Arithmetic Mean Part 1

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Arithmetic Mean Part 1

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1. Statistics For Economics अर्थशास्त्र में साांख्यिकी MEASURES OF CENTRAL TENDENCY (PART-1)
2. ARITHMETIC MEAN समाांतर माध्य Calculation of arithmetic mean of discrete series. व्यख्यिगत श्रृखांला का समाांतर माध्य का पररकलन Direct method, Assumed Mean Method, Step deviation method प्रत्यक्ष विवि, कख्यित माध्य विवि, पद-विचलन विवि 2
3. Some Realistic Observations क ु छ िास्तविक अिलोकन Students scoring between 34% to 70% Students scoring less than 33% Students scoring more than 70% Measure of Central Tendency Higher Lower Income Group Middle class Income Group
4. MEASURE OF CENTRAL TENDENCY (AVERAGE) (औसत) An average value is a single value within range of data which represent all the value of the series. औसत मान, आँकड़ोांकी सीमाओां में ख्यथर्त िह एकल मान है जो श्रृांखला क े सभी मूल्य का प्रवतवनवित्व करता है। Objectives of Average औसत क े उद्देश्य ❖ To get the single value that describes the characteristics of the entire data . एक ऐसा मान प्राप्त करना जो समस्त आँकड़ोांकी विशेषताओांको व्यि करता हो। ❖ To facilitate comparison तुलना में सहायक
5. Characteristics Of Measure Of Central Tendency क े न्द्रीय माप की विशेषताएँ ❖ It should be based on all the items. ❖ यह श्रृांखला की प्रत्येक मद पर आिाररत होना चावहए। ❖ It should be easy to compute and easy to understand. ❖ इसकी गणना सरल है ि यह समझने में आसान होना चावहए। ❖ It should not be unduly affected by extreme values. ❖ इसका मान अनािश्यक रृप से चरम मूल्योांसे प्रभावित नहीांहोना चावहए। ❖ It should be defined by rigid mathematical formula, so it is capable of further algebraic treatment . ❖ इसका गवणतीय सूत्र ख्यथर्र रृप से पररभावषत होना चावहए वजससे इस पर आगे भी गवणतीय- वििेचन वकया जा सक े ।
6. Three Commonly Used Measures of Central Tendency (Averages ) are - तीन सिाथविक प्रचवलत क े न्द्रीय प्रिृवि (औसत) क े माप हैं - (i) Arithmetic Mean समाांतर माध्य (ii) Medianमख्यध्यका (iii) Modeबहुलक
7. ARITHMETIC MEAN समाांतर माध्य It is defined as the value obtained after the sum of all observations divided by the number of observations and is usually denoted by X. समाांतर माध्य को, सभी प्रेक्षणोांक े मूल्योांक े योग को क ु ल प्रेक्षण सांिाओां से विभावजत करने क े उपराांत प्राप्त मान, क े रृप में पररभावषत वकया जाता है और सामान्यत: X से वनदेवशत वकया जाता है । Notation for sum of observation प्रेक्षण ोंक े य ग का सोंक े त चिन्ह ΣX = x1+ X2+ X3+…..+ Xn
8. Calculation of Arithmetic Mean समाांतर माध्य का पररकलन Individual Series व्यख्यिगत श्रृांखला Discrete Series विविि श्रृखांला Continuous Series सांतत श्रृखांला
9. INDIVIDUAL SERIES व्यक्तिगत श्रोंखला The individual series is the series when data are expressed in such a way that each unit of experiment has its own unique value. व्यख्यिगत श्रृांखला िह श्रृांखला है, जब आँकड़ोांको इस प्रकार से व्यि वकया जाता है, वक अनुसांिान से जुड़ी प्रत्येक इकाई का, अपना एक विशेष मूल्य होता है। For example उदाहरणतः Marks obtained by 10 students in Economics Test Roll no. 1 14 10 2 3 4 5 6 8 7 4 8 9 10 16 Marks obtained 12 20 10 18 16
10. प्रत्यक्ष विवि लघु विवि पद विचलन विवि
11. Direct Method प्रत्यक्ष विवि इस चिचि क े पद :- 1) सभी मद ोंका य ग ज्ञात कीचिए। 2) मद ोंकी सोंख्या ज्ञात कीचिए। 3) चदए गए सूत्र का उपय ग कीचिए। Steps involved in this method :- 1) Add all the observations. 2) Find out the number of observations. 3) Apply the given formula . माध्य =x Mean = X Sum of observation Σ X = = Number of observation N मद ोंका क ु ल य ग = मद ोंकी सोंख्या
12. Numerical Example सांिात्मक उदाहरण X 5 15 20 25 35 Step 1) Sum of observation मद ोंका य ग Σ X = 5 + 15 + 20 + 25 + 35 = 5 100 15 Step 2) Number of observation मद ोंकी सोंख्या N =5 N=5 20 25 Mean(माध्य )= Σ X/ N 35 Sum of observation मद ोंका य ग Σ X = 100 = Number of observation मद ोंकी सोंख्या 100/5=20
13. NUMERICAL EXAMPLE Calculate Arithmetic Mean from the data showing marks of students in a class in an economics test: 40, 50, 55, 78, 58. X N 40 +50 + 55 + 78 + 58 281 5 5 = = X= =Σ X 56.2
14. SOME USEFUL OBSERVATIONS क ु छ उपयोगी अिलोकन Mean = 100/5 = 20 5 15 20 25 35 If each of the observation of the given data increased by 5 unit. यचद आँकड़ ोंकी प्रत्येक मद क 5 इकाई बढ़ा दे। Mean = 125/5 = 25 5+5 =10 15+5 =20 20+5 =25 25+5 =30 35+5 =40 Original Mean is also increased by 5 If each of the observation of the given data are multiplied by 2 unit. यचद आँकड़ ोंकी प्रत्येक मद क 2 इकाई से गुना कर दे। Original Mean is also multiplied by 2 Mean = 200/5 = 40 5x2 =10 15x2 =30 20x2 =40 25x2 =50 35x2 =70
15. SOME USEFUL OBSERVATIONS क ु छ उपयोगी अिलोकन Mean = 100/5 = 20 5 15 20 25 35 If each of the observation of the given data decreased by 5 unit. यचद आँकड़ ोंकी प्रत्येक मद क 5 इकाई घटा दे। 20-5 =15 5-5 =0 15-5 =10 25-5 =20 35-5 =30 Original Mean is also decreased by 5 Mean = 75/5 = 15 If each of the observations of the given data divide by 2. यचद आँकड़ ोंकी प्रत्येक मद क 2 इकाई से भाग कर दे। Mean = 50/5 5/2 =2.5 15/2 =7.5 20/2 =10 25/2 =12.5 35/2 =17.5 Original Mean is also divided by 2 = 10
16. Ques : Which of the following is not a measure of Central Tendency ? वनम्नवलख्यखत में से कौनसा, क े न्द्रीय प्रिृवि का माप नहीांहै ? (i) Mean माध्य (ii) Mode बहुलक (iii) Frequency Table आिृवत सारणी (iv) Median मख्यध्यका Ans(iii) Frequency Table आिृवत सारणी
17. Ques 2: Formula for finding Arithmetic Mean by Direct Method is – प्रत्यक्ष विवि द्वारा समाांतर माध्य ज्ञात करने का सूत्र है– (i) X = ∑ X (ii) X = ∑ X.N (iii) X = N / ∑ X Ans : X = ∑ X / N (iv) X = ∑ X / N
18. SHORT CUT METHOD लघु विवि Step 1 – Take any middle value of the series as an assumed mean - (A). श्रृांखला क े मध्य से कोई एक कख्यित माध्य (A) लीवजयें । Step 2 - Take the deviation of assumed mean from all the items and denote these deviations by d. हर मद से कख्यित माध्य का विचलन ज्ञात कीवजयें ि उन्हें d से वनदेवशत कीवजयें। Step 3- Obtained the sum of deviation, ∑dx विचलनोांका योग (∑dx ) प्राप्त कीवजयें Where d = (X – A) And A = Assumed mean Step 4- Apply the formula सूत्र का प्रय ग कीचियें
19. SHORT CUT METHOD (By Assumed Mean) लघु विवि (कख्यित माध्य द्वारा) Step 1 – Take any middle value as assumed mean -(A) Income X 165 225 295 315 145 d = X – A Income(X) A A = 225 Step 2 – find deviations from assumed mean d = X – 225 145 –225 = - 80 145 165 225 295 315 - 140 165 – 225 = -60 225 -225 = 0 295 – 225 = 70 315 – 225 = 90 ∑d = 20 Step 3 - find sum of deviation, ∑dx ∑d = 20 A 160 Step 4 - Apply the formula = 225 + (20/ 5) = 225 + 4 = 229
20. Numerical Example सांिात्मक उदाहरण Families Income (X) d = X - 850 A B C D E F G H I J 80 100 360 400 420 700 750 850 2500 5000 11160 -770 -750 -490 - 450 - 430 -150 - 100 0 1650 +4150 +2660 = 850 + 2660 10 = 850 + 266 = 1166 TOTAL
21. Ques : In which method, to calculate arithmetic mean we used deviations from assumed mean so that we can simplify calculations? वकस विवि में, समाांतर माध्य की गणना करने क े वलए हम कख्यित माध्य का प्रयोग करते है, तावक गणना को सरल की जा सक े ? (i)Direct Method प्रत्यक्ष विवि (ii)Short cut Method लघु विवि Ans : (ii) Short cut Method लघु विवि
22. Merits of Arithmetic Mean समाांतर माध्य क े गुण 3) It is defined by rigid mathematical formula, so it is capable of further algebraic treatment . इसका गवणतीय सूत्र ख्यथर्र रृप से पररभावषत है अतः इस पर आगे भी गवणतीय- वििेचन वकया जा सकता है। 2) It is based on all the values of the series. यह श्रृांखला की प्रत्येक मद पर आिाररत होता है। 1) It is simple to compute and understand. इसकी गणना सरल है ि समझने में आसान है।
23. Limitations Of Arithmetic Mean समाांतर माध्य की सीमाऐां 1) It is affected by extreme values (too large or too small ) of the series. यह श्रृांखला क े चरम मूल्योां(अत्याविक बड़े या छोटे) से प्रभावित होता है । 2) In the distribution with open-end classes the value of mean cannot be computed without making assumptions regarding the size of the class interval of the open-end classes. खुले िगथ अांतराल िाली श्रृांखला में माध्य की गणना नहीांकी जा सकती जब तक खुले िगथ-अांतराल क े आकार को लेकर मान्यताएँ न बनाई जाए। 3) Can give absurd conclusion. वदए गए वनष्कषथ अस्पष्ट हो सकते है।
24. Calculate Arithmetic mean if marks of 10 students are as follows: अांकगवणतीय माध्य की गणना करें यवद10 छात्रोांक े अांक इस प्रकार हैं: 15, 24, 9, 18, 32, 35, 41, 31, 43, 27 Answer: (Arithmetic Mean)
25. Daily pocket expenditure of 5 students are as follows: 20,25,X,40,45. If mean is Rs32 then find X. 5 छात्रोांका दैवनक जेब खचथ इस प्रकार है: 20, 25, X, 40, 45। यवद माध्य 32 रू है, तो X का मान ज्ञात कीवजए। Answer: (A.M.) 160=130+X X=30
26. Q1) What do you understand by the term ‘’Measures of central Tendencies’’? State its main objectives. ‘’क े न्द्रीय प्रिृवत की माप’’ से आप क्या समझते है? इसक े मुि उद्देश्य वलख्यखए। Q2) State the merit and limitations of “Arithmetic Mean’’ as a measure of central value.? क े न्द्रीय प्रिृवत क े मान क े रृप में समाांतर माध्य क े गुण ि सीमाओांका िणथन कीवजए । Q3) Calculate Arithmetic mean of the given data by (i) Direct method (ii) Shortcut method. वदए गए आकड़ोांसे समाांतर माध्य की गणना प्रत्यक्ष विवि ि लघु विवि द्वारा कीवजए। Incentives given by firm to their 10 employees. एक फमथ द्वारा अपने 10 कमथचाररयोांको दी गई प्रोत्साहन रावश 180 150 160 170 150 190 210 250 160 230 Q4) Find X if mean of observation is 9 . यवद मदोांका समाांतर माध्य 9 है तो X का मान ज्ञात कीवजए X , X+ 3 , X + 5 , X + 7 , X + 10

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