Statistics Part IIStatistics Part II

Math 416Math 416

Game PlanGame Plan

Creating QuintileCreating Quintile Decipher QuintileDecipher Quintile Per Centile CreationPer Centile Creation Per Centile Interpretation Per Centile Interpretation Statistical Interpretation Statistical Interpretation

QuintilesQuintiles We have looked at quartiles where We have looked at quartiles where

we divide the data into quarters to we divide the data into quarters to look at the “spread” of datalook at the “spread” of data

Now we will be looking at dividing Now we will be looking at dividing the data into fifthsthe data into fifths

We are now concerned with the We are now concerned with the position of dataposition of data

Eg The data in the group is very Eg The data in the group is very concentrated. concentrated.

QuintilesQuintiles We callWe call First RFirst R11

Second RSecond R22

Third RThird R33

Fourth RFourth R44

Fifth RFifth R55

The top / best

The bottom / worstReport Card Handout

To divide data into fifths there are To divide data into fifths there are certain guidelines certain guidelines not rulesnot rules that that need to be followed.need to be followed.

Creating QuintilesCreating Quintiles

1) The data must be in order (this 1) The data must be in order (this can be a pain!)can be a pain!)

2) Roughly divide them into fifths 2) Roughly divide them into fifths called the 1called the 1stst try try

* 3) Keep numbers the same together* 3) Keep numbers the same together * 4) Keep the groups the same size* 4) Keep the groups the same size * 5) keep number close together - * 5) keep number close together -

togethertogether

Creating QuintilesCreating Quintiles Guidelines #3,4,&5 probably #3 is the Guidelines #3,4,&5 probably #3 is the

most importantmost important Realize that these are guidelines you may Realize that these are guidelines you may

need to make decisions that could be need to make decisions that could be different from medifferent from me

The other factor is size of sampleThe other factor is size of sample If you have a sample of 5000 & you need If you have a sample of 5000 & you need

to move several points… to move several points… RR

55

995995

RR

44

10010055

RR

33

10010000

RR

22

990990

RR

11

10110100

Does not look bad

RR

55

33

RR

44

88

RR

33

66

RR

22

55

RR

11

88

with a sample of 30 points

It can look unbalanced

PracticePractice

Below are a set of numbers… put Below are a set of numbers… put them into quintilesthem into quintiles

(32, 43, 47, 44, 45, 31, 39, 39, 32, (32, 43, 47, 44, 45, 31, 39, 39, 32, 4949

44,40, 49, 36, 49, 48, 40, 41, 43, 44, 44,40, 49, 36, 49, 48, 40, 41, 43, 44,

44, 43, 39, 36, 33, 55, 50, 38, 31, 44, 43, 39, 36, 33, 55, 50, 38, 31, 45)45)

11stst step – get them in order! step – get them in order!

Quintiles Quintiles In order… 31, In order… 31,

(31,31,32,32,33,36,36,38,39,39,39,40,40,40,40,4(31,31,32,32,33,36,36,38,39,39,39,40,40,40,40,41,43,43,1,43,43,

43,44,44,44,44,45,45,47,48,49,49,49,49,50,55)43,44,44,44,44,45,45,47,48,49,49,49,49,50,55)

22ndnd step: step: 11stst try through… try through…31,31,32,32,33,36 *31,31,32,32,33,36 *36,38,39,39,39,40 *36,38,39,39,39,40 *40,40,41,43,43,4340,40,41,43,43,4344,44,44,44,45 *44,44,44,44,45 *45,47,48,49,49,49,50,5545,47,48,49,49,49,50,55

QuintilesQuintiles

RR5 5 31, 31, 32, 32, 33, 36,3631, 31, 32, 32, 33, 36,36

RR4 4 38, 39, 39, 39, 40, 40, 38, 39, 39, 39, 40, 40, 40, 40 40, 40

RR3 3 41, 43, 43, 43 41, 43, 43, 43

RR2 2 44, 44, 44, 44, 44 44, 44, 44, 44, 44

RR1 1 45, 45, 47, 48, 49, 49, 45, 45, 47, 48, 49, 49, 49,50,5549,50,55

MY QUINTILES

Interpretation of Interpretation of QuintilesQuintiles

Basically, since we need to rank data Basically, since we need to rank data from best to worst we create a from best to worst we create a hierarchyhierarchy

Be careful – statistics do not create Be careful – statistics do not create good and badgood and bad

The Quintile RangeThe Quintile Range We sometimes only look at the We sometimes only look at the

endpoints of the quintileendpoints of the quintile Thus Rx [8, 41] means all the numbers Thus Rx [8, 41] means all the numbers

from 8 to 41 including 8 and 41. from 8 to 41 including 8 and 41.

ExampleExample A class has the following quintile structure A class has the following quintile structure RR5 5 [40,50][40,50] RR4 4 [51,60] [51,60] RR3 3 [61,75] [61,75] RR2 2 [76,90] [76,90] RR1 1 [90,99][90,99] What do you notice about this class?What do you notice about this class? The class has students close to the bottom and The class has students close to the bottom and

top and a wide spread in middle (teacher’s top and a wide spread in middle (teacher’s nightmare)nightmare)

If Billy has a mark of 81, where would he fit If Billy has a mark of 81, where would he fit in?in?

Into the second quintile Into the second quintile

PercentilesPercentiles If dividing data into four was fun and five If dividing data into four was fun and five

was more fun, let’s try 100! was more fun, let’s try 100! If we say there are n data points and we are If we say there are n data points and we are

trying to figure out the percentile, the trying to figure out the percentile, the formula isformula is

P P BillyBilly = ( = (number below + ½ number equalnumber below + ½ number equal) ) x 100x 100

n n oror PPx x = = NB + ½ NENB + ½ NE x 100 x 100 n n

Example #1Example #1 Consider 6,9,15,23,27,29,41,52,58)Consider 6,9,15,23,27,29,41,52,58) What is PWhat is P24 24 ?? PP24 24 = = NB + ½ NENB + ½ NE x 100 x 100 n n PP2424 = = 4 + ½ (0)4 + ½ (0) x 100 x 100 9 9 PP2424 = 44 = 44 This means that 24 is in the 44This means that 24 is in the 44thth

percentile and that 44% of the scores percentile and that 44% of the scores fall below fall below

Example #2Example #2 Consider Consider

(5,9,9,18,23,24,24,24,61,63,65)(5,9,9,18,23,24,24,24,61,63,65) What is PWhat is P2424 ? ?

PP2424 = = NB + ½ NENB + ½ NE x 100 x 100 n n PP2424 = = 5+ ½ (3) 5+ ½ (3) x 100 x 100 11 11 PP2424 = 59 = 59

Example #3Example #3 Consider (6,9,11,12,14,18,22,28,29)Consider (6,9,11,12,14,18,22,28,29) Where does the 63Where does the 63rdrd percentile begin? percentile begin? We use the formulaWe use the formula BVP = BVP = n (Px)n (Px)(Beginning Value Position) 100(Beginning Value Position) 100 BVP = BVP = 9 x 639 x 63 100100 BVP = 5.67 ; Thus we need 5 belowBVP = 5.67 ; Thus we need 5 below Therefore, 18 is where the 63Therefore, 18 is where the 63rdrd

percentile beginspercentile begins Note: These are calculations; the size of Note: These are calculations; the size of

the sample make them at best the sample make them at best approximationapproximation

Statistical AnalysisStatistical Analysis

You need to see what all these You need to see what all these techniques mean. Let’s put it all techniques mean. Let’s put it all together… together…

For the following data find the For the following data find the a) mean b) median c) mode d) range e) a) mean b) median c) mode d) range e)

box and whisker and f) quintile rangebox and whisker and f) quintile range(50, 50, 55, 55, 55, 55, 55, 60, 62, 63, (50, 50, 55, 55, 55, 55, 55, 60, 62, 63, 65, 67, 70, 72, 72, 73, 74, 75, 76, 77, 65, 67, 70, 72, 72, 73, 74, 75, 76, 77, 78, 80, 89, 90, 90, 90, 91, 93, 95, 98) 78, 80, 89, 90, 90, 90, 91, 93, 95, 98) n = 30 n = 30

Statistical AnalysisStatistical Analysis

x = 72.5x = 72.5 Median = 72.5Median = 72.5 Mode = 55Mode = 55 Range = 48Range = 48

30

40

50

60

80

70

100

90

Statistical AnalysisStatistical Analysis

11stst try try

50, 50, 55, 55, 55, 55 *50, 50, 55, 55, 55, 55 *

55, 60, 62, 63, 65, 6755, 60, 62, 63, 65, 67

70, 72, 72, 73, 74, 7570, 72, 72, 73, 74, 75

76, 77, 78, 80, 89, 89, 90 *76, 77, 78, 80, 89, 89, 90 *

90, 90, 91, 93, 95, 98 90, 90, 91, 93, 95, 98

Statistical Analysis Statistical Analysis

RR55 50, 50, 55, 55, 55, 55, 55 50, 50, 55, 55, 55, 55, 55

RR44 60, 62, 63, 65, 67 60, 62, 63, 65, 67

RR33 70, 72, 72, 73, 74, 75 70, 72, 72, 73, 74, 75

RR22 76, 77, 78, 80, 89 76, 77, 78, 80, 89

RR11 90, 90, 90, 91, 93, 95, 98 90, 90, 90, 91, 93, 95, 98

Statistical Analysis Statistical Analysis The mean and the median close The mean and the median close

together means it is pretty normaltogether means it is pretty normal The mode being very different may The mode being very different may

be a problembe a problem R = 48 n = 30 look at 48 / 30 = 1.6 R = 48 n = 30 look at 48 / 30 = 1.6

(small spread) (small spread) Box & Whisker looks pretty normal Box & Whisker looks pretty normal

(same size box)(same size box) Quintiles worked out pretty normal Quintiles worked out pretty normal Overall nothing spectacular Overall nothing spectacular

Strange Things You May Strange Things You May EncouterEncouter X = 50; Median = 12… something funnyX = 50; Median = 12… something funny

R = 100 n = 3 100 / 3 = 33 big spread!R = 100 n = 3 100 / 3 = 33 big spread! Box and Whisker will have a very large Box and Whisker will have a very large

box with the above… not normalbox with the above… not normal Consider Quintiles Consider Quintiles RR55 = 17 = 17 RR44 = 10 = 10 RR33 = 16 = 16 RR22 = 4 = 4 RR11 = 20 = 20 Big difference; something funnyBig difference; something funny

Final WordFinal Word Try to live by “statistics never lie – so Try to live by “statistics never lie – so

do not make them”do not make them” Avoid making definite statement (it is)Avoid making definite statement (it is) Use words like seem like, appears to Use words like seem like, appears to

be… be… The teacher is bad because the class The teacher is bad because the class

average is low… is a definite statement average is low… is a definite statement and should be rephrased and should be rephrased

Good luck and study hard! Good luck and study hard!