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Describing Data: The Role of Graphical StatisticsHossein GhaderiWestern UniversitySeptember 9, 2024
Introduction▶Graphical statistics are essential tools in predicting orforecasting variables such as:▶Sales of a new product▶Construction costs▶Customer satisfaction levels▶The weather▶Election results▶University enrollment figures▶Grade point averages▶Interest rates and currency exchange rates▶These variables have significant effects on our daily lives.
The Need for Data Interpretation▶Governments, businesses, and researchers spend billionscollecting data.▶Once data are collected, the real challenge begins:▶How do we interpret the data?▶What insights can be derived from the data to supportdecision-making?▶This is where graphical statistics play a key role.
Graphical Statistics in Decision Making▶Graphical statistics help visualize large datasets in an easilyinterpretable form.▶They reveal patterns, trends, and outliers that may not beimmediately apparent in raw data.▶By visualizing data, decision-makers can:▶Identify trends and forecast future behavior▶Understand variability and uncertainty▶Make informed decisions based on insights
Understanding Data Through Statistics▶In our study of statistics, we learn many tools to help us:▶Process, summarize, analyze, and interpret data▶Make better decisions in an uncertain environment▶Understanding statistics allows us to make sense of all thedata.
Tables and Graphs in Statistics▶Tables and graphs introduced in this chapter help us:▶Gain a better understanding of data▶Provide visual support for improved decision making▶Reports are enhanced by appropriate tables and graphs,including:▶Frequency distributions, bar charts, pie charts, Pareto diagrams▶Line charts, histograms, stem-and-leaf displays, ogives▶Visualization of data is important for effective communication.
Statistical Thinking in Decision Making▶Decisions are often made based on limited information.▶Examples include:▶Accountants selecting records for auditing purposes▶Financial investors understanding market fluctuations▶Managers using surveys to assess customer satisfaction▶Marketing executives gathering customer preferences ordemographics▶Even without certainty, decisions need to be made, such asbalancing portfolios while future market movements remainunknown.
Statistical Thinking in Practice▶In each situation, the process includes:▶Defining the problem▶Determining the data needed▶Collecting and summarizing the data▶Making inferences and decisions based on the data▶Statistical thinking is essential from problem definition todecision making, potentially leading to:▶Reduced costs▶Increased profits▶Improved processes▶Enhanced customer satisfaction
Population and Sample in Market Research▶Before bringing a new product to market, manufacturersassess likely demand by conducting market research surveys.▶They are interested in the entire population of potentialbuyers.▶However, analyzing large populations is often impractical dueto cost or time constraints.▶Instead, manufacturers collect data from a sample (a subsetof the population).▶A population is the complete set of all items of interest,denoted asN, which can be large or even infinite.▶A sample is an observed subset of the population, with samplesize denoted asn.
Examples of Populations▶Examples of populations include:▶All potential buyers of a new product▶All stocks traded on the NYSE Euronext▶All registered voters in a particular city or country▶All accounts receivable for a corporation▶Our goal is to make statements about the population basedon the sample.▶We need a representative sample, and randomness in sampleselection is crucial.
Random Sampling▶Simple random sampling selects a sample ofnobjects suchthat:▶Each member is chosen strictly by chance.▶The selection of one member does not influence the selectionof another.▶Every possible sample of sizenhas the same chance of beingselected.▶The term ”random sample” often refers to simple randomsampling.
Systematic Sampling▶In systematic sampling, everyj-th item is selected from thepopulation.▶The ratiojis calculated asj=Nn, whereNis the populationsize andnis the desired sample size.▶Randomly select a number from 1 tojto get the first item,then select everyj-th item afterward.▶Example: IfN= 5000,n= 100, thenj= 50. If the firstrandomly selected number is 20, select 20, 70, 120, and so on.
Systematic Sampling Considerations▶Systematic sampling assumes the population is in randomorder.▶If there is an unknown link between the ordering and thesubject of study, bias could be introduced.▶Systematic samples provide a good representation of thepopulation if there is no cyclical variation in the data.
Parameter and Statistic▶Suppose we want to know the average age of registered votersin the United States.▶The population is too large to analyze fully, so we might takea random sample, e.g., 500 voters, and calculate their averageage.▶The average based on the sample data is called a**statistic**.▶If we could calculate the average age of the entire population,that would be called a **parameter**.
Parameter and Statistic: Definitions▶A **parameter** is a numerical measure that describes aspecific characteristic of a population.▶A **statistic** is a numerical measure that describes aspecific characteristic of a sample.▶Throughout this course, we will study how to make decisionsabout a population parameter based on a sample statistic.▶We must accept an element of uncertainty since we do notknow the exact value of the parameter.
Sampling and Nonsampling Errors▶**Sampling error** occurs because the information is onlyavailable for a subset of the population (the sample).▶**Nonsampling errors** can occur even in a complete censusof the population.▶Examples of nonsampling errors include:▶Sampling from the wrong population▶Survey subjects giving inaccurate or dishonest answers▶No response from survey subjects
Example of Nonsampling Error: Wrong PopulationSampled▶In 1936, Literary Digest magazine predicted Alfred Landonwould win the U.S. presidential election.▶The prediction was wrong because their sample was takenfrom telephone directories, magazine subscription lists, andcar registrations.▶This sample underrepresented the poor, who were mostlyDemocrats and voted for Franklin Roosevelt.▶Conclusion: To make valid inferences, the sample mustrepresent the correct population.
Example of Nonsampling Error: Inaccurate Responses▶Survey subjects may give inaccurate or dishonest answers.▶This could be due to:▶Poorly worded questions that are hard to understand or biasthe response▶Sensitive questions leading to dishonest answers▶For example, a plant manager asking employees what theyhave stolen may not yield reliable answers.
Example of Nonsampling Error: Nonresponse▶Survey subjects may not respond at all or may omit answersto certain questions.▶This leads to:▶**Sampling error** due to a smaller sample size▶**Nonsampling error** if the respondents differ in importantways from the larger population.▶Nonresponse can induce bias if the respondents are notrepresentative of the population of interest.
Thinking Statistically: Problem Definition▶To think statistically begins with problem definition:1.What information is required?2.What is the relevant population?3.How should sample members be selected?4.How should information be obtained from the samplemembers?▶Once we have sample information, we use it to make decisionsabout the population.▶Finally, we draw conclusions about the population based onthe sample.
Descriptive and Inferential Statistics▶**Descriptive statistics** focus on graphical and numericalmethods to summarize and process data.▶**Inferential statistics** use data to make predictions,forecasts, and estimates to support decision-making.▶Both are essential tools for analyzing and interpreting data.
Variable Classification▶A **variable** is a specific characteristic of an individual orobject, such as age or weight.▶Variables can be classified into two types:1.**Categorical variables**: Responses that fall into groups orcategories.2.**Numerical variables**: Measured quantities that can bediscrete or continuous.
Categorical Variables▶Categorical variables produce responses that belong topredefined groups or categories.▶Examples include:▶Yes/No questions: ”Are you a business major?” or ”Do youown a car?”▶Gender, marital status, or the type of errors in health careclaims (procedural, diagnostic, etc.).▶Faculty evaluation responses ranging from ”strongly disagree”to ”strongly agree.”
Numerical Variables▶Numerical variables can be either **discrete** or**continuous**:▶**Discrete numerical variables**: Often countable and finite.▶Examples: Number of students enrolled in a class, number ofuniversity credits, number of stocks in an investor’s portfolio.▶**Continuous numerical variables**: Can take any valuewithin a given range.▶Continuous variables are typically measured quantities, such asweight or height.
Continuous Numerical Variables▶A **continuous numerical variable** may take on any valuewithin a given range of real numbers.▶It usually arises from a measurement process.▶Example: Height can be 72.1 inches or 71.8 inches dependingon the measurement accuracy.▶Other examples: Weight of a cereal box, time to run a race,distance between cities, temperature.▶Continuous variables are often truncated in daily conversationand treated like discrete variables.
Measurement Levels▶Data can be classified as either **qualitative** or**quantitative**.▶**Qualitative data** have no measurable meaning to thedifference in numbers.▶**Quantitative data** have a measurable meaning to thedifference in numbers.▶Examples:▶Qualitative: Football players’ jersey numbers (7 vs. 10) do notimply skill differences.▶Quantitative: Exam scores (90 vs. 45) have measurablemeaning in terms of performance.
Nominal and Ordinal Levels of Measurement▶**Nominal data**: Responses to categorical questions withno implied ranking.▶Examples:▶Gender (1 = Male, 2 = Female)▶Car ownership (1 = Yes, 2 = No)▶**Ordinal data**: Rank ordering of items, but with nomeasurable meaning to the difference between ranks.▶Examples:▶Product quality (1: poor, 2: average, 3: good)▶Satisfaction rating (1: very dissatisfied to 5: very satisfied)▶Consumer preference among soft drinks (1: most preferred, 2:second choice, 3: third choice)
Interval and Ratio Levels of Measurement▶**Interval data**: Provide rank and distance from anarbitrary zero.▶Example: Temperature measured in Celsius or Fahrenheit.▶The difference between 30°C and 10°C is 20°, but it isincorrect to say that 30°C is three times warmer than 10°C.▶**Ratio data**: Have a meaningful zero point and provideboth rank and meaningful differences.▶Example: Height or weight, where zero has a natural meaning(absence of the quantity).
Examples: Nominal and Ordinal Data▶**Nominal data**:▶Gender: Male/Female▶Political affiliation▶Car ownership▶**Ordinal data**:▶Quality rating: (1: poor, 2: average, 3: good)▶Preference rankings: First, second, third choice.
Examples: Interval and Ratio Data▶**Interval data**:▶Temperature in Celsius/Fahrenheit▶Years (e.g., 2023 vs. 1990)▶**Ratio data**:▶Height, weight, or distance between cities.▶Time to complete a task or run a race.
Ratio Data▶**Ratio data** indicate both rank and distance from anatural zero point.▶Ratios between two measures are meaningful.▶Example: A person weighing 200 pounds is twice as heavy assomeone weighing 100 pounds.▶Another example: A person aged 40 is twice the age ofsomeone who is 20 years old.▶Ratio data allows us to make meaningful comparisons andproportions.
Classifying Data by Type▶After collecting data, responses are classified as:▶**Categorical** (e.g., gender, political affiliation)▶**Numerical** (e.g., age, income, weight)▶Data can also be classified by the measurement scale:▶Nominal, Ordinal, Interval, Ratio▶After classification, we assign an arbitrary ID or code to eachresponse for easier data handling.
Graph Selection for Data Types▶Different types of graphs are used based on the data type:▶**Categorical variables**: Bar charts, pie charts▶**Numerical variables**: Histograms, line graphs, scatter plots▶Choosing the right graph helps visualize and interpret datamore effectively.
Handling Missing Values▶Data files often contain **missing values**, especially insurvey responses.▶Respondents may skip sensitive questions about gender, age,or income.▶Missing values require special codes in the data entry processto ensure accuracy.▶Failure to properly handle missing values can lead toerroneous results in analysis.▶Different statistical software packages handle missing values invarious ways.
Example: Handling Missing Values▶Suppose a survey asks respondents about their income, butsome people choose not to answer.▶In the data file, these missing values should be coded (e.g.,-999 or NaN) to differentiate them from valid responses.▶Statistical software may exclude these missing values fromcalculations or treat them as zero if not handled correctly.▶Correct handling of missing values is crucial for accuratestatistical output.
Describing Categorical Variables▶Categorical variables can be described using:▶**Frequency distribution tables**▶**Bar charts**, **pie charts**, and **Pareto diagrams**▶These tools are commonly used by managers and marketingresearchers to describe data collected from surveys andquestionnaires.
Frequency Distribution▶A **frequency distribution** is a table used to organize data.▶The left column (classes or groups) includes all possibleresponses for a categorical variable.▶The right column lists the frequencies or number ofobservations for each class.▶A **relative frequency distribution** is obtained by dividingeach frequency by the total number of observations andmultiplying by 100
Tables and Charts for Categorical Data▶The classes used for constructing frequency distribution tablesare the possible responses to a categorical variable.▶**Bar charts** and **pie charts** are commonly used torepresent categorical data.▶A **bar chart** uses the height of rectangles to representeach frequency, and the bars do not need to touch.
Example 1.1: Activity Level (Frequency Distribution andBar Chart)▶The U.S. Department of Agriculture (USDA) and NationalCenter for Health Statistics (NCHS) conducted surveys toassess the health and nutrition of the U.S. population.▶One variable in the **Healthy Eating Index (HEI-2005)**study is a participant’s activity level, coded as:▶1 = Sedentary▶2 = Active▶3 = Very active▶We set up a frequency distribution and bar chart of activitylevel for the HEI–2005 participants during their first interview.
Table 1.1: HEI–2005 Participants’ Activity Level: FirstInterviewParticipantsFrequencyPercentSedentary2,18348.9%Active75717.0%Very Active1,52034.1%Total4,460100.0%Table:HEI–2005 Participants’ Activity Level: First Interview
Figure 1.1: Bar Chart of Activity LevelSedentaryActiveVery Active05001,0001,5002,0002,1837571,520Activity LevelNumber of ParticipantsHEI–2005Participants’ Activity Level: First Interview (Bar Chart)
Cross Tables▶A **cross table** (or crosstab) lists the number ofobservations for every combination of values for twocategorical or ordinal variables.▶The combination of all possible intervals for the two variablesdefines the cells in a cross table.▶A cross table withrrows andccolumns is referred to as anr×ccross table.
Example 1.2: Cross Tables and Bar Charts▶Cross tables are useful for describing relationships betweencategorical or ordinal variables.▶Example: Comparing participants’ activity levels (sedentary,active, very active) with other categorical variables, such asage groups or educational levels.▶Component bar charts and cluster bar charts extend thesimple bar chart for multiple variables.
Example 1.2: Activity Level and Gender (Component andCluster Bar Charts)▶This example compares activity levels (sedentary, active, veryactive) with gender (male, female) using data from the**HEI–2005** study.▶We will use component (stacked) and cluster (side-by-side)bar charts to visualize the comparison.
Table 1.2: HEI–2005 Participants’ Activity Level by GenderActivity LevelMalesFemalesTotalSedentary9571,2262,183Active340417757Very Active8426781,520Total2,1392,3214,460Table:HEI–2005 Participants’ Activity Level (First Interview) by Gender
Figure 1.2: Component (Stacked) Bar ChartMaleFemale05001,0001,5002,0002,5008426783404179571,226GenderNumber of ParticipantsSedentaryActiveVery ActiveHEI–2005Participants’ Activity Level by Gender (Component Bar Chart)
Figure 1.3: Cluster (Side-by-Side) Bar ChartSedentaryActiveVery Active02004006008001,0001,2009573408421,226417678Number of ParticipantsMaleFemaleHEI–2005Participants’ Activity Level by Gender (Cluster Bar Chart)
Conclusion▶Component (stacked) bar charts show the contribution ofeach activity level within a gender group.▶Cluster (side-by-side) bar charts compare male and femaleparticipants directly for each activity level.▶Both charts provide insights into the distribution of activitylevels by gender for HEI–2005 participants.
Pie Charts and Their Use▶**Pie charts** are useful for showing proportions and sharesof a whole.▶The circle represents the total, and the segments depict theshares of different categories.▶The area of each segment is proportional to the correspondingfrequency or share.
Example 1.3: Browser Wars - Market Share (Pie Charts)▶In February 2011, the browser market in Europe and NorthAmerica showed different preferences.▶We will visualize the market shares for both regions using piecharts.
Table 1.3: Market Shares of Browsers (February 2011)BrowserEuropean Market (%)North American MarketFirefox37.6926.24Internet Explorer36.5448.16Google Chrome16.0313.76Safari4.9010.58Opera4.260.58Others0.580.68Table:Market Shares of Browsers in Europe and North America(February 2011)
Figure 1.4: European Market Share (Pie Chart)37.69%36.54%16.03%4.90%4.26%0.58%FirefoxInternet ExplorerGoogle ChromeSafariOperaOthersBrowser Market Share in Europe (February 2011)
Figure 1.5: North American Market Share (Pie Chart)26.24%48.16%13.76%10.58%0.58%0.68%FirefoxInternet ExplorerGoogle ChromeSafariOperaOthersBrowser Market Share in North America (February 2011)
Conclusion▶The European market was dominated by Firefox (37.69%) andInternet Explorer (36.54%) in February 2011.▶In North America, Internet Explorer held the largest sharewith 48.16%, while Firefox had 26.24%.▶Pie charts are effective tools for visualizing the distribution ofmarket shares among competing categories.
Pareto Diagrams and Their Use▶**Pareto diagrams** are bar charts used to emphasize themost frequent causes of defects.▶The diagram is used to separate the ”vital few” from the”trivial many.”▶The Italian economist **Vilfredo Pareto** observed that asmall number of factors are responsible for most of theproblems—commonly known as the **80–20 rule**.▶Bars are arranged from the most frequent cause to the leastfrequent cause from left to right.
Example 1.4: Health Care Claims Processing Errors▶A health insurance company set a goal to reduce errors by 50▶After auditing 1,000 claims, the team identified the mostfrequent errors in the claims processing system.▶The data are listed in Table 1.4, and the Pareto diagram helpsidentify the most significant factors contributing to theseerrors.
Table 1.4: Frequency of Claims Processing ErrorsError TypeFrequencyProcedural and Diagnostic Codes40Contractual Applications37Pricing Schedules17Provider Information9Provider Adjustments7Patient Information6Program and System Errors4Table:Health Care Claims Processing Errors
Figure 1.6: Pareto Diagram for Health Care ClaimsProcessing ErrorsProcedural CodesContractual AppsPricing SchedulesProvider InfoProvider AdjPatient InfoProgram Errors0102030404037179764Error TypeFrequency of Errors
Cumulative Percentages for Pareto Diagram0020406080100Error TypeCumulative PercentageCumulative Percentage of Errors
Conclusion▶The Pareto diagram shows that **Procedural and DiagnosticCodes** and **Contractual Applications** account for over60% of the errors.▶By addressing these ”vital few” errors, the company canachieve significant reductions in total errors.▶This aligns with the **80–20 rule**, where most of theproblems come from a few key factors.
Introduction to Time-Series Plots▶A **time-series plot** (also called a line chart) is used todisplay data points collected or measured at successive pointsin time.▶In a time series, the sequence of observations is important.▶Time-series plots are useful for identifying trends, cycles, orseasonal patterns in data.▶Examples of time-series data include GDP, currency exchangerates, stock prices, and corporate earnings.
Example 1.5: Gross Domestic Product (Time-Series Plot)▶The U.S. **Bureau of Economic Analysis (BEA)** providesannual GDP data from 1929 through 2009.▶We will plot the GDP data to identify long-term trends ineconomic growth.
Figure 1.7: Time-Series Plot of GDP (1929-2009)1,9291,9491,9691,9892,0000.40.81.21.4·104YearBillions of Real 2005 DollarsGross Domestic Product (1929–2009)Gross Domestic Product (GDP) from 1929–2009
Example 1.6: Currency Exchange Rates (Time-Series Plot)▶Exchange rates fluctuate over time and are important forinvestors, travelers, and businesses.▶We will plot the exchange rates between USD and EUR, andUSD and GBP, for the 6-month period from **August 22,2010** to **February 17, 2011**.
Figure 1.8: Time-Series Plot of USD to EUR (Aug 2010 -Feb 2011)AugOctDecJanFeb1.251.31.351.41.45Date (Aug 2010 - Feb 2011)Exchange Rate (USD to EUR)Currency Exchange Rates: USD to EUR
Figure 1.9: Time-Series Plot of USD to GBP (Aug 2010 -Feb 2011)AugOctDecJanFeb1.51.551.61.65Date (Aug 2010 - Feb 2011)Exchange Rate (USD to GBP)Currency Exchange Rates: USD to GBP
Conclusion▶**Time-series plots** allow us to observe trends over time fordifferent variables.▶The **GDP time-series plot** shows steady growth from1929 to 2009.▶The **currency exchange rate plots** reveal fluctuationsbetween USD and both EUR and GBP from August 2010 toFebruary 2011.▶Time-series analysis is essential for understanding trends,cycles, and seasonal patterns in economic data.
Introduction to Frequency Distributions▶A **frequency distribution** is a table that summarizes databy listing the classes and the number of observations in eachclass.▶For numerical data, we use formulas to determine the numberof classes and the class width.▶A **cumulative frequency distribution** adds the frequenciesof all classes up to the current class.▶These tools help summarize data and improve communicationof results.
Table 1.6: Completion Times (Seconds)271236294252254263266222262278288262237247282224263267254271278263262288247252264263247225281279238252242248263255294268255272271291263242288252226263269227273281267263244249252256263252261245252294288245251269256264252232275284252263274252252256254269234285275263263246294252231265269235275288294263247252269261266269236276248299Table:Completion Times (in Seconds) for Employees
Calculating Class Width▶The number of classes,k, is chosen based on the size of thedataset. In this case, we have chosenk= 8 classes.▶The class widthwis determined using the formula:w=Largest Observation−Smallest ObservationNumber of Classes▶From the data in **Table 1.6**, the largest observation is**299** and the smallest observation is **222**.▶Substituting into the formula:w=299−2228= 9.625▶Since class width must be rounded upward, the class width is**10**.▶The first class interval is from **220 to less than 230**, andsubsequent classes are created by adding the class width.
Table 1.7: Frequency and Relative Frequency DistributionsCompletion Times (in Seconds)FrequencyPercent220 less than 23054.5%230 less than 24087.3%240 less than 2501311.8%250 less than 2602220.0%260 less than 2703229.1%270 less than 2801311.8%280 less than 290109.1%290 less than 30076.4%Table:Frequency and Relative Frequency Distributions for CompletionTimes
Table 1.8: Cumulative Frequency and Relative CumulativeFrequencyCompletion Times (in Seconds)Cumulative FrequencyCumulaLess than 2305Less than 24013Less than 25026Less than 26048Less than 27080Less than 28093Less than 290103Less than 3001101Table:Cumulative Frequency and Relative Cumulative Frequency forCompletion Times
Cumulative Frequency Distribution Plot22023024025026027028029030020406080100120Completion Times (in Seconds)Cumulative FrequencyCumulative Frequency Distribution for Completion TimesCumulative Frequency Distribution for Completion Times
Conclusion▶A frequency distribution summarizes numerical data bygrouping observations into classes.▶Cumulative frequency distributions help identify the totalnumber of observations below certain thresholds.▶In the example, 72.7% of the employees completed the taskwithin the goal of 270 seconds.▶These tools provide a clearer picture of the data, allowing thesupervisor to make informed decisions.
Figure 1.13: Histogram of Completion Times (ConnectedBars)22023024025026027028029005101520253035Completion Time (Seconds)Frequency
Shape of a Distribution▶The shape of a distribution can be visually described using ahistogram.▶**Symmetry**: A distribution is said to be symmetric if theobservations are balanced or evenly distributed about thecenter.▶**Skewness**: A distribution is skewed if the observations arenot evenly distributed.▶A **skewed-right distribution** (positively skewed) has alonger tail on the right.▶A **skewed-left distribution** (negatively skewed) has alonger tail on the left.▶The following slides illustrate the three shapes:▶Symmetric distribution▶Skewed-right distribution▶Skewed-left distribution
Figure 1.15(a): Symmetric Distribution123456780510ValueFrequencySymmetric Distribution
Figure 1.15(b): Skewed-left Distribution123456780510ValueFrequencySkewed-left Distribution
Figure 1.15(c): Skewed-right Distribution12345670510ValueFrequencySkewed-right Distribution
Stem-and-Leaf Displays▶A **stem-and-leaf display** is an alternative to a histogram,grouping data based on their leading digits (**stems**) andarranging the final digits (**leaves**).▶Each stem represents a class of data values, while the leavesrepresent individual data points within each class.▶The leaves are displayed in ascending order after thecorresponding stem.▶Stem-and-leaf displays help reveal the internal structure of thedata, making it easy to identify patterns and outliers.
Example 1.11: Stem-and-Leaf DisplayCompletion Times (in Seconds)StemLeaves222234677824224556778925122222445566692612222233334667799992711123455568899281145588882914449Stem-and-Leaf Display of Completion Times
Data: Accounting Final Exam Grades▶A random sample of 10 final exam grades for an introductoryaccounting class is as follows:88,51,63,85,79,65,79,70,73,77▶We will use a stem-and-leaf display to describe thedistribution of these grades.
Example 1.11: Grades on an Accounting Final ExamStem-and-Leaf DisplayStemLeaves51635703799858Stem-and-Leaf Display of Accounting Final Grades
Scatter Plot: Explanation▶A **scatter plot** is used to visualize the relationshipbetween two numerical variables.▶In business and economics, scatter plots are often used toinvestigate the relationship between variables such as:▶The effect of advertising on total profits▶The change in quantity sold as a result of a change in price▶In this example, we will analyze the relationship between**SAT Math scores** and **GPA** for 11 students.▶The **SAT Math score** is the independent variable (labeledasX) and the **GPA** is the dependent variable (labeled asY).
Figure 1.18: GPA vs. SAT Math Scores (Scatter Plot)400450500550600650700752.62.833.23.43.63.84SAT Math ScoreGPAGPA vs. SAT Math Scores
Conclusion▶Categorical variables can be effectively described usingfrequency distribution tables and bar charts.▶Cross tables provide insights into relationships between twocategorical variables and are often used with component orcluster bar charts.▶Proper representation of categorical data helps in makinginformed decisions based on survey and questionnaire results.
Case Study: Impact on Business Decisions▶Example: A company launching a new product▶Use historical sales data and trends to:▶Forecast product demand▶Identify the right pricing strategy▶Graphical statistics help maximize profits and minimize risks.
Types of Graphical Statistics▶Common graphical methods include:▶Histograms: Show distribution of data▶Box plots: Highlight quartiles, median, and outliers▶Scatter plots: Explore relationships between two variables▶Line graphs: Track trends over time▶Bar charts: Compare categorical data▶These methods provide a range of perspectives to understanddata.
Case Study: Impact on Business Decisions▶Consider a company launching a new product:▶Historical sales data and customer trends can be visualized toforecast demand.▶Graphical statistics can help identify the right pricing strategyby analyzing customer satisfaction and market behavior.▶In this way, graphical statistics provide critical support inmaximizing profits and minimizing risk.
Conclusion▶Graphical statistics are a powerful tool for interpreting largeamounts of data.▶They play a crucial role in decision-making processes acrossindustries.▶In an era of data-driven decision making, understanding andusing graphical statistics is essential.