless you are systematic enough. Traditionally, we used "tally mark" to count the frequency. Now you can use some software programs (e.g. Excel). Let me give you a method, using calculator (TI-83).

    1. Press "stat",
    2. to input data enter "edit",
    3. enter your data (say in L1),
    4. press "stat"
    5. enter "sortA" L1
    6. press "stat" and then enter "edit". Now on L1 you will see the data is sorted in an increasing order.
    7. Now you can count the frequencies.

    Suggested Problems: Example 1 (pp. 456), Ex.1 (pp. 475), Ex 11 (pp. 477)

     

    Class Intervals and Bar Graphs

    Some time, it is impossible and unreasonable to try to erect one bar above each data value. This is particularly so if the number of data values present in the data is too large (say more than 20). In this case, we divide the whole range of data into class intervals. The class frequency of a class interval is the number of data values that fall in the class interval. The table that gives the class intervals and corresponding class frequencies is called class frequency table. We can construct a bar graph by erecting bars above the class intervals, of height equal to the corresponding class frequencies. Normally we take 5 to 20 class intervals.

    Example: Following is the class frequency table of hourly wages (in nearest dollar) of 30 workers. Construct a bar graph.
    Class 8-9 10-11 12-13 14-15 16-17 18-19
    Frequency 5 5 8 5 2 5

    The following is the bar graph of the above data:


    Picture 7

    Suggested Problems: Example 5 (pp. 480).

    Qualitative Data and Bar Graphs

    If we have a set of qualitative data (say blood groups of a group of people) then for each variable value the class frequency is the number of times the value is present. (So, the frequency of blood group "A" is the number of people in the group with blood group "A"). The frequency table of such a data set is given by listing the variable values and the corresponding frequencies. The following is an example.

    Example: The following is a the data on the blood group of 36 patients in a hospital:
    O A B O A A A O O
    O A O A B O O O AB
    B A A O O A A O AB
    O A A B A O A O O

    The variable blood group can assume 4 values O, A, B, AB. The frequency table of this data is given as follows:
    Blood Group O A B AB
    Frequency 16 14 4 2

    Following is the bar gragh of this qualitative data:


    Picture 8

    I used Excel to sketch these bar graphs. You could experiment with any kind of software programs or calculators.

    Continuous Variables and Histograms

    If the data type is continuous then the "bar graph like" representation of data is called histogram. Since the variable is continuous and there is no "gap" between them, there will not be any gap between the bars. Given a set of data, histograms are constructed as follows:

    1. Decide how many class intervals (or bars) you want to have. Again this will be between 5 to 20.
    2. Pick a number l smaller than the lowest data value and pick a number u bigger than the largest data value.
    3. Now divide the range from l to u into the number of class intervals you want to have. (In this class, we will only have class intervals of equal length.)
    4. Make a reasonable decision which class (right of left) you want to include the data values that falls on the boundary between two class intervals. (In contrast, this is not a problem for bar graphs or discrete data.)
    5. Construct the frequency table, as before. (You can use your calculator or any software.) Now you complete the histogram by erecting bar of appropriate height above the class intervals.

    Exercise : The following table gives frequency table of the weight (in ounces) of certain number of babies in some city.
    Class Interval
    above-below
    (in ounces)    		Frequency
    48-60 15
    60 - 72 24
    72 - 84 41
    84 - 96 67
    96 - 108 119
    108 - 120 184
    120 -132 142
    132 - 144 26
    144 - 156 5
    156 - 168 2

    Construct the histogram. (Following is the bar graph. There Should not be any gap in histogram.)

    Picture 9.

    Suggested Problems: Example 7 (pp. 461), Ex. 19-22 (pp. 478)

    Pie Charts

    We see pie charts almost everyday. Latest I saw them in last March-April, in the instruction booklet for IRS Form1040. Pie Charts are self explanatory, we will not go further into them. As a matter of fact there are many ways to pictorially represent data and they are so natural that we need not go through all of them. Because of the availability of many software programs, people are using all kinds of innovative and colorful pictorial representations of data. You could also experiment with them (e. g. use Excel).
    Two Pie Charts reproduced from IRS Booklets

    Picture 10     		 
    Picture 11