Chapter 2 Charts and Graphs

Learning ObjectivesDistinguish between grouped and ungrouped dataProduce graphical summaries of univariate data - histograms, frequency polygons, ogives, pie charts, stem and leaf plots and Pareto chartsProduce graphical summaries of two-variable continuous data scatter plots

Ungrouped Versus Grouped DataUngrouped dataAlso called raw data Data that have not been summarised in any wayGrouped dataData that have been organised into a frequency distributionA frequency distribution is a summary of data presented in the form of class intervals and their corresponding frequency

Frequency DistributionsTable 2.1

Frequency Distribution of per capita GDP dataTable 2.2

Data RangelargestsmallestTable 2.1

Number of Classes and Class WidthThe number of classes should be between 5 and 15Fewer than 5 classes cause excessive summarisationMore than 15 classes leave too much detailClass WidthDivide the range by the number of classes for an approximate class widthRound up to a convenient number(20000, 40000] means that the lower end is NOT included in this class interval, but the upper end IS included

Class Width = 20000

Class Midpoint10000= = widthclass21 + point beginning class = MidpointClass

Relative and Cumulative FrequencyThe ratio of the frequency of the class interval to the total frequency.A running total of frequencies through the classes of a frequency distribution.

Table 2.3

Demonstration Problem 2.1 The price of a Big Mac in US dollars in 37 countries is given below. Construct a frequency distribution with class midpoints, relative and cumulative frequencies.

Solution to Demonstration Problem 2.1The frequencies and relative frequencies real the most prevalent price interval. More than half theCountries (57%) have the price in the $US2 to $US4 interval.

Common Statistical GraphsHistogram -- vertical bar chart of frequenciesFrequency Polygon -- line graph of frequenciesOgive -- line graph of cumulative frequenciesPie Chart -- proportional representation of categories of a wholeStem & Leaf Plot Pareto ChartScatter Plot

HistogramA type of vertical bar where the area of each bar is equal to the frequency of the corresponding intervalFig. 2.1

HistogramAnother histogram of the same data with a different selection of interval width.Fig. 2.2

HistogramThe national debt as a percentage of GDP for 132 countries.Fig. 2.3

HistogramIt is often useful to determine whether data are approximately normally distributed. The histogram in figure 2.3 indicates that the national debt data are not normally distributed.Fig. 2.4

Frequency PolygonA graph constructed by plotting a dot for the frequencies at the class midpoints and connecting the dotsFig. 2.5

OgiveA cumulative frequency polygonPlotted by graphing a dot at each class endpoint for the cumulative frequency value and connecting the dotsFig. 2.6

Pie ChartA circular display of data where the area of the whole pie represents 100% of the data being studied and slices represent a percentage breakdown of the sublevels

Stem and Leaf PlotsA plot of numbers constructed by separating each number into two groups, a stem and a leaf. The leftmost digits are the stems and the rightmost digits are the leaves

Fig. 2.7

Safety Examination Scores for Plant TraineesFig. 2.7Table 2.4Raw DataStem & Leaf Plot

Construction of Stem and Leaf PlotStemLeafStemLeaf

Pareto ChartsA vertical bar chart in which the number and types of defects for a product or service are graphed in order of magnitude from greatest to leastFig. 2.8

Scatter PlotA plot or graph of pairwise data from two continuous variables, to explore the relationship between themFig. 2.9

Demonstration problem 2.8

The data are the measurements of gas usage and minimum outside temperature ()

Solution to demonstration problem 2.8

Summary

Graphical analysis and summary of data is primarily used to highlight key aspects of the data to reveal any special features, such as outliers or other unusual observations.Histograms show the spread and shape of a data distribution. A stem and leaf plot is similar, but it plots the actual data. Categorical data can be presented as a pie chart or a bar chart.Scatter plots are used to investigate relationships between two variables.

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