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Chapter I : Describing Data With Graphs Kian Jahromi May 31, 2012Kian Jahromi () Chapter I : Describing Data With Graphs May 31, 2012 1 / 19
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Table of contents1 VARIABLES AND DATA TYPES OF VARIABLES2 GRAPHS FOR CATEGORICAL DATA3 GRAPHS FOR QUANTITATIVE DATA4 Interpreting Graphs with a Critical Eye Kian Jahromi () Chapter I : Describing Data With Graphs May 31, 2012 2 / 19
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VARIABLES AND DATADeﬁnitionsDeﬁnitionA Variable is a characteristic that changes or varies over time and/or fordiﬀerent individuals or objects under consideration.DeﬁnitionAn experimental unit is the individual or object on which a variable ismeasured. A single measurement or data value results when a variable isactually measured on an experimental unit.DeﬁnitionA population is the set of all measurements of interest to the investigator.DeﬁnitionA sample is a subset of measurements selected from the population ofinterest. Kian Jahromi () Chapter I : Describing Data With Graphs May 31, 2012 3 / 19
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VARIABLES AND DATAExampleIdentify the experimental units on which the following variables aremeasured:a. Gender of a studentThe studentb. Number of errors on a midterm examThe midterm examc. Age of a cancer patientThe patiente. Colour of a car entering a parking lotThe Car Kian Jahromi () Chapter I : Describing Data With Graphs May 31, 2012 4 / 19
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VARIABLES AND DATADeﬁnitionUnivariate data result when a single variable is measured on a singleexperimental unit.DeﬁnitionBivariate data result when two variables are measured on a singleexperimental unit. Multivariate data result when more than two variablesare measured. Kian Jahromi () Chapter I : Describing Data With Graphs May 31, 2012 5 / 19
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VARIABLES AND DATAThe following data set is a multivariate data set. Each column itself is aUnivariate data set. Kian Jahromi () Chapter I : Describing Data With Graphs May 31, 2012 6 / 19
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VARIABLES AND DATA TYPES OF VARIABLESDeﬁnitionQualitative variables measure a quality or characteristic on eachexperimental unit. Quantitative variables measure a numerical quantityor amount on each experimental unit.DeﬁnitionDeﬁnition A discrete variable can assume only a ﬁnite or countablenumber of values. A continuous variable can assume the inﬁnitely manyvalues corresponding to the points on a line interval. Kian Jahromi () Chapter I : Describing Data With Graphs May 31, 2012 7 / 19
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GRAPHS FOR CATEGORICAL DATAGraphs for Categorical DataAfter the data have been collected, they can be consolidated andsummarized to show the following information: (i) What values of the variable have been measured (ii) How often each value has occurred For this purpose, you can construct a statistical table that can be used to display theExampleA bag contains 25 candies: Kian Jahromi () Chapter I : Describing Data With Graphs May 31, 2012 8 / 19
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GRAPHS FOR CATEGORICAL DATASo, the Statistical table for last page example is as follows:Also, it is possible to express the frequency of each categories usingfollowing formulas: (i) Relative frequency= frequency (n is the total number of n measurements) (ii) Percent= 100 × Relative frequencyThe following table contain the relative frequency and percent for eachcategories of last example: Kian Jahromi () Chapter I : Describing Data With Graphs May 31, 2012 9 / 19
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GRAPHS FOR CATEGORICAL DATAKian Jahromi () Chapter I : Describing Data With Graphs May 31, 2012 10 / 19
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GRAPHS FOR CATEGORICAL DATAExampleFifty people are grouped into four categories A, B, C, and D and thenumber of people who fall into each category is shown in the table:The following ﬁgure is the bar chart for upper table: Kian Jahromi () Chapter I : Describing Data With Graphs May 31, 2012 11 / 19
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GRAPHS FOR CATEGORICAL DATAand the pie chart is as follows: Kian Jahromi () Chapter I : Describing Data With Graphs May 31, 2012 12 / 19
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GRAPHS FOR QUANTITATIVE DATAGRAPHS FOR QUANTITATIVE DATALine ChartsWhen a quantitative variable is recorded over time at equally spacedintervals (such as daily, weekly, monthly, quarterly, or yearly), the data setforms a time series. Time series data are most eﬀectively presented on aline chart with time as the horizontal axis. The idea is to try to discern apattern or trend that will likely continue into the future, and then to usethat pattern to make accurate predictions for the immediate future.Example Kian Jahromi () Chapter I : Describing Data With Graphs May 31, 2012 13 / 19
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GRAPHS FOR QUANTITATIVE DATADotplotsMany sets of quantitative data consist of numbers that cannot easily beseparated into categories or intervals of time. You need a diﬀerent way tograph this type of data! The simplest graph for quantitative data is thedotplot. For a small set of measurements for example, the set 2, 6, 9, 3, 7,6 you can simply plot the measurements as points on a horizontal axis.Example Kian Jahromi () Chapter I : Describing Data With Graphs May 31, 2012 14 / 19
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GRAPHS FOR QUANTITATIVE DATAStem and Leaf PlotsAnother simple way to display the distribution of a quantitative data set isthe stem and leaf plot. This plot presents a graphical display of the datausing the actual numerical values of each data point.How Do I Construct a Stem and Leaf Plot? 1. Divide each measurement into two parts: the stem and the leaf . 2. List the stems in a column, with a vertical line to their right. 3. For each measurement, record the leaf portion in the same row as its corresponding stem. 4. Order the leaves from lowest to highest in each stem. 5. Provide a key to your stem and leaf coding so that the reader can recreate the actual measurements if necessary. Kian Jahromi () Chapter I : Describing Data With Graphs May 31, 2012 15 / 19
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GRAPHS FOR QUANTITATIVE DATAExample Kian Jahromi () Chapter I : Describing Data With Graphs May 31, 2012 16 / 19
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GRAPHS FOR QUANTITATIVE DATAExample Kian Jahromi () Chapter I : Describing Data With Graphs May 31, 2012 17 / 19
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Interpreting Graphs with a Critical EyeDeﬁnitionA distribution is symmetric if the left and right sides of the distribution,when divided at the middle value, form mirror images.DeﬁnitionA distribution is skewed to the right if a greater proportion of themeasurements lie to the right of the peak value. Distributions that areskewed right contain a few unusually large measurements. Kian Jahromi () Chapter I : Describing Data With Graphs May 31, 2012 18 / 19
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Interpreting Graphs with a Critical EyeDeﬁnitionA distribution is skewed to the left if a greater proportion of themeasurements lie to the left of the peak value. Distributions that areskewed left contain a few unusually small measurements.DeﬁnitionA distribution is unimodal if it has one peak; a bimodal distribution hastwo peaks.Bimodal distributions often represent a mixture of two diﬀerentpopulations in the data set. Kian Jahromi () Chapter I : Describing Data With Graphs May 31, 2012 19 / 19
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