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Central Tendency, Variance, and Variability

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Central Tendency, Variance, and Variability © Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION © Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC © Mad Dog/Shutterstock CHAPTER 2 NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION Central© Jones & Bartlett Learning, Tendency, LLC © Jones & Bartlett Learning, LLC Variance,NOT FOR SALE OR DISTRIBUTION and VariabilityNOT FOR SALE OR DISTRIBUTION Facts are stubborn, but statistics are more pliable. © Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION—Mark Twain CHAPTER OUTLINE© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC Introduction NOT FOR SALE OR DISTRIBUTIONInterquartile Range NOT FOR SALE OR DISTRIBUTION Measures of Central Tendency Standard Deviation Mean Variance Median Data Harvesting Mode© Jones & Bartlett Learning, LLC Global Perspective© Jones & Bartlett Learning, LLC FrequencyNOT Distribution FOR SALE OR DISTRIBUTION Chapter SummaryNOT FOR SALE OR DISTRIBUTION Variance and Measures of Dispersion or Variability Apply Your Knowledge Min and Max References Range Web Links Outlier Data © Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION LEARNING OUTCOMES After completing this chapter, you should be able to do the following: 1. Discuss the uses of mean, median, and mode in a hospital setting. © Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC 2. Discuss the uses of measures of central tendency. 3. Demonstrate howNOT to find FOR mean, SALE median, OR DISTRIBUTIONmode, and standard deviation. NOT FOR SALE OR DISTRIBUTION 4. Describe the difference between continuous data and discrete data. 5. Describe the term skewed. KEY ©TERMS Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION Continuous data Max Outlier data Descriptive statistics Mean Range Discrete data Median Skewed © JonesFrequency & Bartlett distribution Learning, LLC Min © Jones & BartlettStandard Learning, deviation LLC NOT FORInterquartile SALE OR range DISTRIBUTION Mode NOT FOR SALE ORVariance DISTRIBUTION 14 All Microsoft screenshots used with permission from Microsoft. © Jones & Bartlett Learning LLC, an Ascend Learning Company. NOT FOR SALE OR DISTRIBUTION. Measures of Central Tendency 15 © Jones & Bartlett Learning, LLC © Jones & BartlettHow Learning, Does Your LLC Hospital Rate? NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION Dr. Barker, chief of the medical staff at his hospital, is reviewing cases of heart failure and the length of stay, readmission rates, and average age of patients with this condition. The purpose of this review is to improve patient care, reduce readmission rates, and potentially reduce length of stay in the hospital.© Jones Heart & Bartlett failure Learning,is the heart’s LLC inability to pump enough© Jonesblood and & Bartlettoxygen to Learning, LLC support the otherNOT organs FOR SALEin the body.OR DISTRIBUTION Heart failure expenditures averageNOT about FOR $32 billionSALE each OR DISTRIBUTION year in the United States. This cost includes healthcare services, medication, and missed days of work. Dr. Barker went to the website of the Centers for Disease Control and Prevention (CDC) to find data on his county (CDC Interactive Heart Disease site: https://nccd.cdc.gov/DHDSPAtlas/). Consider the following: © Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC 1. If you were in Dr. Barker’s position, what specific goals might you have to improve patient care NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION in this situation? 2. What might a high readmission rate indicate, and what are measures that could be taken to reduce this number? 3. Why would Dr. Barker want to reduce length of stay among heart failure patients? How does © Jones & Bartlettlength Learning, of stay relate LLC to the quality of patient© Jones care? & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION or measures of central location and summary statistics. Introduction Each is a valid measure of central tendency but is used © Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC The Mark Twain quote that introduces this chapter is under different conditions. You will recall from the first NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION quite telling in that although the data you have may chapter how mode was used to estimate the height of a be reliable and accurate, the interpretation of the data brick wall, which is only one example of this measure. can be questionable, as this and other chapters will If all the data are perfectly normal, the mean, reveal. In this chapter, you will learn more about key median, and mode will be identical in reliability and statistical© Jones methods & Bartlett and provide Learning, simple LLC and real-world will represent© Jones summary & Bartlett values Learning,accurately in LLC a given examplesNOT FOR of each.SALE These OR examples DISTRIBUTION will involve com- dataset.NOT Follow FOR along SALE and ORtry theDISTRIBUTION examples to see puting your results with a Microsoft Excel spreadsheet how these methods work. or other software tools. First you will learn about measures of central tendency, including mode, mean, median, variables, and frequency distribution. These Mean © Jones & Bartlettmeasures Learning, are part of descriptive LLC statistics and© are Jones Mean & Bartlett is a measure Learning, of central LLC tendency that can be NOT FOR SALEconcerned OR DISTRIBUTION with summarizing and interpreting someNOT FORdetermined SALE ORby mathematically DISTRIBUTION calculating the average of the properties of sets of data, but they do not sug- of observations (e.g., data elements) in a frequency gest necessarily the properties of the entire popula- distribution. Mean is the most common measure of tion from which the sample was taken (which would central tendency. The mean can be used with discrete require determining© Joneswhether & your Bartlett collected Learning, data were LLCdata (e.g., “choose 1, 2,© or Jones 3”) but is& mostBartlett commonly Learning, LLC in fact representativeNOT of FORthe entire SALE population). OR DISTRIBUTION used with continuous dataNOT (e.g., FOR weight SALE of a ORpatient). DISTRIBUTION Following the discussion of measures of central ten- In this case, the mean is just a model of the dataset. dency, you will learn about measures of difference among One of the most important factors relating to the the numbers in a dataset. Standard deviation, standard mean is that it minimizes errors in predicting any one error, range, and variance are covered in this section and © Jones & Bartlett Learning, LLC value in© theJones dataset. & Bartlett In fact, it Learning, produces the LLC fewest are known as measures of dispersion or variability. NOT FOR SALE OR DISTRIBUTION errors NOTcompared FOR with SALE median OR DISTRIBUTIONand mode. Another important characteristic of this measure is that it Measures of Central includes all values in a dataset. Remember that a pop- ulation is all of your data, whereas a sample is just a Tendency subset, preferably a random but representative one. © Jones & BartlettStatistical Learning, methods used LLC to determine the shape of a ©dis Jones- &However, Bartlett using Learning, the mean LLC has one disadvantage: it NOT FOR SALEtribution OR of DISTRIBUTION data are mean, median, and mode. TheseNOT FORis susceptible SALE OR to DISTRIBUTIONthe influence of outlier data, or data methods are known as measures of central tendency, elements that are very different or far away from most © Jones & Bartlett Learning LLC, an Ascend Learning Company. NOT FOR SALE OR DISTRIBUTION. 16 Chapter 2 Central Tendency, Variance, and Variability of the other data elements. Outlier data, discussed There are actually several types of calculation for © Jones & Bartlett Learning, LLC © Joneslater & Bartlettin the chapter, Learning, may also LLC be referred to as flier, mean. We have already covered the standard mean, NOT FORmaverick SALE, aberrant OR DISTRIBUTION, or straggler, although outlier is NOTalso FOR known SALE as the arithmeticOR DISTRIBUTION mean. Another is the sam- most commonly used. Outlier data may be detected ple mean, also known as the average. By using the sam- by statistical tests (such as a Dixon or Grubbs test) or ple mean, outlier data can be used. Skewed data will by a graphical display of the data. make no difference in the sample mean. Sample mean Moreover, the more© Jonesskewed & Bartlett(or varied) Learning, the is LLCan estimate of the population,© and Jones the dataset & Bartlett may be Learning, LLC data become, the less effectiveNOT FOR the meanSALE is ORin locat DISTRIBUTION- quite varied, such as 8, 34, 56,NOT 25, 41, FOR 2, 17, SALE 25. OR DISTRIBUTION ing a central tendency and typical value. Skewed Other means are the harmonic mean, for rate or simply means that the dataset contains both very speed measures, and the geometric mean, for when small and very large numerical values. Mean is best the ranges you are comparing are different and you used with© Jones data that & Bartlettare not skewed, Learning, such LLCas in the need to equalize© Jones them. Next & Bartlett we will computeLearning, a sam LLC- set 1, 3, 2, 1, 3. ple mean. NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION Hands-on Statistics 2.1: Use Excel to Find Mean © Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC To compute the mean, add all items together and divide by the number of elements in the dataset. NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION To perform this calculation in Excel, examine Figure 2.1 and follow these directions: 1. Open a new, blank spreadsheet in Excel. 2. Key in the following data about inpatient hospital days, entering one numeral per cell in consecutive cells of the same row, beginning in row 2, column C (C2), and ending in cell G2: © Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION1, 3, 2, 1, 3 NOT FOR SALE OR DISTRIBUTION 3.
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