# Chapter 2: Frequency Distributions and Graphs (or making pretty tables and pretty pictures)

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1 Chapter 2: Frequency Distributions and Graphs (or making pretty tables and pretty pictures) Example: Titanic passenger data is available for 1310 individuals for 14 variables, though not all variables are recorded for all individuals. Consider the following variables: Survival, Sex, Number of relatives on board, Age Who wants to stare at a big dataset? If you have 1310 people measured for 14 variables, how much information are we going to get by looking at the data set? See for yourself: That s where tables that summarize the data and graphs of these summaries come in handy! Ch2: Frequency Distributions and Graphs Santorico -Page 26

2 Section 2-1 Organizing Data Data must be organized in a meaningful way so that we can use it effectively. This is often a pre-cursor to creating a graph. Frequency distribution the organization of raw data in table form, using classes and frequencies. Class a quantitative or qualitative category. A class may be a range of numerical values (that acts like a category ) or an actual category. Frequency the number of data values contained in a specific class. Ch2: Frequency Distributions and Graphs Santorico -Page 27

3 There are 3 types of frequency distributions: Categorical frequency distributions Ungrouped frequency distributions Grouped frequency distributions Qualitative Variables Quantitative Variables Let s start with Categorical frequency distributions frequency distribution for qualitative data. Review: What is qualitative data? Ch2: Frequency Distributions and Graphs Santorico -Page 28

4 Titanic Example: Survival status and sex are qualitative variables. The following tables give their categorical frequency distributions. Survival Status Frequency Sex Frequency Yes 500 Female 466 No 809 Male 843 We ll come back for graphs which can include a pie graph, bar chart or Pareto chart. Example: Areas of study for students in our class Area of Study Medical Sciences Public Health Biology Education Geography Other Frequency Ch2: Frequency Distributions and Graphs Santorico -Page 29

5 For quantitative variables we have grouped and ungrouped frequency distributions. An Ungrouped Frequency Distribution is a frequency distribution where each class is only one unit wide. Meaningful when the data does not take on many values. Each class is constructed using a single data value for each class, e.g., 0, 1, 2, 3,, 10 Class boundaries will be defined to separate the classes (when graphing) so there are no gaps in the frequency distribution. o Should have one additional decimal place and end in a 5. o The lower boundary will round to the lower class limit. o The upper boundary will round to the next class o Another way of thinking about this: draw the boundary half way between consecutive classes. Ch2: Frequency Distributions and Graphs Santorico -Page 30

6 Titanic example: Number of relatives on board Number of Relatives on Board Class Boundaries Frequency Is this an ungrouped frequency distribution? Ch2: Frequency Distributions and Graphs Santorico -Page 31

7 Grouped frequency distribution frequency of a quantitative variable with a large range of values, so the data must be grouped into classes that are more than one unit in width. Class Limits Age Group in Years (Lower, Upper) Class Boundaries (Lower, Upper) Frequency Cumulative Frequency Age of Passengers on the Titanic Classified into 17 Age Groups with a class wide of 5 years Ch2: Frequency Distributions and Graphs Santorico -Page 32

8 Guidelines: There should be between 5 and 20 classes. The classes must be mutually exclusive (nonoverlapping). Makes placing observations into classes unambiguous. The classes must be continuous. There should be no gaps in the frequency distribution. The classes must be exhaustive. The classes should accommodate all the data. The classes must be equal in width. Avoids a distorted view of the data. Ch2: Frequency Distributions and Graphs Santorico -Page 33

9 TERMINOLOGY Class Limits the range of values to be included in a class. Will have the same decimal place value as the data. Class Boundaries used to separate the classes (when graphing) so there are no gaps in the frequency distribution. Should have one additional decimal place and end in a 5. The lower boundary will round to the lower class limit. The upper boundary will round to the next class Another way of thinking about this: draw the boundary half way between consecutive classes. Class width found by subtracting the lower class limit of one class from the lower class limit of the next class (OR likewise by subtracting an upper class boundary from the corresponding lower class boundary.) Ch2: Frequency Distributions and Graphs Santorico -Page 34

10 Steps for Constructing a Grouped Frequency Distribution: 1. Determine the classes Find the range of the data = largest value minus the smallest value Find the class width by dividing the range by the number of classes and rounding up. (Add 1 if this value is a whole number). Select a starting point (usually the lowest value); add the width to get the lower limits of all subsequent classes. Find the upper class limit for each class by subtracting 1 unit from the last decimal place of the lower class limit of the next class. o If the lower class limit of the next class is 12.5, then the upper class limit for the previous class is o If the lower class limit of the next class is 11, then the upper class limit of the previous class is 10. o If the lower class limit of the next class is 18.55, then the upper class limit of the previous class is Find the boundaries. 2. Tally (count) the data in each class. 3. Find the numerical frequencies from the tallies. Ch2: Frequency Distributions and Graphs Santorico -Page 35

11 Titanic Example: Ages of passengers in first class Ch2: Frequency Distributions and Graphs Santorico -Page 36

12 Let s use 8 age classes: Class Class Cumulative Limits Boundaries Frequency Frequency Sanity check: make sure that the last cumulative frequency actually matches with the number of observations! Ch2: Frequency Distributions and Graphs Santorico -Page 37

13 Cumulative frequency distribution a distribution that shows the number of observations less than or equal to a specific value. These are easy to determine from a frequency distribution. Simply add the number of observations for each class with the previous classes. Start with the first lower class boundary and then use every upper class boundary. Ch2: Frequency Distributions and Graphs Santorico -Page 38

14 Continuing the titanic first class passenger age example: Less than Less than Less than Less than Less than Less than Less than Less than Less than Less than Less than Cumulative Frequency ` Ch2: Frequency Distributions and Graphs Santorico -Page 39

15 Section 2-2 Histograms, Frequency Polygons, and Ogives (making pretty pictures for counts from quantitative variables) Pronounced O-JIVE Ch2: Frequency Distributions and Graphs Santorico -Page 40

16 Histogram a graph that displays quantitative data by using contiguous vertical bars (unless the frequency of a class is 0) of various heights to represent the frequencies of the classes. Steps: 1. Draw and label the x and y axes. 2. Represent the frequency on the y axis and the class boundaries on the x axis. 3. Draw vertical bars which have height corresponding to the frequency of each class. Ch2: Frequency Distributions and Graphs Santorico -Page 41

17 Example: Use the frequency distribution for the ages of titanic first class passengers to create a histogram. Ch2: Frequency Distributions and Graphs Santorico -Page 42

18 Frequency Polygon a graph that displays the data by using lines that connect points plotted for the frequencies at the midpoints of the classes. The frequencies are represented by the heights of the points. Steps: 1. Find the midpoint of each class (by averaging the class boundaries). 2. Draw the x and y axes. Label the x axis with the midpoints of each class and use a suitable scale for the y axis frequencies. 3. Using the midpoints of the classes for the x value and the frequencies for the y value, plot the points. 4. Connect the adjacent points with line segments Draw a line back to the x-axis at the beginning and end of the graph, at the same distance that the previous and next midpoints would be located. a. To calculate these points, subtract the class width from the midpoint of the first class and add the class width to the midpoint of the last class. Ch2: Frequency Distributions and Graphs Santorico -Page 43

19 Example: Draw a frequency polygon for the ages of titanic first class passengers. Ch2: Frequency Distributions and Graphs Santorico -Page 44

20 Frequency polygons vs histograms: They are two different ways to represent the same data set. The choice is at the researcher s discretion. They show the overall shape of the distribution of the variable. Ch2: Frequency Distributions and Graphs Santorico -Page 45

21 Ogive ( o-jive ) a graph that represents the cumulative frequencies for the classes in a frequency distribution. Ogives are used to visually represent how many values are below a certain upper class boundary. Steps: 1. Draw the x and y axes. Label the x axis with the class boundaries. Use an appropriate scale for the y axis. 2. Plot the cumulative frequency at each class boundary. 3. Connect the points with lines. Ch2: Frequency Distributions and Graphs Santorico -Page 46

22 Example: Draw an ogive for the ages of titanic first class passengers. Ch2: Frequency Distributions and Graphs Santorico -Page 47

23 Relative frequency graph a distribution graph that uses proportions instead of frequencies. Histograms, frequency polygons, and ogives can be converted into relative frequency graphs by converting the frequencies into proportions/percents. Divide the frequency (or cumulative frequency) by the total number of observations. Use the relative frequencies on the y-axis rather than the frequencies Ch2: Frequency Distributions and Graphs Santorico -Page 48

24 DISTRIBITION SHAPE When describing data, it is important to be able to recognize the shapes of the distribution values. Being able to identify the shape of the distribution will be important later when we are choosing appropriate statistical tests. Histograms and frequency polygons allow us to see the distribution of the data. [Type text] Page 49

25 Section 2-3 Other Types of Graphs Bar Graph represents the data by using vertical or horizontal bars whose heights or lengths represent the frequencies of the data. Used for qualitative data. The bars of a horizontal bar graph are horizontal. Relevant to qualitative variables. Steps: 1. Draw the x and y axes. Label the x axis and y axis. 2. Draw of the appropriately sized bars for each category. Ch2: Frequency Distributions and Graphs Santorico - Page 50

26 The following bar graph is for the survival by sex groups on the titanic: F/Dead F/Alive M/Dead M/Alive Ch2: Frequency Distributions and Graphs Santorico - Page 51

27 Pareto Chart used to represent a frequency distribution for a categorical variable, and the frequencies are displayed by the heights of vertical bars, which are arranged in order from highest to lowest. This is a bar chart with a special ordering. The Pareto chart has the qualitative variable on the x axis. The highest frequency categories are on the left and the lowest frequency categories are on the right. Suggestions for Drawing Pareto Charts Make the bars the same width. Arrange the data from largest to smallest according to frequency. Make the units that are used for the frequency equal in size. Ch2: Frequency Distributions and Graphs Santorico - Page 52

28 Example: Construct a horizontal bar graph and a Pareto chart for the passenger class by survival categories Group Frequency 1 st Class, Survived st Class, Dead nd Class, Survived nd Class, Died rd Class, Survived rd Class, Dead 528 Ch2: Frequency Distributions and Graphs Santorico - Page 53

29 Ch2: Frequency Distributions and Graphs Santorico - Page 54

30 Pie Graph a circle that is divided into sections or wedges according to the percentage of frequencies (relative frequency) in each category of the distribution. When analyzing a pie graph, examine the size of the sections of the pie graph and compare them to other sections and how they compare to the whole. Ch2: Frequency Distributions and Graphs Santorico - Page 55

31 TWO OTHER USEFUL GRAPHS: Time series graph represents data that occur over a specific period of time (time is your x-axis). Steps: 1. Draw and label the x and y axes. 2. Plot a point for each time at the corresponding value in the y direction. 3. Connect the points. Notes on Time Series Graphs: 1. Do not extend the ends of the times series graph to the x- axis unless it makes sense that the data would fall to zero (which is rarely the case)! 2. Two data sets can be compared on the same graph, called a compound time series graph. 3. When you analyze a time series graph, look for patterns or trends that occur over time. Ch2: Frequency Distributions and Graphs Santorico - Page 56

32 Example: Draw a time series graph to represent the data for the number of airline departures (in millions) for the given years. Over the years, is the number of departures increasing, decreasing, or about the same? Year # of Departures (in millions) Ch2: Frequency Distributions and Graphs Santorico - Page 57

33 Ch2: Frequency Distributions and Graphs Santorico - Page 58

34 Stem and leaf plot a data plot that uses part of the data value as the stem and part of the data value as the leaf to form groups or classes. Steps: 1. Order the data. 2. Separate the data by the first digit. 3. Use the first digit as the stem and trailing digit as a leaf. Notes on Stem and Leaf Plots: When you analyze a stem and leaf plot, look for peaks and gaps in the distribution. See if the distribution is symmetric or skewed. Check to see how variable the data is by examining how spread out the data is. A back-to-back stem and leaf plot can be created to compare two data sets using the same digits as stems. Ch2: Frequency Distributions and Graphs Santorico - Page 59

35 Example: Consider the following data on car thefts in large cities and draw a stem and leaf plot: Ch2: Frequency Distributions and Graphs Santorico - Page 60

36 You can construct all of the graphs we have discussed in Excel. See pages of the textbook for a description. Ch2: Frequency Distributions and Graphs Santorico - Page 61

37 Misleading Graphs When creating graphs we must be careful not to misrepresent the data by inappropriately drawing the graphs. When the graphs misrepresent the data they can lead the reader to draw false conclusions. Let s look at a few examples. Why are they misleading and how should they be correctly portrayed? Ch2: Frequency Distributions and Graphs Santorico - Page 62

38 Example 1: Using a scale that doesn t go from 0 to 100%. The graph below shows the percentage of the manufacturer s automobiles still on the road and the percentage of its competitors automobiles still on the road. Is there a large difference? Versus On the full scale, it is clear that the manufacturer s numbers are comparable! Ch2: Frequency Distributions and Graphs Santorico - Page 63

39 Example 2: Changing the units or starting point on they-axis. Versus Looks like much more change as occurred. That said, which way is the correct way to plot the data? Ch2: Frequency Distributions and Graphs Santorico - Page 64

40 Example 3: Not labeling or not putting units on the y-axis. With no given scale it is impossible to tell if the differences are meaningful. Not to mention, what do N, E, S and W represent? Ch2: Frequency Distributions and Graphs Santorico - Page 65

41 Summary of Graphs Ch2: Frequency Distributions and Graphs Santorico - Page 66

42 What graphs are used for qualitative data? Histogram Bar chart Freqency polygon Pareto chart Ogive What graphs are used for quantitative data? Stem and leaf plot Time series graph Pie chart Ch2: Frequency Distributions and Graphs Santorico - Page 67

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