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Файл:Ординатура / Хирургия / Библиотека им академика М.И. Перельмана / Книга_155_библиотеки_им_акад_М_И_Перельмана
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22 2 Tables and Charts
there are 10 infants weighing between 3.50 and 3.99 kg whose mothers did not smoke
at all during pregnancy.
2.5 Graphs
For many people, graphical presentation is more tangible and easier than presenting
tables and raw numbers and data. This is because they may not have the statistical
knowledge or patience to calculate percent and …. Therefore, the information
obtained by looking at a chart is much easier than that obtained by reading and
explaining. For this reason, it will be useful to present a graph of the study results.
Before explaining in detail the pie chart, the bar chart, the histogram, the polygon,
the stem-and-leaf diagram, the boxplot, and the scattergram, we will outline the
common features to be observed in these diagrams.
1. Each diagram must have a title.
2. Include the source of the information.
3. A guideline should be included on the top corner.
4. Unit of measurement and scale should also be given.
5. The beauty and neatness of the chart are important for the attention of the viewer.
6. If the distance of the smallest value observed in each of the axes from the origin
of coordinates (zero) is large, it is necessary to put a cut sign (
) in front of it.
Pie Chart
A frequently used graphical representation for illustrating the proportions within
categorical data is the pie chart, wherein a circle is segmented into sectors, each
representing a distinct category. The size of each sector corresponds to the frequency
count of its respective category, or its relative frequency. For instance, if a category
possesses a relative frequency of 0.18, its sector would occupy 18% of the circle. If
the data is on qualitative scale (nominal and ordinal), the observations themselves
can be used directly to draw a pie chart. But if the data is quantitative, they should
be categorized first and then the categorized data used for the pie chart.
To draw a pie chart, the area of the circle (360° of central angle) is divided
according to the relative frequency of each group. To do this, we multiply the relative
frequency of each category by 360° and separate the corresponding sectors. For
the above example that has 18% of the total frequencies, the sector should have
0.18 × 360 = 64.8
◦
.
Different sectors can be prominent by different colors or patterns. Software usually
places a color guide in the upper corner. This chart can be simple or three-dimensional
and prominent. Sometimes, one of its sectors is raised a little to be shown separately.
The steps of drawing pie chart with SPSS software for 200 people with blood
type (A = 59, B = 49, AB = 16, O = 76) are shown in the figure below. To do this,
choose path graphs\legacy dialogs\pie to open another sub-window (Fig.
2.1).
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2.5 Graphs 23
Fig. 2.1 Pie chart of blood types for a sample of 200
Bar Graph/Bar Chart
A bar chart serves as a visual depiction of the frequency distribution of categorical
data, offering an alternative to the pie chart. Comprised of equally wide bars
(rectangles), each corresponding to a category, the bar graph illustrates the
frequencies of categorical data. The height of each rectangle reflects the frequency
(or relative frequency) of its respective category. To prevent visual errors and
ambiguity and misinterpretation, the width of the bars should be equal and it is
recommended that the distance between the bars be equal.
Tip
It is easier to compare different categories with this chart than with a pie chart.
To draw this graph, the Y-axis is divided according to the absolute frequencies (or
relative frequency or percentage). The X-axis should also be according to the different
classes of the variable under study. The height of the bars should also be proportional
to the frequency of each category. The use of different colors and patterns such as
hatching and shading makes the graph more beautiful.
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24 2 Tables and Charts
The steps for drawing a bar chart with SPSS software are shown in Fig. 2.2.To
do this, first select the path graphs\legacy dialogs\bar to open the bar charts window.
Pareto Chart
When a bar graph arranges its bars in descending order of frequency or relative
frequency, placing the highest frequency or relative frequency on the left and the
lowest on the right, it is termed as a Pareto chart. These charts are valuable when
clarity is sought regarding the most commonly occurring categories.
Fig. 2.2 Bar chart of blood types for a sample of 200
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2.5 Graphs 25
Stem-and-Leaf Plot/Stemplot
This type of chart is employed to showcase quantitative data, whether continuous or
discrete. Stem-and-leaf displays serve to illuminate the relative density and shape of
the data, providing readers with a swift overview of its distribution. In a stemplot
(or stem-and-leaf plot), each quantitative value is separated into two components:
the stem (typically the leftmost digit) and the leaf (usually the rightmost digit). The
advantage of this plot lies in its ability to not only exhibit chart-like features but also
allow readers to grasp the original data and interpret indicators such as the mode and
median easily. Examples of stem-and-leaf diagrams can be seen in Figs.
3.2 and 4.2
.
To construct a stem-and-leaf display, the observations must first be arranged in
ascending order. Subsequently, the stems and leaves must be defined. Typically, the
leaf comprises the last digit of each number (the rightmost digit), while the stem
encompasses all other digits. For very large numbers or those with multiple digits,
values may be rounded to a specific place value (e.g., hundreds), with the remaining
digits on the left serving as the stem. It’s worth noting that some of the remaining
stems may be duplicates or like each other.
Then we draw a vertical bar and write the stems on the left side of this Bar. Then
we read the sorted numbers in order and placed each leaf in front of its stem (to the
right of the vertical line). If the numbers are repeated, the leaves should be written
on the right side of the vertical line as many times as repeats. If the value of the leaf
(the digit on the right) is zero, zero must also be written.
If the data includes two subgroups (such as sick and healthy or male and female),
a back-to-back stemplot can be used to better compare these subgroups.
Example 8 In example 2, suppose the first 50 patients are female and the next 40
patients are male. Draw a back-to-back stemplot.
Solution: To draw this plot, we first sort 50 female patients and 40 male patients
separately. Then in each group, we separate the right digit as “leaf”. What remains
is “stems”, which include Figs. 1 to 9. We write them in a column in the middle of
the page and, as in the previous exercise, we write down the values for women on
the right and the values for men on the left side of the column (stem).
As in the previous exercise, we write the values for women on the right and the
values for men on the left side of the stems’ column.
Female 29, 31, 32, 35, 36, 36, 37, 38, 40, 41, 42, 45, 45, 47, 48, 48, 49, 51, 53, 55, 58, 59, 59,
60, 61, 61, 62, 63, 65, 66, 67, 68, 68, 69, 69, 72, 75, 79, 80, 80, 80, 81, 82, 83, 84, 84,
84, 85, 86, 88
Male 16, 21, 22, 26, 27, 27, 32, 34, 37, 38, 40, 42, 42, 45, 46, 47, 48, 50, 50, 52, 53, 54, 56,
57, 61, 61, 61, 62, 63, 64, 65, 66, 71, 72, 73, 76, 80, 81, 89, 90
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26 2 Tables and Charts
Male age Female Age
6
1
776 21
2
9
87 42
3
1256678
87 6 52 2 0
4
0125578899
76 4 32 0 0
5
135899
654 3 2 1 1 1
6
011235678899
63 21
7
259
910
8
000123444568
0
9
To draw this plot in SPSS, first choose the path analyze\descriptive
statistics\explore to open a window like below. Then enter the desired quantitative
variable in the “dependent list” box and the categorized quality variable in the
“factor list” box. Click the plots button to open another window and select the
stem-and-leaf option to draw a chart.
Note: To activate the plots option, both or plots options must be checked at the
bottom of the left window.
Note: To activate the plots option, one of the both or plots options must be marked
at the bottom left of this window (Fig.
2.3).
Fig. 2.3 Stem-and-leaf diagram of patients referred to a health center
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2.5 Graphs 27
Fig. 2.4 Stem-and-leaf plot
with frequencies
It is also recommended that on the left side of this diagram, a column is allocated
to the cumulative frequencies of data up to the median. In this way, we add the
frequencies of each row from the top with the frequencies of the previous row and
write in the corresponding cell until we reach the median of the distribution. Then,
from the bottom, we add the frequency of each row to the frequencies of the next
row and write it in its corresponding cell to reach to the median from this side as
well (the median is the value of an observation from which half of the observations
are less and the other half are larger than it).
Example 9 For example, for patients referred to the health center, the eleventh value
(25) is the median and is located in stem 2, and the absolute frequency of that stem
(6) is written in brackets (Fig.
2.4).
Histogram
This chart is used for classifying quantitative data (such as age). In this diagram,
the Y -axis is scaled in terms of frequency (absolute frequency or relative frequency
or percentage of frequency) and the X-axis is scaled according to the boundaries
of the classes. In this diagram, the rectangles are glued together, the width of each
rectangle is proportional to the width of the classes, and their area is proportional to
the observed frequency of the class. In this diagram, the height of the rectangles can
only be proportional to the frequency of observations when the width of the classes
is equal.
To avoid any ambiguity and to obtain the height and area of each rectangle relative
to the frequency classes, it is recommended that the frequency of each class be divided
by the width of its class.
To draw a histogram with SPSS, first choose the path graphs\legacy
dialogs\histogram to open a window (Fig. 2.5).
Polygon and Ogive
These two similar charts are suitable for representing quantitative data. Firstly, let’s
discuss how to create a polygon diagram and then an Ogive.
Tip
The midpoint of a class is the average of its lower limit and the upper limit of
the next class.
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28 2 Tables and Charts
Fig. 2.5 Histogram of the age of 120 nurses working in the hospital
To draw a polygon diagram, we start by calculating the class midpoints, which
are the averages of the lower and upper boundaries of each class. Next, we position
each class along the X-axis based on its midpoint. The Y-axis is scaled according to
the observed frequency ( either absolute frequency, relative frequency, or frequency
percentage). After determining the points where the X-coordinates represent the class
midpoints and the Y-coordinates represent the frequencies, we connect these points
sequentially with straight lines to form the frequency polygon.
Class midpoint =
Lower limit + Lower limit of next class
2
.
An Ogive plot, pronounced “oh jive”, is another type of polygon that utilizes
cumulative frequencies and cumulative relative frequencies instead of absolute
frequency and relative frequency. To draw a polygon with SPSS, first choose the
path graphs\legacy dialogs\line to open a window. After the new window was
opened, if you choose one of the two options in the first row (% of cases or no. of
cases), a polygon plot is drawn, and if you choose one of the two options in the next
row (Cum N or Cum%), the Ogive plot is drawn (Fig.
2.6).
Scatterplot
A scatter plot, also known as a scatterplot, scatter graph, scatter chart, point graph,
X–Y plot, or scattergram, is a type of plot used to depict the relationship between two
quantitative variables (x, y ). It offers a visual representation of how these variables
are related and can suggest the presence and nature of a linear relationship between
them.
To construct a scatterplot, the horizontal X-axis is dedicated to the first variable
(x), while the vertical Y-axis represents the second variable (y). If both variables
share the same measurement scale, it is advisable to maintain uniform scaling on
both the X and Y axes.
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2.5 Graphs 29
Fig. 2.6 Polygon and Ogive diagram for hemoglobin level of 70 females
Then we find the coordinates of all the observations in pairs (Y and X )onthe
coordinate system and mark it with a symbol. The set of specified points will give a
general glance of the relationship between the two variables.
Obviously, if the values of two or more pairs of observations are the same and
equal, there will be the same number of overlapping or very close points on the
graph. For this reason, some points on the chart may appear more colorful than other
non-duplicate points.
This scatterplot can be plotted for different subgroups of a dataset with different
symbols to reveal if the relationship between two variables is different from each
other in various subgroups. This plot can be drawn for a variable that has been
measured at two different times (before and after an event).
To draw the scatterplot with SPSS, choose the path graphs\legacy dialogs\scatter/
dot to open the following window (Fig.
2.7).
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30 2 Tables and Charts
Fig. 2.7 Weight and arm circumference scatterplot of 50 persons
Boxplot
This plot is used for quantitative data (continuous or discrete). To draw the graph,
calculating of central indices of the data is required and only one axis is scaled (Y -
axis for vertical presentation and X-axis for horizontal presentation). A boxplot, also
known as a box-and-whisker diagram, is a graphical representation of a dataset. It
includes a line that extends from the minimum value to the maximum value, along
with a box that encompasses the interquartile range (IQR)—the distance between
the first quartile Q
1
and the third quartile Q
3.
Additionally, lines are drawn at the first
quartile Q
1
, the median, and the third quartile Q
3
.
Tip
An outlier is a value that is considerably larger or considerably smaller than
most of the values in a dataset. In other words, observed values that lie very
far away from the vast majority of the other sample values.
To draw the boxplot, we have to calculate first quartile (Q
1
), second quartile or
median (
Q
2
= Med) and the third quartile (Q
3
) of the dataset.
Then we mark the location of the first, second and third quartiles on the Y -axis
(these indices are non-parametric and do not depend on the value of observations
and their distribution, but only depend on the order in which they are placed. If
the assumption of normality is met for the distribution of data, mean and standard
deviation can be used instead). Then specify the maximum and minimum values of
the data and draw a line from each one to the box.
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2.5 Graphs 31
Steps of Construction of a Boxplot
Step 1: Compute the first quartile, the median, and the third quartile.
Step 2: Draw X-axis and then vertical lines at the first quartile, the second and
the third quartile.
Connect horizontal lines between the first and third quartiles to complete
the box and create a rectangle.
Step 3: Compute the lower outlier boundaries (LOB) and upper outlier
boundaries (UOB).
Step 4: Find the largest data value that is less than the UOB. Draw a horizontal
line from the third quartile to this value. This horizontal line is called a whisker.
Step 5: Find the smallest data value that is greater than the LOB. Draw a
horizontal line (whisker) from the first quartile to this value.
Step 6: Determine and plot any values, which are outliers, separately.
The width of the box in a boxplot is often proportional to the sample size, with
a wider box indicating a larger sample size. It is common to relate the width of the
box to the square root of the sample size, although this proportionality is not always
strictly followed. Some statistical software packages offer modified boxplots that
represent outliers as distinct points, such as asterisks or dots, and the solid horizontal
line extends only to the minimum and maximum data values that aren’t outliers.
The most commonly used method for identifying outliers is the interquartile range
(IQR) method. The outlier boundaries are determined as follows:
•
Lower outlier boundary = Q
1
− 1.5 × IQR
•
Upper outlier boundary = Q3 + 1.5 × IQR.
Any observed data value falling below the lower outlier boundary or exceeding
the upper outlier boundary is classified as an outlier.
When data can be partitioned into subsets according to another qualitative feature
(such as gender or marital status), it is very useful to draw a boxplot for each subset
and compare the subgroups.
In the boxplot, the height or length of the box contains 50% of the middle of the
dataset and shows the interquartile range (Q
3
–Q
1
).
To draw boxplot with SPSS, first choose the path graphs\legacy dialogs\boxplot
to open a window (Fig.
2.8).
Although a boxplot is determined by the five indexes (minimum, first quartile,
median, third quartile, and maximum value), it does not provide detailed information
about the distribution of the data compared to histograms or stemplots. However,
boxplots excel in comparing several datasets. To effectively compare two or more
different datasets using boxplots, it is essential for them to share the same scale,
facilitating straightforward comparisons.
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