Graphic representations in statistics. Graphic representation and graphic analysis n Graphic...

Graphic representations in

statistics

Graphic representation and graphic analysis

Graphic representations are used for evident representation of statistical quantities they allow to analyze them deeper.

The graphic representation can be built both after absolute and after relative quantities.

Using the graphic method, it is important to know that the type of graphic representation must strictly answer the maintenance of every index.

Source data and graph types

Relative quantities:

intensive indices extensive indices index of correlation index of evidence


Intensive quantities - 4 types of diagrams:- column- linear- mapgram - mapdiagram


Extensive quantities: (they characterize the structure) sector or inwardly-column diagram.

Indices of correlation: the same diagrams, that for intensive quantities (column and linear diagrams, mapgram, mapdiagram).


Indices of evident: the principles of graphic representation are the same, that for intensive quantities.

Column diagrams – for illustration of homogeneous, but not interconnected indices. They represent the static’s of the phenomena.


Linear diagrams – for the representations of dynamics of that or other phenomenon (a typical example is a temperature curve, change of birth-rate, death rate level).

Radial diagram – is built on the system of polar co-ordinates of the phenomenon representations during the close cycle of time (days, week, year). For example: structure of morbidity or cause of mortality, where in a circle every cause of mortality, depending on its percent occupies a certain sector.


Mapgram is the representations of statistical quantities on a geographical map (or scheme of card). Absolute and other indices can be marked.

Mapdiagram is the representations of different types of diagrams on the geographical map.

Common rules of construction of graphs

every graphic representation must have a name, where its contest, time and place is mentioned;


it must be built to a certain scale;


for every graphic representation explanation of colored application must be given (as conditional denotations or shading).

Types of graphs

Linear diagram is used for illustration of the frequency phenomena which changes with time, that for the representations of the phenomena dynamics.

The base of this diagram is the rectangular system of co-ordinates. For example: on abscising axis – Х - segments are put aside on a scale, on a y-axis – indices of morbidity (х : y = 4: 3).

Types of graphs

Column diagram (rectangular) is used for illustration of homogeneous, but not connected between themselves intensive indices. It represents dynamics or static of the phenomena.

At construction of this kind of diagram columns are drawn, the height of which must suit the quantities of the represented indices taking the scale into account. It is necessary to take into account that the wideness of all the columns and also the distance between them must be identical and arbitrary. Columns on a diagram can be vertical or horizontal. For example: growth of number of beds in permanent establishment from 1990 to 2003 year.

Types of graphs

Sector diagram is used for illustration of extensive indices, which characterize the structure of the phenomenon, thus they give imagination about specific gravity of the phenomena in common.

The circle is taken as 100 % (if indices are shown in %) thus 1 % equal to 3,60 circumferences. With the help of protractor the segments, which suit the size of an index are put aside on a circle.

Types of graphs

Radial diagram is the type of the linear diagram built on polar co-ordinates.

At the construction of radial diagram in the role of abscising axis - Х is the circle divided on the identical number of parts, according to the spans of time of that or another cycle.

Types of graphs

Linear diagrams are made according to system of rectangular coordinates where the horizontal scale is postponed at the left - to the right on a line of abscissas (X), and vertical - from below - upwards on a line which is called as ordinate (Y). The obligatory requirement of construction of any schedule is scale, that is the image on drawing should be reduced, compared with corresponding figures.

Types of graphs

Long-pillar diagrams are for comparison of several sizes. It is possible to rectangular which represent sizes, it is possible to place also on the plane diagram not on a vertical, and across and then there will be a tape diagram (Fig.4). In some cases the image of sizes as tapes (stirs) is more convenient, than as columns because it is easier to sign with each tape by a horizontal inscription.

Quantitative types of connection

functional one is the connection, at which the strictly defined value of the second sign answers to any value of one of the signs (for example, the certain area of the circle answers to the radius of the circle)

Quantitative types of connection

correlation - connection at which a few values of one sign answer to the value of every average size of another sign associated with the first one (for example, it is known that the height and mass of man’s body are linked between each other; in the group of persons with identical height there are different valuations of mass of body, however, these valuations of body mass varies in certain sizes – round their average size).

Correlative connection

Correlative connection foresees the dependence between the phenomena, which do not have clear functional character.

Correlative connection is showed up only in the mass of supervisions that is in totality. The establishment of correlative connection foresees the exposure of the causal connection, which will confirm the dependence of one phenomenon on the other one.

Correlative connection

Correlative connection by the direction (the character) of connection can be direct and reverse. The coefficient of correlation, that characterizes the direct communication, is marked by the sign plus (+), and the coefficient of correlation, that characterizes the reverse one, is marked by the sign minus (-).

By the force the correlative connection can be strong, middle, weak, it can be full and it can be absent.

Estimation of correlation by coefficient of correlation

Force of connection Line (+) Reverse (-)

Complete +1

Strong From +1 to +0,7 From -1 to -0,7

Average from +0,7 to +0,3 from –0,7 to –0,3

Weak from +0,3 to 0 from –0,3 to 0

No connection 0 0

Types of correlative connection

By direction

direct (+) – with the increasing of one sign increases the middle value of another one;

reverse (-) – with the increasing of one sign decreases the middle value of another one;

Types of correlative connection

By character rectilinear - relatively even changes of

middle values of one sign are accompanied by the equal changes of the other (arterial pressure minimal and maximal)

curvilinear – at the even change of one sing there can be the increasing or decreasing middle values of the other sign.

Terms Used To Describe The Quality Of Measurements

Reliability is variability between subjects divided by inter-subject variability plus measurement error.

Validity refers to the extent to which a test or surrogate is measuring what we think it is measuring.

Measures Of Diagnostic Test Accuracy

Sensitivity is defined as the ability of the test to identify correctly those who have the disease.

Specificity is defined as the ability of the test to identify correctly those who do not have the disease.

Predictive values are important for assessing how useful a test will be in the clinical setting at the individual patient level. The positive predictive value is the probability of disease in a patient with a positive test. Conversely, the negative predictive value is the probability that the patient does not have disease if he has a negative test result.

Likelihood ratio indicates how much a given diagnostic test result will raise or lower the odds of having a disease relative to the prior probability of disease.

Measures Of Diagnostic Test Accuracy

Expressions Used When Making Inferences About Data

Confidence Intervals- The results of any study sample are an estimate of the true value

in the entire population. The true value may actually be greater or less than what is observed.

Type I error (alpha) is the probability of incorrectly concluding there is a statistically significant difference in the population when none exists.

Type II error (beta) is the probability of incorrectly concluding that there is no statistically significant difference in a population when one exists.

Power is a measure of the ability of a study to detect a true difference.

Multivariable Regression Methods

Multiple linear regression is used when the outcome data is a continuous variable such as weight. For example, one could estimate the effect of a diet on weight after adjusting for the effect of confounders such as smoking status.

Logistic regression is used when the outcome data is binary such as cure or no cure. Logistic regression can be used to estimate the effect of an exposure on a binary outcome after adjusting for confounders.

Survival Analysis

Kaplan-Meier analysis measures the ratio of surviving subjects (or those without an event) divided by the total number of subjects at risk for the event. Every time a subject has an event, the ratio is recalculated. These ratios are then used to generate a curve to graphically depict the probability of survival.

Cox proportional hazards analysis is similar to the logistic regression method described above with the added advantage that it accounts for time to a binary event in the outcome variable. Thus, one can account for variation in follow-up time among subjects.

Kaplan-Meier Survival Curves

Why Use Statistics?

Cardiovascular Mortality in Males

0

0.2

0.4

0.6

0.8

1

1.2

'35-'44 '45-'54 '55-'64 '65-'74 '75-'84

SMR Bangor

Roseto

Descriptive Statistics

Identifies patterns in the data Identifies outliers Guides choice of statistical test

Percentage of Specimens Testing Positive for RSV (respiratory syncytial virus)

Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr May Jun

South 2 2 5 7 20 30 15 20 15 8 4 3

North-east

2 3 5 3 12 28 22 28 22 20 10 9

West 2 2 3 3 5 8 25 27 25 22 15 12

Mid-west

2 2 3 2 4 12 12 12 10 19 15 8

Descriptive Statistics

Percentage of Specimens Testing Postive for RSV 1998-99

0

5

10

15

20

25

30

35

Jul Sep Nov Jan Mar May Jul

SouthNortheastWestMidwest

Distribution of Course Grades

0

2

4

6

8

10

12

14

Number of Students

A A- B+ B B- C+ C C- D+ D D- F

Grade

Describing the Data with Numbers

Measures of Dispersion• RANGE • STANDARD DEVIATION• SKEWNESS

Measures of Dispersion

• RANGE • highest to lowest values

• STANDARD DEVIATION• how closely do values cluster around the

mean value• SKEWNESS

• refers to symmetry of curve

The Normal Distribution

Mean = median = mode

Skew is zero 68% of values fall

between 1 SD 95% of values fall

between 2 SDs

.

Me

an

, Med

ian

, Mo

de

1

2

Statistics graph

Data recorded in surveys are displayed by a statistical graph. There are some specific types of graphs to study in the data statistics graphs. There are eleven type of graphs used in Data statistics graphs

Type of graphsType of graphs Box plot, Stem and leaf plot, Frequency polygon, Scatter plot, Line graph, Bar graph, Histogram, Pictograph, Map chart, Pie chart, Line plot.

Introduction to types of statistical graphs:

Let us see some content about types of statistical graphs. The statistical data's are used to represent by some graphs format. The statistical data's are either continuous or discontinuous. There are more types of statistical graphs. Each and every graph is used in various purposes. The graphs and charts are commonly used by the business trends.

Types of Statistical Graphs:

There are several types of statistical data. Each graph is used to show the data levels of the business without the any calculations. The types of statistical graphs are following below:

Statistical graph 1: Line graph.Statistical graph 2: Bar graph.Statistical graph 3: Pie chart.Statistical graph 4: Histogram.Statistical graph 5: Scatter plot.These are the types of the statistical graphs

used to draw the statistical data's.

Statistical graph 1: Line graph:

The line graph is used to conspire the continuous data. The data's are plotted as points. The points are joined by the lines. This graph is used to compare multiple data sets. But the line graph only using continuous data's.

This is the symbolization of the line graph.

Linear diagram

Statistical graph 2: Bar graph:

The bar graph displays discrete data in disconnect columns. A double bar graph can be used to evaluate two data sets. It is visually muscular. It is used to easily evaluate two or three data sets. It uses only discrete data.

This is the symbolization of the bar graph.

Statistical graph 3: pie chart:

The pie chart displays data as a percentage of the whole. It displays the percentage of each category. But, it has no exact data's.

This is the symbolization of the pie chart.

Sector diagram

Statistical graph 4: Histogram:

The histogram shows the continuous data in order to columns. It is same as the line graph but it is represented as in column format.

This is the histogram for the given statistical data.

Graphical Representation of Statistical Data

A histogram is a two-dimensional

graphical representation of a

continuous frequency

distribution.A histogram is a

special type of bar diagram.

HistogramA histogram is a way of summarizing data

pictorially. Histograms show the distribution of the data.

They are constructed from a frequency table, which is a summary of the data. The general format

for a histogram is a vertical scale that demonstrates frequencies and a horizontal scale that

represents the individual intervals, sometimes called classes. Bars are used to represent each

individual interval with the height of the bar corresponding to the frequency.

A histogram is drawn according to the steps given below.

Prepare a grouped frequency distribution table of the given data.

Show class-limits on X-axis with a suitable scale.

Show frequencies on Y-axis with a suitable scale.

Draw rectangles having base equal to the class limits and heights proportional to the frequencies. These rectangles should be joined to each other.

A frequency distribution table is shown below

Class ( Cost of saree in Rs.)

Frequency ( No. of sarees sold in a week)

100 – 200 12

200 – 300 28

300 – 400 37

400 – 500 23

500 – 600 20

600 – 700 14

700 – 800 09

Histogram

The following points can be inferred

from the above histogram. It is not necessary that the scale on

the X-axis and the Y-axis be the same. Different scales may be taken on the axes considering the nature of the data, size of the paper etc. A histogram should look neat and attractive.

The position of origin on the Y-axis is according to the scale, which is not so on the X - axis. This is indicated by drawing √mark on

The X-axis near the origin. If necessary, the mark can be made on the Y-axis or on both axes.

In a histogram, it is necessary that the adjacent rectangles be attached to each other. Therefore, if the given classes are not continuous, it is necessary to make them continuous e.g.; if the classes are 2 to 5, 6 to 9, 10 to 13,…. It should be as 1.5 to 5.5, 5.5 to 9.5, 9.5 to 13.5…

Statistical graph 5: Scatter plot:

The scatter pot shows the relationship between two factors of the experiment. It displays the relationship between two data's.

This is the example of the scatter plot.

WHAT IS A CARTOGRAM?

A cartogram is a colored map that gives a graphical representation of statistical data. It is used for an immediate view of a phenomenon or behavior.

CARTOGRAM

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