B Graphing Representation of Data The second way of displaying data is by use of graphs Although such visual aids are even easier to read than tables, they often do not give the same detail It is essential that each graph be self-explanatory that is, it should has a descriptive title, labeled axes, An indication of the units of observation An effective graph should not attempt to present so much information that it is difficult to comprehend المدرج التكراري 1 Histograms A histogram is a graphical display of a frequency distribution that uses classes and vertical bars (rectangles) of various heights to represent the frequencies Histograms are useful when the data values are quantitative A histogram gives an estimate of the shape of the distribution of the population from which one sample was taken To make a histogram Make frequency table that shows class intervals and class frequencies Determine the class boundaries for each class interval Draw both abscissa (X or horizontal axis), which depicts the class boundaries (not limits), and a perpendicular ordinate (Y or vertical axis), which depicts the frequency (or relative frequency) of observations Begin the vertical scale at zero Once the scales have been laid out, a vertical bar is constructed above each class interval equal in height to its class frequency When the size of class intervals is equal, frequencies are represented by both the height and the area of each bar The total area represents 1% For example: The 1 scores in the class interval 895-195 represent a 16% of the area and that 38% of the area corresponds to the 24 observations in the second bar Class interval (Systolic Blood Pressure*) Class boundaries f (frequency) 9-19 895-195 1 11-129 195-1295 24 13-149 1295-1495 18 15-169 1495-1695 9 17-189 1695-1895 2 19-29 1895-295 Total n = 63 18
Note that, the height of the vertical scale should equal to approximately three-fourths the length of the horizontal scale Otherwise, the histogram may appear to be out of proportion with reality 25 2 15 1 5 695 895 195 1295 1495 1695 1895 Systolic blood pressure (mmhg) Figure 31 Histogram Illustrating the Systolic Blood Pressure of a Sample of 63 Nonsmokers المضلع التكراري 2 Polygons A histogram gives the impression that frequencies jump suddenly from one class to the next If you want to emphasize the continuous rise or fall of the frequencies, you can use a frequency polygon or line graph polygon uses the same axes as the histogram It is constructed by marking a dot at the class midpoint of the class interval at the height of the class frequency lower class boundary (limit) + upper class boundary (limit) class midpoint = 2 The coordinates of these dots are the class midpoint and the class frequency These points are then connected with straight lines 25 2 15 1 5 795 995 1195 1395 1595 1795 1995 Systolic blood pressure (mmhg) Figure 32 polygon of systolic blood pressure of 63 Nonsmokers 19
Note that the polygon is brought down to the horizontal axis at both ends at points that would be the midpoints if there were additional classes with zero frequency In this case, the midpoints are 795 and 1995 (Figure 32) polygons are superior, to histograms in providing a means of comparing two frequency distributions In frequency polygons, the frequency of observations in a given class interval is represented by the area contained beneath the line segment and within the class interval This area is proportional to the total number of observations in the frequency distribution polygons should be used to graph only quantitative (numerical) data, never qualitative (ie, nominal or ordinal) data since these latter data are not continuous المضلع التكراري التراآمي (Ogive) 3 Cumulative Polygons Ogive can be used to determine how many scores are above or below a set level To make an ogive Make a frequency table showing class boundaries and cumulative frequencies Class Interval (Systolic Blood Pressure*) Cumulative Relative (%) Nonsmokers Smokers 895-195 16 14 195-1295 54 55 1295-1495 83 82 1495-1695 97 9 1695-1895 1 95 1895-295 1 1 Use the same horizontal scale as that for a histogram, whereas the vertical scale indicates cumulative frequency or cumulative relative frequency For each class interval, make a dot over the upper class boundary at the height of the cumulative class frequency The coordinates of the dots are the upper class boundary and the cumulative class frequency Connect these dots with straight line segments (see the next Figure) By convention, an ogive begins on the horizontal axis at the lower class boundary of the first class interval Significance Ogives are useful in comparing two sets of data, as, for example, data on healthy and diseased individuals In the next Figure we can see that 9% of the nonsmokers and 86% of the smokers had systolic blood pressures below 16 mmhg 2
Cumulative relative frequency 1 9 8 6 5 4 2 Nonsmoker Smoker 895 195 1295 1495 1695 1895 295 Systolic blood pressure It also provides a class of important statistics known as percentiles The 9th percentile, for example, is the numerical value (16 mmhg for nonsmokers) that exceeds 9% of the values in the data set and is exceeded by only 1% of them and so on for other percentiles The 5th percentile is commonly called the median In the above Figure the median systolic blood pressure for smokers (or nonsmokers) was about 125 To get the median, we start at the 5% point on the vertical axis and go horizontally until meeting the cumulative frequency graph; the projection of this intersection on the horizontal axis is the median Other percentiles are obtained similarly 4 Stem-and-leaf Displays (طريقة الساق و الورقة ( distributions and histograms provide a useful organization and summary of data However, in a histogram, we lose most of the specific data values A stem-and-leaf plot has an advantage over a grouped frequency distribution, since a stem-and-leaf plot retains the actual data by showing them in graphic form Steps to follow in constructing a Stem and Leaf Display 1 Divide each observation in the data set into two parts, the leftmost part is called the Stem and the rightmost part is called the Leaf Note: Stems may have as many digits as needed, but each leaf contains only a single digit For grouped data, the stem represents the class intervals while the leaves are the strings of values within each class interval 2 List the stems in order from smallest to largest in a vertical column Draw a vertical line to the right of the stems 21
Stems Stems Stems Stems (Intervals) 1 3 1 9-99 2 3 4 5 2 3 1-19 3 Place all the leaves with the same stem on the same row as the stem 4 Proceed through the data set, placing the leaf for each observation in the appropriate stem row 5 Arrange the leaves in increasing order Note: The leaves (strings of observations) portray a histogram laid on its side; each leaf reflects the values of the observations, from which it is easy to note their size and frequencies Consequently, we have displayed all observations and provided a visual description of the shape of the distribution It is often useful to present the stem-and leaf display together with a conventional frequency distribution Significance From the stem-and-leaf display of the systolic blood pressure data we can see that the range of measurements is 92 to 172 The measurements in the 12s occur most frequently, with 128 being the most frequent We can also see which measurements are not represented Table Stem and leaf Display of systolic blood pressure of 63 Nonsmoker (Data from Table ) Stems (Intervals) Leaves (Observation) (f) 9-99 2 4 6 8 4 1-19 4 6 8 8 8 6 11-119 2 2 4 4 8 8 8 8 8 9 12-129 2 2 2 2 4 4 8 8 8 8 8 8 8 8 15 13-139 2 2 4 4 4 4 4 4 8 12 14-149 2 4 4 6 6 15-159 2 2 4 4 4 4 6 7 16-169 2 2 2 17-179 2 2 18-189 Total 63 22
5 Bar Graph (Chart) A bar graph is a graph composed of bars whose heights are the frequencies of the different categories in a data set Typically used for displaying categorical or qualitative (nominal or ordinal) data - shows in tabular form like blood type, ethnicity, sex, and treatment category To construct a bar graph, the categories are placed along the horizontal axis and frequencies are marked along the vertical axis A bar is drawn for each category such that the height of the bar is equal to the frequency for that category To prevent any impression of continuity, it is important to leave a small gap of equal width between the bars Bar graphs can also be constructed by placing the categories along the vertical axis and the frequencies along the horizontal axis Example The distribution of the blood type of the 25 blood donors is given in the following table Class (Blood Type) 1 8 A 5 B 8 O 8 AB 4 Total 25 6 4 2 A B O AB Blood Type (Group) Figure Bar chart of blood type It is essential that the scale on the vertical axis begin at zero If that is impractical, one should employ broken bars (or a similar device), as shown in Figure below 12 Excess mortality (%) 1 8 6 4 Less than 1 1-19 2-39 4or more Number of cigarettes smoked per day Figure Bars broken to show vertical scale does not begin at zero 23
6 Pareto Chart A Pareto chart is a type of bar chart in which the horizontal axis represents categories of interest The bars are ordered from largest to smallest in terms of frequency counts for the categories A Pareto chart can help you determine which of the categories make up the critical few and which are the insignificant many 14 12 1 8 6 4 7 Time plot A time plot is a graph display how data change over time To make a time plot, we put time on the horizontal scale and the variable being measured on the vertical scale In a basic we connect the data points by lines It is best if the units of time are consistent in a given plot For instance, measurements taken every day should not be mixed on the same plot with data taken every week Example How does average height for boys changes as the boy gets older? According to Physician s Handbook, the heights at the different ages are as follows: Age (year) Height (inches) 5 26 1 29 2 33 3 36 4 39 5 42 6 45 7 47 8 5 9 52 1 54 11 56 12 58 13 6 14 62 2 Primary None Height (inches) Intermediate Education level 65 6 55 5 45 4 35 3 25 Senior High Technical School 2 1 2 3 4 5 6 7 8 9 1 11 12 13 14 15 Age (year) 24
8 Pie Charts A pie graph or pie chart is a circle that is used to graphically display either qualitative or quantitative data A pie chart allows us to observe the proportions of sectors relative to the entire data set Constructing a pie chart To construct a pie chart, a circle is divided into portions that represent the relative frequencies or percentages belonging to different categories Example 1 To construct a pie chart, construct a frequency table that gives frequency, relative frequency and the angle sizes for each category The Table below shows the determination of the angle sizes for each of the categories (primary sites for cancer of 75 patients) The 36 o in a circle are divided into portions that are proportional to the category sizes Primary Site Relative frequency Angle size Digestive system 2 26 36 x 26 = 936 o Respiratory system 3 4 36 x 4 = 144 o Breast 1 13 36 x 13 = 468 o Genitals 5 7 36 x 7 = 252 o Urinary tract 5 7 36 x 7 = 252 o Others 5 7 36 x 7 = 252 o Total 75 1 Urinary tract 7% Genitals 7% Breast 13% Others 7% Digestive system 26% Respiratory system 4% Example 2, if you spend 7 hours at school and 55 minutes of that time is spent eating lunch, then 131% of your school day was spent eating lunch (55/42 x 1 =131) To present this in a pie chart, you would need to find out how many degrees represent 131% This calculation is done by developing the equation: (percent 1) x 36 degrees = the number of degrees This ratio works because the total percent of the pie chart represents 1% and there are 36 degrees in a circle Therefore 471 degrees of the circle (131%) represents the time spent eating lunch 25