Describing Distributions - With Graphs or Tables

1 Describing DistributionsWith Graphs or TablesApril 16, 2012 distribution of a VariableScales of MeasurementDistribution of a Qualitative VariableHistogramsWhat to look for in a HistogramVariation on Frequency HistogramsStem and Leaf distribution of a VariableDefinitionThedistribution of a variabletells us what values ittakes and how often it takes these distribution of a VariableTable:On a scale of 1-10, how nervous are you about takingSTAT 220? (1=very nervous, 10=not nervous)Level ofNumber ofPercentNervousnessStudentsof Scales of MeasurementAcategoricalorqualitativevari able places anindividual into one of several groups or categories may have ordering in some cases.

2 OrderedcategoricalAquantitativevariable takes numerical values forwhich arithmetic operations such as adding andsubtracting makes variables can becontinuousordiscrete. For a continuous variable, the values candiffer by any amount. For a discrete variable, the valuescan only differ by fixed Scales of Measurement Types of VariablesQualitative/CategoricalQuantita tiveContinuousDiscreteOrderedUnordered Age Time Weight Height Number of siblings Pairs of shoes Likert scales Economic status (rich, middle, poor) Letter grade (A, B, C) Eye color Political Affiliations (Dem, Ind, Rep)

3 Categorizing a QuantitativeVariableA study of the age distribution of the audience for socialnetworking sites categorized age as under 25 years, 25 to 34years, 35 to 49 years and over 49 years. Is age a quantitativeor a qualitative variable in this context?FacebookMySpaceAge groupvisitorsvisitorsunder 25 to 34 to 49 49 distribution of a QualitativeVariableThe distribution of a categorical or qualitative variable lists thecategories and gives either the count or the percent ofindividuals who fall in each ways to display the distribution of a categoricalvariable are:ITablesIPie chartsIBar Graphs (or plots) Practicing Making a DistributionTableA survey of college freshmen in 2001 asked what fieldthey planned to study.

4 The results: , arts andhumanities; , business; , education; ,engineering and science; , professional; ,social science; and , are the observational units? What is the variable onwhich data has been collected? What is its scale ofmeasurement?Make a table showing the distribution of the What Makes a Clear table ?A caption that tells the content of the within the table identify the variable distribution of the variable is shown in numbers andalso percents (or rates), if source of the data at the foot of the table distribution table for TwoQualitative VariablesIn the survey of college freshmen, suppose in addition tothe field of study, gender of the freshmen was the distribution of choice of field of StudyMaleFemaleTotal PercentArts and and % Pie Charts for QualitativeVariablesOn a scale of 1-10, how nervous are you about taking STAT220?

5 (1=very nervous, 10=not nervous) This area is 13%of the pie because13% of students gave theanswer ``10''.Pie charts show thedistribution of acategorical variable asslices of a pie .Use a pie chart onlywhen you want toemphasize eachcategory s relation tothe Pie Charts for QualitativeVariablesThe Higher Education Research Institute s Freshman Surveyincludes over 200,000 full-time freshmen who entered collegein 2009. The survey reports the following data on the sourcesstudents use to pay for college for college expensesStudentsFamily not to be to be why it is not correct to use a pie chart to display Bar Graphs for Qualitative DataOn a scale of 1-10, how nervous are you about taking STAT220?

6 (1=very nervous, 10=not nervous) Bar Graphs representeach category as a bar heights showthe category counts Graphs can comparequantities that are notpart of a Pareto bar graph hasthe bars ordered fromtallest to HistogramsTables work for categorical variables because thesevariables take relatively few variables can take so many values that anymeaningful display must group nearby values togetherinto class most common graph of the distribution of aquantitative variable is Bar Graphs versus HistogramsA histogram displays the

7 distribution ofa quantitative variable. The horizontalaxis is marked in marked in the units ofmeasurement of the histograms with no space, toindicate that all values of the variableare bar graph comparesthe sizes of differentquantities. Thehorizontal axis need nothave any bar Graphs withspace in between How to Make A HistogramHow many hours per week (h) will you study for STAT 220?quantitativeHoursCountProportion0 h< h< h< h< h< h< h< h< h< h How to Make a HistogramTheheightof each bar in a histogram corresponds tothe count in each bin.

8 This is called afrequencyhistogram. What to Look For in a HistogramDetectoutliers, if at it symmetric? skewed?Where is itcentered? Where is the mid-point?Howspreadout is it? Basic Shapes of a Histogram Describing A distribution Ex. 1 Mystery Question: How old do you think Prof. Grove is? n= Describing a distribution Ex. 1 Mystery Question: How old do you think Prof. Grove is?(minus the outlier) 41315103n=45 Shape:The distributionhas a single peak in themiddle representingguesses of 35-40 seems :Arranging theobservations in order ofsize shows the mid-pointis roughly 37 :The range isfrom 27 years to Describing a distribution Ex.

9 2 Mystery Question: How many facebook friends do you have? 1615932n=45 n= Describing a distribution Ex. 2 Mystery Question: How many facebook friends do you have?When the class width is halved ( figure to right), theindividual with 1000 friends stands a bit they an outlier or just the largest observation?We will only flag strong the distribution of the number of Describing a distribution Ex. 3 Mystery Question: What was your Math score? n=81 Single peak, somewhatskewed to the at a score of is Describing a distribution Ex.

10 4 The values that a statistic takes in many random samplesfrom the same population form a distribution with apattern. sampling distribution Populationp= 1: n=2527 p-hat= 2: n=2527 phat= 3: n=2527 phat= peak at lack ofbiasSpread is - Describing a distribution Ex. 5 Two distinct at around 3 and18 is 0-35 Variation on Frequency HistogramTable: distribution of family income in the , 1973 Income levelPercent$0 - $1,0001$1,000 - $2,0002$2,000 - $3,0003$3,000 - $4,0004$4,000 - $5,0005$5,000 - $6,0005$6,000 - $7,0005$7,000 - $10,00015$10,000 - $15,00026$15,000 - $25,00026$25,000 - $50,0008$50,000 and Variation on FrequencyHistogramsSince the class intervals are not of equal width, it ismisleading to plot the counts versus class intervals.