Some Measures of SpreadBar Graphs1. Data are divided into groups2. For each group draw a rectanglea. Width same for each groupb. Height proportional to frequency3. Rectangles are equally spacedExample: The strength of reaction of a blood specimen to a certain antigen can be categorized according to degree of clumping.Strength of ReactionFrequencyI6II35III15IV3V3VI8Total70Ages of 30 patients seen in the ER of a hospital on a Friday night353221433960361254453753452364103422364555445546223835564557Ordered Array102336434555123236444656213437455357223538455460223539455564Sturges RuleWidth of Intervalk = number of intervalsW = R / k = (64 – 10) / 6 = 9Intervals10 – 18 vs 10 – 19 Smallest19 – 27 20 – 2928 – 36 30 – 3937 – 45 40 – 4946 – 54 50 – 5955 – 63 60 – 69 LargestMultiples of 5 or 10 are usually more meaningfulMidpoint of lowest interval = (10 + 19) / 2 = 14.5Frequency Distribution of Ages of 30 Patients Seen in the ER on a Friday NightClass IntervalTallyFrequency10-1920-2930-3940-4950-5960-69TotalFrequency Distribution of Ages of 30 Patients Seen in the ER on a Friday NightClass IntervalFreq.Cumulative Freq.Relative Freq.Cumulative Relative Freq.10-19220.06670.066720-29460.13330.200030-399150.30000.500040-497220.23330.733350-596280.20000.933360-692300.06671.0000Total301.0000Measures of Dispersion (Spread)1. Range—difference between largest and smallest values--unstableR = xL - xS2. Percentiles or quantilespth percentile is the value Vp such that p percent of the sample points are  Vp.P50 = medianDef: After sorting observations from smallest to largest, pth percentile =a. (k+1)th largest sample point if np/100 is not an integer (k = largest integer less than np/100)b. average of the (np/100)th and (np/100 +1)th largest observations if np/100 is an integerExample: For ER data, find the 25th percentile.n = 30, p = 25  np/100 = 30*25/100 = 7.5(not an integer)Therefore, 25th percentile = (7 + 1)th largest value = 34Find the 50th percentile.n = 30, p = 50  np/100 = 30*50/100 = 15 (an integer)Therefore, 50th percentile = average of (30*50/100)th and (30*50/100 +1)th largest values = average of 15th and 16th largest values= average of 39, 43 = 41Find P40.Some Measures of SpreadP90 – P10P75 – P25 – Called Interquartile Range3. Variance 1122nxxsniiExample: Fasting blood glucose levels of six children74686065718270xxixxi 2xxi74 74 – 70 = 4 1668 -2 460 -10 10065 -5 2571 1 182 12 144Total 0 290 =  2xxis2 = 290 / 5 = 58Use an alternative formula if the data set is large or if the mean has a long decimal.112122nnxxsniniiixixi274 547668 462460 360065 422571 504182 5724Total = 420 29,690585290529,40029,6905642029,690s224. Standard Deviation7.62582ssSome Properties of the Variance and StandardDeviationTranslation: x1, x2, …, xn – original sample.x1+c, x2+c, …, xn+c – translated sampleDef: yi = xi + c, i = 1, … , nTherefore, 22xyss Scaling:x1, x2, …, xn – original sample.cx1, cx2, …, cxn – rescaled sampleDef: yi = cxi, i = 1, … , nxyxycssscs222Original sample Scaled sample0.013 130.024 240.010 100.009 90.014 14 = 0.070  = 700140.x xc*.*y  014010001422222222000000100003555354142414141413000035540001424014014014013*cs,,*..)(...)(s..).(....).(.sxyx5. Coefficient of Variation%*xsCV 100Example:Blood Glucose in Children:62770 .s,x Blood Glucose in Adults:6518110 .s,x %.%*.CV%.%*.CVAC017100110651891010070627Bar Graphs1. Data are divided into groups2. For each group draw a rectanglea. Width same for each groupb. Height proportional to frequency3. Rectangles are equally spacedExample: The strength of reaction of a blood specimen to a certain antigen can be categorized according to degree of clumping.Strength ofReactionFrequencyI 6II 35III 15IV 3V 3VI 8Total 70Ages of 30 patients seen in the ER of a hospital on a Friday night35 32 21 43 39 6036 12 54 45 37 5345 23 64 10 34 2236 45 55 44 55 4622 38 35 56 45 57Ordered Array10 23 36 43 45 5512 32 36 44 46 5621 34 37 45 53 5722 35 38 45 54 6022 35 39 45 55 64Sturges RuleK = 1 + 3.322(log10 n) = 1 + 3.322(log10 30) = 1 + 4.91 = 5.91 ~ 6Width of IntervalLet R = Range (largest value – smallest value) k = number of intervalsW = R / k = (64 – 10) / 6 = 9Intervals10 – 18 vs 10 – 19 Smallest19 – 27 20 – 2928 – 36 30 – 3937 – 45 40 – 4946 – 54 50 – 5955 – 63 60 – 69 LargestMultiples of 5 or 10 are usually more meaningfulMidpoint of lowest interval = (10 + 19) / 2 = 14.5Frequency Distribution of Ages of 30Patients Seen in the ER on a Friday NightClass IntervalTally Frequency10-1920-2930-3940-4950-5960-69TotalFrequency Distribution of Ages of 30Patients Seen in the ER on a Friday NightClass IntervalFreq. CumulativeFreq.RelativeFreq.CumulativeRelativeFreq.10-19 2 2 0.0667 0.066720-29 4 6 0.1333 0.200030-39 9 15 0.3000 0.500040-49 7 22 0.2333 0.733350-59 6 28 0.2000 0.933360-69 2 30 0.0667 1.0000Total 30 1.0000Class IntervalTrue ClassLimitsFrequency10-19 9.5-19.5 220-29 19.5-29.5 430-39 29.5-39.5 940-49 39.5-49.5 750-59 49.5-59.5 660-69 59.5-69.5 2Stem and Leaf PlotStem Leaf1 0 22 1 2 2 33 2 4 5 5 6 6 7 8 94 3 4 5 5 5 5 5 65 3 4 5 5 6 76 0