Statistics Part II Math 416

Game Plan Creating Quintile Creating Quintile Decipher Quintile Decipher Quintile Per Centile Creation Per Centile Creation Per Centile Interpretation Per Centile Interpretation Statistical Interpretation Statistical Interpretation

Quintiles We have looked at quartiles where we divide the data into quarters to look at the spread of data We have looked at quartiles where we divide the data into quarters to look at the spread of data Now we will be looking at dividing the data into fifths Now we will be looking at dividing the data into fifths We are now concerned with the position of data We are now concerned with the position of data Eg The data in the group is very concentrated. Eg The data in the group is very concentrated.

Quintiles We call We call First R 1 First R 1 Second R 2 Second R 2 Third R 3 Third R 3 Fourth R 4 Fourth R 4 Fifth R 5 Fifth R 5 The top / best The bottom / worst Report Card Handout To divide data into fifths there are certain guidelines not rules that need to be followed.

Creating Quintiles 1) The data must be in order (this can be a pain!) 1) The data must be in order (this can be a pain!) 2) Roughly divide them into fifths called the 1 st try 2) Roughly divide them into fifths called the 1 st 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 - together * 5) keep number close together - together

Creating Quintiles Guidelines #3,4,&5 probably #3 is the most important Guidelines #3,4,&5 probably #3 is the most important Realize that these are guidelines you may need to make decisions that could be different from me Realize that these are guidelines you may need to make decisions that could be different from me The other factor is size of sample The other factor is size of sample If you have a sample of 5000 & you need to move several points… If you have a sample of 5000 & you need to move several points… R5R5R5R5995 R4R4R4R41005 R3R3R3R31000 R2R2R2R2990 R1R1R1R11010 Does not look bad R5R5R5R53 R4R4R4R48 R3R3R3R36 R2R2R2R25 R1R1R1R18 with a sample of 30 points It can look unbalanced

Practice Below are a set of numbers… put them into quintiles Below are a set of numbers… put them into quintiles (32, 43, 47, 44, 45, 31, 39, 39, 32, 49 (32, 43, 47, 44, 45, 31, 39, 39, 32, 49 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, 45) 44, 43, 39, 36, 33, 55, 50, 38, 31, 45) 1 st step – get them in order! 1 st step – get them in order!

Quintiles In order… 31, (31,31,32,32,33,36,36,38,39,39,39,40,40,40,40,41,43,43, In order… 31, (31,31,32,32,33,36,36,38,39,39,39,40,40,40,40,41,43,43, 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) 2 nd step: 2 nd step: 1 st try through… 1 st try through… 31,31,32,32,33,36 * 36,38,39,39,39,40 * 40,40,41,43,43,43 44,44,44,44,45 * 45,47,48,49,49,49,50,55

Quintiles R 5 31, 31, 32, 32, 33, 36,36 R 5 31, 31, 32, 32, 33, 36,36 R 4 38, 39, 39, 39, 40, 40, 40, 40 R 4 38, 39, 39, 39, 40, 40, 40, 40 R 3 41, 43, 43, 43 R 3 41, 43, 43, 43 R 2 44, 44, 44, 44, 44 R 2 44, 44, 44, 44, 44 R 1 45, 45, 47, 48, 49, 49, 49,50,55 R 1 45, 45, 47, 48, 49, 49, 49,50,55 MY QUINTILES

Interpretation of Quintiles Basically, since we need to rank data from best to worst we create a hierarchy Basically, since we need to rank data from best to worst we create a hierarchy Be careful – statistics do not create good and bad Be careful – statistics do not create good and bad The Quintile Range The Quintile Range We sometimes only look at the endpoints of the quintile We sometimes only look at the endpoints of the quintile Thus Rx [8, 41] means all the numbers from 8 to 41 including 8 and 41. Thus Rx [8, 41] means all the numbers from 8 to 41 including 8 and 41.

Example A class has the following quintile structure A class has the following quintile structure R 5 [40,50] R 5 [40,50] R 4 [51,60] R 4 [51,60] R 3 [61,75] R 3 [61,75] R 2 [76,90] R 2 [76,90] R 1 [90,99] R 1 [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 top and a wide spread in middle (teachers nightmare) The class has students close to the bottom and top and a wide spread in middle (teachers nightmare) If Billy has a mark of 81, where would he fit in? If Billy has a mark of 81, where would he fit in? Into the second quintile Into the second quintile

Percentiles If dividing data into four was fun and five was more fun, lets try 100! If dividing data into four was fun and five was more fun, lets try 100! If we say there are n data points and we are trying to figure out the percentile, the formula is If we say there are n data points and we are trying to figure out the percentile, the formula is P Billy = (number below + ½ number equal) x 100 P Billy = (number below + ½ number equal) x 100 n or or P x = NB + ½ NE x 100 P x = NB + ½ NE x 100 n

Example #1 Consider 6,9,15,23,27,29,41,52,58) Consider 6,9,15,23,27,29,41,52,58) What is P 24 ? What is P 24 ? P 24 = NB + ½ NE x 100 P 24 = NB + ½ NE x 100 n P 24 = 4 + ½ (0) x 100 P 24 = 4 + ½ (0) x P 24 = 44 P 24 = 44 This means that 24 is in the 44 th percentile and that 44% of the scores fall below This means that 24 is in the 44 th percentile and that 44% of the scores fall below

Example #2 Consider (5,9,9,18,23,24,24,24,61,63,65) Consider (5,9,9,18,23,24,24,24,61,63,65) What is P 24 ? What is P 24 ? P 24 = NB + ½ NE x 100 P 24 = NB + ½ NE x 100 n n P 24 = 5+ ½ (3) x 100 P 24 = 5+ ½ (3) x P 24 = 59 P 24 = 59

Example #3 Consider (6,9,11,12,14,18,22,28,29) Consider (6,9,11,12,14,18,22,28,29) Where does the 63 rd percentile begin? Where does the 63 rd percentile begin? We use the formula We use the formula BVP = n (Px) BVP = n (Px) (Beginning Value Position) 100 BVP = 9 x 63 BVP = 9 x BVP = 5.67 ; Thus we need 5 below BVP = 5.67 ; Thus we need 5 below Therefore, 18 is where the 63 rd percentile begins Therefore, 18 is where the 63 rd percentile begins Note: These are calculations; the size of the sample make them at best approximation Note: These are calculations; the size of the sample make them at best approximation

Statistical Analysis You need to see what all these techniques mean. Lets put it all together… You need to see what all these techniques mean. Lets put it all together… For the following data find the For the following data find the a) mean b) median c) mode d) range e) box and whisker and f) quintile range (50, 50, 55, 55, 55, 55, 55, 60, 62, 63, 65, 67, 70, 72, 72, 73, 74, 75, 76, 77, 78, 80, 89, 90, 90, 90, 91, 93, 95, 98) n = 30

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

Statistical Analysis 1 st try 1 st try 50, 50, 55, 55, 55, 55 * 55, 60, 62, 63, 65, 67 70, 72, 72, 73, 74, 75 76, 77, 78, 80, 89, 89, 90 * 90, 90, 91, 93, 95, 98

Statistical Analysis R 5 50, 50, 55, 55, 55, 55, 55 R 5 50, 50, 55, 55, 55, 55, 55 R 4 60, 62, 63, 65, 67 R 4 60, 62, 63, 65, 67 R 3 70, 72, 72, 73, 74, 75 R 3 70, 72, 72, 73, 74, 75 R 2 76, 77, 78, 80, 89 R 2 76, 77, 78, 80, 89 R 1 90, 90, 90, 91, 93, 95, 98 R 1 90, 90, 90, 91, 93, 95, 98

Statistical Analysis The mean and the median close together means it is pretty normal The mean and the median close together means it is pretty normal The mode being very different may be a problem The mode being very different may be a problem R = 48 n = 30 look at 48 / 30 = 1.6 (small spread) R = 48 n = 30 look at 48 / 30 = 1.6 (small spread) Box & Whisker looks pretty normal (same size box) Box & Whisker looks pretty normal (same size box) Quintiles worked out pretty normal Quintiles worked out pretty normal Overall nothing spectacular Overall nothing spectacular

Strange Things You May Encouter X = 50; Median = 12… something funny X = 50; Median = 12… something funny R = 100 n = / 3 = 33 big spread! R = 100 n = / 3 = 33 big spread! Box and Whisker will have a very large box with the above… not normal Box and Whisker will have a very large box with the above… not normal Consider Quintiles Consider Quintiles R 5 = 17 R 5 = 17 R 4 = 10 R 4 = 10 R 3 = 16 R 3 = 16 R 2 = 4 R 2 = 4 R 1 = 20 R 1 = 20 Big difference; something funny Big difference; something funny

Final Word Try to live by statistics never lie – so do not make them Try to live by statistics never lie – so do not make them Avoid making definite statement (it is) Avoid making definite statement (it is) Use words like seem like, appears to be… Use words like seem like, appears to be… The teacher is bad because the class average is low… is a definite statement and should be rephrased The teacher is bad because the class average is low… is a definite statement and should be rephrased Good luck and study hard! Good luck and study hard!