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S1 Averages and Measures of Dispersion

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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

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Can you work out the rule for finding median and quartiles from discrete data? LQ Median UQ Can you spot any rules for n amount of numbers in a list?

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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

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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

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Calculate the mean, median and inter quartile range from a table of discrete data Number of CDs(x) Number of students (f) Mean = Σfx Σf

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Calculate the mean, median and inter quartile range from a table of discrete data Number of CDs(x) Number of students (f) Cumulative frequency Median = n/2 Median = 95/2 = 47.5 = 48 th value Median = 37 CDs LQ = 95/4 = LQ = 24 th value LQ (Q1) = 37 CDs UQ (Q3) = 95/4 x 3 = UQ = 72 nd value UQ (Q3) = 38 CDs IQR = Q3-Q1 = 38-37=1

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Calculate the mean, median and inter quartile range from a table of continuous data Length of flower stem (mm) Number of flowers (f) Cumulative frequency Median = n/2 We do not need to do any rounding because we are dealing with continuous data Median = 70/2 = 35 th value This lies in the class but we dont know the exact value of the term

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Using interpolation to find an estimate for the median 33.5mm36.5mm m m – 33.5 = – 33.5 = m – 33.5 = m – 33.5 = 0.26 x 3 m = = 34.3

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Using interpolation to find an estimate for the lower quartile 31.5mm33.5mm Q1 Q1 – 31.5 = – 31.5 = Q1 – 31.5 = Q1 – 31.5 = 0.62 x 2 Q1 = = LQ = 70/4 = 17.5 (in the group)

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Using interpolation to find an estimate for the upper quartile 33.5mm36.5mm Q3 Q3 – 33.5 = – 33.5 = Q3 – 33.5 = Q3 – 33.5 = 0.85 x 3 Q1 = = UQ = 70/4x3 = 52.5 (in the group)

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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

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The lengths of a batch of 2000 rods were measured to the nearest cm. The measurements are summarised below. Length (nearest cm) Number of rods Cumulative frequency Q1= x Q1=77.06 Q2= x Q2=81.57 Q3= x Q3=85.88 By altering the formula slightly can you work out how to find the 3 rd decile (D3) and the 67 th percentile (P67)?

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Answers D3= x D3=78.09 P67= x P67=84.26

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