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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) 35 3 36 17 37 29 38 34 39 12 Mean = Σfx Σf

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**Calculate the mean, median and inter quartile range from a table of discrete data**

Median = n/2 Number of CDs(x) Number of students (f) 35 3 36 17 37 29 38 34 39 12 Cumulative frequency Median = 95/2 = 47.5 = 48th value 3 20 49 83 95 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 UQ (Q3) = 38 CDs IQR = Q3-Q1 = 38-37=1

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**Length of flower stem (mm)**

Calculate the mean, median and inter quartile range from a table of continuous data Median = n/2 We do not need to do any rounding because we are dealing with continuous data Length of flower stem (mm) Number of flowers (f) 30-31 2 32-33 25 34-36 30 37-39 13 Cumulative frequency 2 27 57 70 Median = 70/2 = 35th value This lies in the class but we don’t know the exact value of the term

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**Using interpolation to find an estimate for the median**

33.5mm m 36.5mm 27 35 57 m – = m – = 8 36.5 – = m – = 0.26 x 3 m = = 34.3

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**Using interpolation to find an estimate for the lower quartile**

LQ = 70/4 = 17.5 (in the group) 31.5mm Q1 33.5mm 2 17.5 27 Q1 – = Q1 – = 15.5 33.5 – = Q1 – = 0.62 x 2 Q1 = = 32.74

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**Using interpolation to find an estimate for the upper quartile**

UQ = 70/4x3 = 52.5 (in the group) 33.5mm Q3 36.5mm 27 52.5 57 Q3 – = Q3 – = 25.5 36.5 – = Q3 – = 0.85 x 3 Q1 = = 36.05

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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 lengths of a batch of 2000 rods were measured to the nearest cm. The measurements are summarised below. Length (nearest cm) Number of rods 60-64 11 65-69 49 70-74 190 75-79 488 80-84 632 85-89 470 90-94 137 95-99 23 Cumulative frequency Q1= x 5 Q1=77.06 11 60 250 738 1370 1840 1977 2000 Q2= x 5 Q2=81.57 Q3= x 5 Q3=85.88 By altering the formula slightly can you work out how to find the 3rd decile (D3) and the 67th percentile (P67)?

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**Answers D3=74.5 + 600-250 x 5 738-250 D3=78.09 P67=79.5 + 1340-738 x 5**

P67=84.26

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