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S1 Averages and Measures of Dispersion S1 Measures of Dispersion Objectives: To be able to find the median and quartiles for discrete data To be able to find the median and quartiles for continuous data using interpolation Can you work out the rule for finding median and quartiles from discrete data? LQ 1.5 Median 3 UQ 4.5 12345 123456 1234567 12345678 123456789 Can you spot any rules for n amount of numbers in a list? LQ n/4 If n/4 is a whole number find the mid point of corresponding term and the term above If n is not a whole number, round the number up and find the corresponding term UQ 3n/4 If 3n/4 is a whole number find the mid point of corresponding term and the term above If n is not a whole number, round the number up and find the corresponding term Median n/2 If n/2 is a whole number find the midpoint of the corresponding term and the term above If n/2 is not a whole number, round up and find the corresponding term Calculate the mean, median and inter quartile range from a table of discrete data Number of Number of CDs(x) students (f) 35 3 36 17 37 29 38 34 39 12 Mean = Σfx Σf Calculate the mean, median and inter quartile range from a table of discrete data Number of CDs(x) Number of students (f) Cumulative frequency 35 3 3 36 17 20 37 29 49 38 34 83 39 12 95 Median = n/2 Median = 95/2 = 47.5 = 48th value Median = 37 CDs LQ = 95/4 = 23.75 LQ = 24th value LQ (Q1) = 37 CDs UQ (Q3) = 95/4 x 3 = 71.25 UQ = 72nd value IQR = Q3-Q1 = 38-37=1 UQ (Q3) = 38 CDs Calculate the mean, median and inter quartile range from a table of continuous data Length of flower stem (mm) Number of flowers (f) 30-31 2 2 32-33 25 27 34-36 30 57 37-39 13 70 Cumulative frequency Median = n/2 We do not need to do any rounding because we are dealing with continuous data Median = 70/2 = 35th value This lies in the 3436 class but we don’t know the exact value of the term Using interpolation to find an estimate for the median 33.5mm 27 m 36.5mm 35 57 m – 33.5 = 35 - 27 36.5 – 33.5 = 57 - 27 m – 33.5 = 8 3 30 m – 33.5 = 0.26 x 3 m = 33.5 + 0.8 = 34.3 Using interpolation to find an estimate for the lower quartile LQ = 70/4 = 17.5 (in the 32-33 group) 31.5mm 2 Q1 33.5mm 17.5 27 Q1 – 31.5 = 15.5 2 25 Q1 – 31.5 = 0.62 x 2 Q1 = 31.5 + 1.24 = 32.74 Q1 – 31.5 = 17.5 - 2 33.5 – 31.5 = 27 - 2 Using interpolation to find an estimate for the upper quartile UQ = 70/4x3 = 52.5 (in the 34-36 group) 33.5mm 27 Q3 36.5mm 52.5 57 Q3 – 33.5 = 25.5 3 30 Q3 – 33.5 = 0.85 x 3 Q1 = 33.5 + 2.55 = 36.05 Q3 – 33.5 = 52.5 - 27 36.5 – 33.5 = 57 - 27 Summary of rules n = total frequency w = class width fB = cumulative frequency below median/lq/uq fU = cumulative frequency above median/lq/uq Median = LB + ½n – fB x w fU - fB LQ = LB + ¼n – fB x w fU - fB UQ = LB + ¾n – fB x w fU - fB The lengths of a batch of 2000 rods were measured to the nearest cm. The measurements are summarised below. Length Number of Cumulative Q1=74.5 + 500-250 x 5 (nearest cm) rods frequency 60-64 65-69 70-74 75-79 80-84 85-89 90-94 95-99 11 49 190 488 632 470 137 23 738-250 11 60 250 738 1370 1840 1977 2000 Q1=77.06 Q2=79.5+1000-738 x 5 1370-738 Q2=81.57 Q3=84.5+1500-1370 x 5 1840-1370 Q3=85.88 By altering the formula slightly can you work out how to find the 3rd decile (D3) and the 67th percentile (P67)? Answers D3=74.5 + 600-250 x 5 738-250 D3=78.09 P67=79.5 + 1340-738 x 5 1370-738 P67=84.26