5Numerical Descriptive Techniques Measures of Central LocationMean, Median, ModeMeasures of VariabilityRange, Standard Deviation, Variance, Coefficient of VariationMeasures of Relative StandingPercentiles, QuartilesMeasures of Linear RelationshipCovariance, Correlation, Determination, Least Squares Line
6The Arithmetic MeanThis is the most popular and useful measure of central locationSum of the observationsNumber of observationsMean =
7Sample and population medians are computed the same way. The MedianThe Median of a set of observations is the value that falls in the middle when the observations are arranged in order of magnitude.Sample and population medians are computed the same way.ExampleFind the median of the time on the internet for the 10 adultsSuppose only 9 adults were sampled (exclude, say, the longest time (33))Comment0, 0, 5, 7, 8, 9, 12, 14, 22, 33Even number of observationsOdd number of observations0, 0, 5, 7, 8, , 12, 14, 22, 338.5,0, 0, 5, 7, 8 9, 12, 14, 228
8The ModeThe Mode of a set of observations is the value that occurs most frequently.Set of data may have one mode (or modal class), or two or more modes.For large data setsthe modal class ismuch more relevantthan a single-valuemode.The modal class
9Example 1The times (to the nearest minute) that a sample of 9 bank customers waited in line were recorded and are listed here.Determine the mean, median, and mode for these data.
11Relationship among Mean, Median, and Mode If a distribution is symmetrical, the mean, median and mode coincideIf a distribution is asymmetrical, and skewed to the left or to the right, the three measures differ.A positively skewed distribution(“skewed to the right”)ModeMeanMedian
12Relationship among Mean, Median, and Mode If a distribution is symmetrical, the mean, median and mode coincideIf a distribution is non symmetrical, and skewed to the left or to the right, the three measures differ.A positively skewed distribution(“skewed to the right”)A negatively skewed distribution(“skewed to the left”)ModeMeanMeanModeMedianMedian
13The rangeThe range of a set of observations is the difference between the largest and smallest observations.Its major advantage is the ease with which it can be computed.Its major shortcoming is its failure to provide information on the dispersion of the observations between the two end points.But, how do all the observations spread out????The range cannot assist in answering this questionRangeSmallestobservationLargestobservation
14Note! the denominator is sample size (n) minus one ! Variance…population meanThe variance of a population is:The variance of a sample is:population sizesample meanNote! the denominator is sample size (n) minus one !
15Variance…As you can see, you have to calculate the sample mean (x-bar) in order to calculate the sample variance.Alternatively, there is a short-cut formulation to calculate sample variance directly from the data without the intermediate step of calculating the mean. Its given by:
16Coefficient of Variation… The coefficient of variation of a set of observations is the standard deviation of the observations divided by their mean, that is:Population coefficient of variation = CV =Sample coefficient of variation = cv =
17The Empirical Rule… Approximately 68% of all observations fall within one standard deviation of the mean.Approximately 95% of all observations fallwithin two standard deviations of the mean.Approximately 99.7% of all observations fallwithin three standard deviations of the mean.4.17
18Chebysheff’s Theorem… A more general interpretation of the standard deviation is derived from Chebysheff’s Theorem, which applies to all shapes of histograms (not just bell shaped).The proportion of observations in any sample that lie within k standard deviations of the mean is at least:For k=2 (say), the theorem states that at least 3/4 of all observations lie within 2 standard deviations of the mean. This is a “lower bound” compared to Empirical Rule’s approximation (95%).4.18
19Example 2Determine the variance, standard deviation, range, and the cv of the following sample.
21Measures of Relative Standing and Box Plots PercentileThe pth percentile of a set of measurements is the value for whichp percent of the observations are less than that value100(1-p) percent of all the observations are greater than that value.ExampleSuppose your score is the 60% percentile of a SAT test. Then60% of all the scores lie here40%Your score
28Interquartile range = Q3 – Q1 This is a measure of the spread of the middle 50% of the observationsLarge value indicates a large spread of the observationsInterquartile range = Q3 – Q1
29Box PlotThis is a pictorial display that provides the main descriptive measures of the data set:L - the largest observationQ3 - The upper quartileQ2 - The medianQ1 - The lower quartileS - The smallest observation1.5(Q3 – Q1)WhiskerSQ1Q2Q3L
30Measures of Linear Relationship… We now present two numerical measures of linear relationship that provide information as to the strength & direction of a linear relationship between two variables (if one exists).They are the covariance and the coefficient of correlation.Covariance - is there any pattern to the way two variables move together?Coefficient of correlation - how strong is the linear relationship between two variables?
31Covariance… population mean of variable X, variable Y sample mean of variable X, variable YNote: divisor is n-1, not n as you may expect.
32Covariance…In much the same way there was a “shortcut” for calculating sample variance without having to calculate the sample mean, there is also a shortcut for calculating sample covariance without having to first calculate the mean:
33Covariance… (Generally speaking) When two variables move in the same direction (both increase or both decrease), the covariance will be a large positive number.When two variables move in opposite directions, the covariance is a large negative number.When there is no particular pattern, the covariance is a small number.
34Coefficient of Correlation… The coefficient of correlation is defined as the covariance divided by the standard deviations of the variables:Greek letter “rho”This coefficient answers the question:How strong is the association between X and Y?
35Coefficient of Correlation… The advantage of the coefficient of correlation over covariance is that it has fixed range from -1 to +1, thus:If the two variables are very strongly positively related, the coefficient value is close to +1 (strong positive linear relationship).If the two variables are very strongly negatively related, the coefficient value is close to -1 (strong negative linear relationship).No straight line relationship is indicated by a coefficient close to zero.
36Coefficient of Correlation… +1-1Strong positive linear relationshipr or r =No linear relationshipStrong negative linear relationship
37Example 5 (Textbook 4.58)Are the marks one receives in a course related to the amount of time spent studying the subject? To analyze this mysterious possibility, a student took a random sample of 10 students who had enrolled in an accounting class last semester. She asked each to report his or her mark in the course and the total number of hours spent studying accounting. These data are listed here.Time SpentStudyingMarksa. Calculate the covarianceb. Calculate the coefficient of correlationc. Determine the least squares lined. What do the statistics calculated above tell you about the relationship between marks and study time?e. Calculate the coefficient of determination