2 Chapter 3 Descriptive Statistics: Numerical Methods Measures of LocationMeasures of VariabilityMeasures of Relative Location and Detecting OutliersExploratory Data AnalysisMeasures of Association Between Two VariablesThe Weighted Mean andWorking with Grouped Data%x
3 Measures of LocationMeanMedianModePercentilesQuartiles
4 Example: Apartment Rents Given below is a sample of monthly rent values ($)for one-bedroom apartments. The data is a sample of 70apartments in a particular city. The data are presentedin ascending order.
5 Mean The mean of a data set is the average of all the data values. If the data are from a sample, the mean is denoted by.If the data are from a population, the mean is denoted by m (mu).
7 MedianThe median is the measure of location most often reported for annual income and property value data.A few extremely large incomes or property values can inflate the mean.
8 MedianThe median of a data set is the value in the middle when the data items are arranged in ascending order.For an odd number of observations, the median is the middle value.For an even number of observations, the median is the average of the two middle values.
9 Example: Apartment Rents MedianMedian = 50th percentilei = (p/100)n = (50/100)70 = Averaging the 35th and 36th data values:Median = ( )/2 = 475
10 ModeThe mode of a data set is the value that occurs with greatest frequency.The greatest frequency can occur at two or more different values.If the data have exactly two modes, the data are bimodal.If the data have more than two modes, the data are multimodal.
12 PercentilesA percentile provides information about how the data are spread over the interval from the smallest value to the largest value.Admission test scores for colleges and universities are frequently reported in terms of percentiles.
13 PercentilesThe pth percentile of a data set is a value such that at least p percent of the items take on this value or less and at least (100 - p) percent of the items take on this value or more.Arrange the data in ascending order.Compute index i, the position of the pth percentile.i = (p/100)nIf i is not an integer, round up. The p th percentile is the value in the i th position.If i is an integer, the p th percentile is the average of the values in positions i and i +1.
14 Example: Apartment Rents 90th Percentilei = (p/100)n = (90/100)70 = 63Averaging the 63rd and 64th data values:90th Percentile = ( )/2 = 585
15 Quartiles Quartiles are specific percentiles First Quartile = 25th PercentileSecond Quartile = 50th Percentile = MedianThird Quartile = 75th Percentile
17 Measures of Variability It is often desirable to consider measures of variability (dispersion), as well as measures of location.For example, in choosing supplier A or supplier B we might consider not only the average delivery time for each, but also the variability in delivery time for each.
18 Measures of Variability RangeInterquartile RangeVarianceStandard DeviationCoefficient of Variation
19 RangeThe range of a data set is the difference between the largest and smallest data values.It is the simplest measure of variability.It is very sensitive to the smallest and largest data values.
20 Example: Apartment Rents RangeRange = largest value - smallest valueRange = = 190
21 Interquartile RangeThe interquartile range of a data set is the difference between the third quartile and the first quartile.It is the range for the middle 50% of the data.It overcomes the sensitivity to extreme data values.
23 VarianceThe variance is a measure of variability that utilizes all the data.It is based on the difference between the value of each observation (xi) and the mean (x for a sample, m for a population).
24 VarianceThe variance is the average of the squared differences between each data value and the mean.If the data set is a sample, the variance is denoted by s2.If the data set is a population, the variance is denoted by 2.
25 Standard DeviationThe standard deviation of a data set is the positive square root of the variance.It is measured in the same units as the data, making it more easily comparable, than the variance, to the mean.If the data set is a sample, the standard deviation is denoted s.If the data set is a population, the standard deviation is denoted (sigma).
26 Coefficient of Variation The coefficient of variation indicates how large the standard deviation is in relation to the mean.If the data set is a sample, the coefficient of variation is computed as follows:If the data set is a population, the coefficient of variation is computed as follows:
27 Example: Apartment Rents VarianceStandard DeviationCoefficient of Variation
28 Measures of Relative Location and Detecting Outliers z-ScoresChebyshev’s TheoremEmpirical RuleDetecting Outliers
29 z-Scores The z-score is often called the standardized value. It denotes the number of standard deviations a data value xi is from the mean.A data value less than the sample mean will have a z-score less than zero.A data value greater than the sample mean will have a z-score greater than zero.A data value equal to the sample mean will have a z-score of zero.
30 Example: Apartment Rents z-Score of Smallest Value (425)Standardized Values for Apartment Rents
31 Chebyshev’s TheoremAt least (1 - 1/k2) of the items in any data set will bewithin k standard deviations of the mean, where k isany value greater than 1.At least 75% of the items must be withink = 2 standard deviations of the mean.At least 89% of the items must be withink = 3 standard deviations of the mean.At least 94% of the items must be withink = 4 standard deviations of the mean.
32 Example: Apartment Rents Chebyshev’s TheoremLet k = 1.5 with = and s = 54.74At least (1 - 1/(1.5)2) = = 0.56 or 56%of the rent values must be between- k(s) = (54.74) = 409and+ k(s) = (54.74) = 573
33 Example: Apartment Rents Chebyshev’s Theorem (continued)Actually, 86% of the rent valuesare between 409 and 573.
34 Empirical Rule For data having a bell-shaped distribution: Approximately 68% of the data values will be within one standard deviation of the mean.
35 Empirical Rule For data having a bell-shaped distribution: Approximately 95% of the data values will be within two standard deviations of the mean.
36 Empirical Rule For data having a bell-shaped distribution: Almost all (99.7%) of the items will be within three standard deviations of the mean.
37 Example: Apartment Rents Empirical RuleInterval % in IntervalWithin +/- 1s to /70 = 69%Within +/- 2s to /70 = 97%Within +/- 3s to /70 = 100%
38 Detecting OutliersAn outlier is an unusually small or unusually large value in a data set.A data value with a z-score less than -3 or greater than +3 might be considered an outlier.It might be an incorrectly recorded data value.It might be a data value that was incorrectly included in the data set.It might be a correctly recorded data value that belongs in the data set !
39 Example: Apartment Rents Detecting OutliersThe most extreme z-scores are and 2.27.Using |z| > 3 as the criterion for an outlier,there are no outliers in this data set.Standardized Values for Apartment Rents
40 Exploratory Data Analysis Five-Number SummaryBox Plot
41 Five-Number Summary Smallest Value First Quartile Median Third QuartileLargest Value
42 Example: Apartment Rents Five-Number SummaryLowest Value = First Quartile = 450Median = 475Third Quartile = Largest Value = 615
43 Box PlotA box is drawn with its ends located at the first and third quartiles.A vertical line is drawn in the box at the location of the median.Limits are located (not drawn) using the interquartile range (IQR).The lower limit is located 1.5(IQR) below Q1.The upper limit is located 1.5(IQR) above Q3.Data outside these limits are considered outliers.… continued
44 Box Plot (Continued)Whiskers (dashed lines) are drawn from the ends of the box to the smallest and largest data values inside the limits.The locations of each outlier is shown with the symbol * .
46 Measures of Association Between Two Variables CovarianceCorrelation Coefficient
47 CovarianceThe covariance is a measure of the linear association between two variables.Positive values indicate a positive relationship.Negative values indicate a negative relationship.
48 CovarianceIf the data sets are samples, the covariance is denoted by sxy.If the data sets are populations, the covariance is denoted by
49 Correlation Coefficient The coefficient can take on values between -1 and +1.Values near -1 indicate a strong negative linear relationship.Values near +1 indicate a strong positive linear relationship.If the data sets are samples, the coefficient is rxy.If the data sets are populations, the coefficient is
50 The Weighted Mean and Working with Grouped Data Mean for Grouped DataVariance for Grouped DataStandard Deviation for Grouped Data
51 Weighted MeanWhen the mean is computed by giving each data value a weight that reflects its importance, it is referred to as a weighted mean.In the computation of a grade point average (GPA), the weights are the number of credit hours earned for each grade.When data values vary in importance, the analyst must choose the weight that best reflects the importance of each value.
52 Weighted Mean x = wi xi wi where: xi = value of observation i wi = weight for observation i
53 Grouped DataThe weighted mean computation can be used to obtain approximations of the mean, variance, and standard deviation for the grouped data.To compute the weighted mean, we treat the midpoint of each class as though it were the mean of all items in the class.We compute a weighted mean of the class midpoints using the class frequencies as weights.Similarly, in computing the variance and standard deviation, the class frequencies are used as weights.
54 Mean for Grouped Data Sample Data Population Data where: fi = frequency of class iMi = midpoint of class i
55 Example: Apartment Rents Given below is the previous sample of monthly rentsfor one-bedroom apartments presented here as groupeddata in the form of a frequency distribution.
56 Example: Apartment Rents Mean for Grouped DataThis approximation differs by $2.41 fromthe actual sample mean of $
57 Variance for Grouped Data Sample DataPopulation Data
58 Example: Apartment Rents Variance for Grouped DataStandard Deviation for Grouped DataThis approximation differs by only $.20from the actual standard deviation of $54.74.