1 THE NORMAL DISTRIBUTION METHOD ARTICLE NO.: 10080 By Ken Stndfield, Director Reserch & Development, KNOWCORP http://www.knowcorp.com Emil: ks@knowcorp.com INTRODUCTION The following methods hve been developed over the yers tht I lectured sttistics. Wht follows is not new discovery, in itself, but rther strict ordering in wht hs remined lrgely unstructured re. It is method designed to provide frmework through which prcticlly every norml distribution problem cn be nlyed. I would like to etend my thnks to my friend nd collegue Dr Roger Wllce t Dekin University. I hve found tht students find difficulty understnding the concept of res under the stndrd norml curve. This problem cn be solved by reding rticle 10150: Understnding Ares Under Curves - ND & CLT. Sttisticl formule nd sttisticl symbols hve lso cused students considerble confusion, rticles 10060: Understnding Sttisticl Formule nd Article 10050: Eplining Sttisticl Symbols should cler up ny problems. THE NORMAL DISTRIBUTION METHOD We solve norml distribution problems by trnslting the norml distribution into stndrd norml distribution through use of the following formul: This formul is methodologicl formule (see Article 10060: Understnding Sttisticl Formule), nd hence it hs definite method of solution TYPES OF PROBLEMS There re two bsic norml distribution problems: (1) more or less thn problems nd (2) between problems. We will investigte ech in turn. Ken Stndfield, KNOWCORP (http://www.knowcorp.com ), 1991-1999 ( ks@knowcorp.com ) No prt of this rticle cn be reproduced without the written consent of the uthor. All rights reserved.
2 MORE THAN OR LESS THAN PROBLEMS It is very esy to identify more or less thn problems. For more thn problems, look for the words "more thn" or "increses" or "eceeds" or other words, in the question, tht hve the sme mening s more thn. For less thn problems, look for the words "less thn" or "decreses" or other words, in the question, tht hve the sme mening s less thn. Before we look t the specific method, we will investigte the two ctegories (more nd less thn) brodly. 1.1 MORE THAN PROBLEM TYPES There re three ctegories tht more thn problems cn fll into: More thn negtive -score More thn -score of ero More thn positive -score Ken Stndfield, KNOWCORP (http://www.knowcorp.com ), 1991-1999 ( ks@knowcorp.com ) No prt of this rticle cn be reproduced without the written consent of the uthor. All rights reserved.
3 2.1 LESS THAN PROBLEM TYPES Less thn problems cn fll into three ctegories: Less thn negtive -score Less thn -score of ero Less thn positive -score 3.1 THE MORE THAN METHOD The following represents the more thn method. The method is divided into four sections: (1) Bsic Mthemtics (2) Drwing the Correct Curve (3) Work out the Approprite Are (4) Conclusion Ken Stndfield, KNOWCORP (http://www.knowcorp.com ), 1991-1999 ( ks@knowcorp.com ) No prt of this rticle cn be reproduced without the written consent of the uthor. All rights reserved.
4 3.1.1 SECTION #1 BASIC MATHEMATICS PN > PSN > ( ) ( ) PSN ( > ) Simp PSN ( > ) P ( > ) SN Let's trnslte the bove section of the method: (1) determine the vlue, (2) proceed to trnslte the -vlue into stndrd norml vlue by pplying the stndrd norml curve formul; substitute the vlue for the vlue of ; simplify (Simp) the numertor of the stndrd norml formul; simplify the whole frction to rrive t the -vlue of. If terminology like numertor nd denomintor confuses you, refer to Article 10050: Eplining Sttisticl Formule, it will cler up ny problems. 3.1.2 SECTION #2 DRAWING THE CORRECT CURVE This is three-stge process. Step 1 is to nlye the vlue of (positive, ero, or negtive). Step 2 involves mrking in the vlue of nd step 3 involves shding in the correct direction nd shding the correct distnce, for emple: Shding from Shding from Shding from negtive -score ero -score positive -score The ppropritely shded digrm is integrted into stge #1 - you will see how lter. 3.1.3 SECTION #3 WORK OUT THE APPROPRIATE AREA Immeditely under the pproprite digrm (determined from Stge #2), we strt determining the pproprite vlue. We do this by drwing our conclusions together in the following formt: Ken Stndfield, KNOWCORP (http://www.knowcorp.com ), 1991-1999 ( ks@knowcorp.com ) No prt of this rticle cn be reproduced without the written consent of the uthor. All rights reserved.
5 P > P > ( ) ( ) P ( > ) N SN SN Simp PSN ( > ) P ( > ) SN Hence: P ( > ) P ( > ) 0. 5000 φ ( ) N SN 0. 5000 det / d det / d Article 10150: Understnding Ares under Curves - ND & CLT, shows how to determine the re under ny curve - which is dding or subtrcting 0.5000 from -vlues (etc). It lso eplins phi-nottion - φ. The det/d prt of the solution bove indictes tht this is vlue tht hs been determined. 3.1.4 SECTION #4 CONCLUSION You hve determined you nswer. To ensure tht you mintin completely logicl pproch to solving problems, you need to drw together your findings using words nd numbers, not just numbers. Ken Stndfield, KNOWCORP (http://www.knowcorp.com ), 1991-1999 ( ks@knowcorp.com ) No prt of this rticle cn be reproduced without the written consent of the uthor. All rights reserved.
6 4.1 THE LESS THAN METHOD We hve investigted the more thn method, now we shll ttention to the less thn method. The less thn method is identicl to the more thn method, we only chnge few symbols round. P < P < ( ) ( ) P ( < ) N SN SN Simp PSN ( < ) P ( < ) SN Hence, P ( < ) P ( < ) 0. 5000 + φ( ) N SN 0. 5000 + det / d det / d Conclusion: Put the mthemtics into words nd nswer the question. 5 ANALYZING BETWEEN PROBLEMS The other mjor type of problem involves "between" problems. These problems re composed of more thn problem nd less thn problem put together to form between problem. Hence between problems re recognied esily - two -vlues must be determined. Ken Stndfield, KNOWCORP (http://www.knowcorp.com ), 1991-1999 ( ks@knowcorp.com ) No prt of this rticle cn be reproduced without the written consent of the uthor. All rights reserved.
7 5.1 BETWEEN A NUMBER AND ZERO. These problems involve problems tht seek us to determine the re (probbility) between the men nd some other number. 5.2 BETWEEN TWO NUMBERS ON DIFFERENT SIDES b 5.3 BETWEEN TWO NUMBERS ON THE SAME SIDE When we hve two numbers on the sme side, we must decompose the problem b b Note tht Article 10150: Understnding Ares under Curves - ND & CLT, shows the solutions to vriety of shding problems. Ken Stndfield, KNOWCORP (http://www.knowcorp.com ), 1991-1999 ( ks@knowcorp.com ) No prt of this rticle cn be reproduced without the written consent of the uthor. All rights reserved.
8 THE BETWEEN METHOD The following is the generl formt used when solving between problems: 1 2 b P b P P N ( < < ) ( SN < < ) ( SN < < ) Simp Simp PSN ( < < ) P ( < < b ) SN b Hence, P ( < < b ) P ( < < b ) φ( ) + φ( ) N SN b det / d + det / d det / d CONCLUSION Put the mthemtics into words nd nswer the question. If between terminology ( < < b or < < b) cuses you confusion, look t Article 10050: Eplining Sttisticl Symbols, it should cler up the confusion. ABOUT THE AUTHOR See http://www.knowcorp.com/developer.html for more informtion. Ken Stndfield, KNOWCORP (http://www.knowcorp.com ), 1991-1999 ( ks@knowcorp.com ) No prt of this rticle cn be reproduced without the written consent of the uthor. All rights reserved.