1. BASIC CONCEPTS OF STATISTICS by : DR. T.K. JAIN AFTERSCHO ☺ OL centre for social entrepreneurship sivakamu veterinary hospital road bikaner 334001 rajasthan, india FOR – PGPSE / CSE PARTICIPANTS [email_address] mobile : 91+9414430763
2. My words..... My purpose here is to give a few questions on fundamentals of statistics. I welcome your suggestions. I also request you to help me in spreading social entrepreneurship across the globe – for which I need support of you people – not of any VIP. With your help, I can spread the ideas – for which we stand....
3. What were the root words of statistics ? Latin = status Germany = statistik Italian = statista french = statistique
4. Who carried out first cencus of the world ? Pharaoh (over 1000 years before Christ)
5. What are the subjects where statistics has application ? Every subject including the following : business management economics commerce industry etc.
6. What are 2 major sources of data? 1. primary data : collected for the first time during the research 2. secondary data : which are already available published data – they were collected for some other purpose, but they can be used for the present research
7. From where can we get secondary data ? 1. industry report 2. previous researches 3. published data 4 annual reports 5. statistical department 6. directories / reports / data bases
8. From where can we get primary data ? 1. interview 2. survey 3. schedule / questionnaire 4. observation 5. experimentation
9. What do we do after collection of data ? Scrutinise = remove data which are defective then we arrange them we try to tabulate them for this we have to fix classification of data then we we can prepare graphs / tables / charts from data and then we can analyse data
10. Why do we classify data ? After classification, data can easily be analysed. We can easily interpret data
11. How can we classify data ? 1. chronologically (data wise / year wise) 2. geographically (north v/s south zone) 3. qualitative ( order = like first, second, third) 4. quantitative data analysis (use of tools for quantitative analysis)
12. Name a few international bodies that publish data (which we can use as secondary source of data)? IBRD, IMF, ADB, ILO, UNO, WTO, WHO etc.
13. What is the difference between primary and secondary data ? Primary data is first hand original in nature whereas secondary data is in the form of compilation of existing data or already published data. The collection of primary data involves huge resources in terms of money and time, finance and energy whereas secondary data is relatively less costly. Primary data is usually collected by keeping in mind the purpose for which it is collected so its suitability will be more in comparison to secondary data
14. What is difference between census and sample survey ? Under the census or complete enumeration method, data are collected for each and every unit of the population or universe which is a complete set of items which are of interest in any particular situation in sample, we pick up only a few items and from them we collect data. So reliability is less comparatively
15. What are the steps in presentation of data ? Classification of data (put data in classes) Tabulation of data (prepare table from data) Frequency distribution of data (identify frequencies) Diagrammatic presentations of data (prepare diagrams) Graphic representation of data (prepare graphs).
16. What do you understand from tabulation ? Tabulation is a systematic and logical arrangement of data in columns and rows in accordance with some salient features and characteristics.
17. What are the parts of a table ? Table Number Title of the Table Sub-title or Head Note Captions and stub Body Footnotes Source Note
18. What is class limit ? The end numbers or the highest and lowest values that can be included in a class interval are known as the class limits of that class. For example, in above table 40-50 and 80-100 are the lower and upper class limits.
19. What is class interval ? It is the difference between the upper limit and lower limit of the same class. The lower limit of a class is usually represented by symbol I1 and upper limit by I 2 .
20. What is Class frequency ? The number of observations included in a particulars class is known as the frequency of that class.
21. It refers to that classification where both the class limits are included in the class itself while determining the class intervals.
22. What are the 3 methods of data presentation ? 1. textual presentation = present data in the form of text – write reports etc. 2. graphical presentation = prepare graphs, pie chart, bar chart, histo gram etc. 3. tabular presentation : prepare tables of data for better analysis
37. Z score (Xi – mean) / standard deviation it is a standardised value = showing difference from mean + & - 1 standard deviation =68.27% + & - 2 standard deviation = 95.45% + & - 3 standard deviation = 99.73%
38. Empirical rule ? In a bell shaped distribution (normal distribution), we have data in 1 or 2 or 3 standard deviation to mean in some % of total data
39. Outlier ? Unusually small or unusually large data
49. Tree diagram A graphical representation helpful in identifying the sample points of an experiment involving multiple steps
50. What is permutation & combination? Permutation = it denotes order / Sequence but combination = it only denotes that some objects are together example : ABC can have only one combination taking all of them together. But permutations are many : - ABC,ACB,BCA,BAC,CBA,CAB
51. What is relative frequency method ? Method of assigning probability on the basis of histrorical data
54. Venn diagram Graphical representation showing sample space and operations involving events sample space = rectangle event = circle within sample space
55. What is formula of permutation ? Npr = n! / (n-r)! p=permutation n= total number of objects r=how many objects you are taking at a time ! = multiply with reducing numbers till it reaches 1 example : 5p5 = 5! / (5-5)! 5!=5*4*3*2*1 0! = 1 thus answer = 120 answer
56. How many different 4 digit letters can you make out of A,B,C,D,E? N = 5 (A,B,C,D,E) R = 4 formula = Npr = n! / (n-r)! =5!/(5-4)! = 120 answer
57. How many different 4 digit numbers can you make out of 1,2,3,4,0? N = 5 (1,2,3,4,0) R = 4 but 0 cannot come in the first digit for first digit we have 4 options (1,2,3,4), for next digits, we can use 0. thus we have 4*4*3*2*1 = 96 options OR formula = Npr = n! / (n-r)! =5!/(5-4)! but this contains all those numbers which start with 0. so let us keep 0 as fixed for 1 st digit and solve it. Now we have to pick up 3 digit out of 4 contd.
58. contd..... If it is not 0, permutation will be : formula = Npr = n! / (n-r)! =5!/(5-4)! = 120 Zero fixed for 1 st potion, we have these options : Npr = n! / (n-r)! n=4,r=3 4!/(4-3)! = 24 deduct this 24 from 120 120 -24 = 96 answer you can use any formula (out of these 2), you get the same answer
59. How many different 4 digit numbers can you make out of 1,2,3,4,0 which are divisible by 2? Start with 96 of the last question now pick up all those which are ending with 1 : 3*3*2*1 = 18 similarly those which are ending with 3 3*3*2*1 = 18 thus 96 – (18+18) = 60 seems to be the answer
60. In how many ways can Raj invite any 3 of his 7 friends? This is a question of combination. Here order (sequence) is not important, his friends can come in any order. Thus this is a case of combination. Formula : N! / ((n-r)!*r!) you can calculate combination by dividing permutation by r! =7! / ((7-3)!*3!) =(7*6*5)/(3*2*1) = 35 answer
61. How many different words can you frame from FUTURE ? Here we have two U total we have 6 digits. Formula : N ! / L! N= total number of digits L = those digits which are repeated. Answer = 6! / 2! = 360 answer
62. How many different words can you frame from DALDA ? Here we have two D & A total we have 5 digits. Formula : N ! / L! N= total number of digits L = those digits which are repeated. Answer = 5! / (2!*2!) = 30 answer
63. In how many ways can 8 person sit around a round table ? For questions relating to round table , we have to use the following formula : (n-1)! So here answer = (8-1)! = 7! =5040 answer
64. How many 4 digit numbers can be formed out of 1,2,3,5,7,8,9 if no digit is repeated. Total number ofdigits = 7 formula = Npr n =7 r 4 7p4 = 7! / 3! =7*6*5*4 = 840
65. How many numbers greater than 2000 can be formed from 1,2,3,4,5. No repeatition is allowed. 5 digit numbers = 5! = 120 4 digit numbers,: we cant take 1 in the beginning. We have 4 options for 1 st digit 4 for 2 nd digit 3 for 3 rd digit ... 4*4*3*2*1 = 96 total = 216 answer
66. There are 6 books on english, 3 on maths, 2 on GK. In how many ways can they be placed in shelf, if books of 1 subject are together? We have 3 subjects so 3! books of same subjects can be interchanged. So answer : 3!*6!*3!*2! =6*720*6*2 = 51840 answer
67. How many words can we make out of DRAUGHT, the vowels are never separated? Number of vowels = 2 other digits = 5 we will treat vowels as 1 word so we have 6!. Vowels can be interchanged so 2! so answer = 6!*2! = 1440 answer
68. In how many ways can 8 pearls be used to form a necklace ? In questions of necklace, we use the following formula : ½ (N-1)! Here we can take reverse order of left to right or right to left, so divide by ½ =1/2 (8-1)! =2520
69. In how many number of ways can 7 boys form a ring ? (7-1) ! = 6! = 720 answer
70. 50 different jewels can be set to form necklace in how many ways ? ½ ( n -1) ! = ½ (50 -1)! =1/2 (49)!
71. How many number of different digits can be formed from 0,2,3,4,8,9 between 10 to 1000? Let us assume that repeatition is not allowed Let us make 2 digit numbers : for first digit we have 5 option, for 2 nd digit also we have 5 options (including 0) = 25 for 3 digit numbers : 5*5*4 = 100 total 125 if repeatition is allowed : for 2 digit : 5 * 6 = 30 for 3 digit : 5*6*6 = 180 total = 210 answer
72. What is the number of permutations of 10 different things taking 4 at a time in which one thing never comes ? = 9 p 4 = (9*8*7*6) =3024
73. There are 5 speakers (A,B,C,D,E) , in how many ways can we arrange their speach that A always speaks before B For A and then B without gap : Let us take A and B as one. 4! = 24 for A and then B let us keep B at 3 rd place and A at 1 st place =3! there are total 6 such possibilities so we have 6*6 = 36 total possibilities = 60 answer
74. 5 persons are sitting in a round table in such a way that the tallest person always sits next to the smallest person? Keep tallest and smallest person as 1. we have (4-1)! = 6 the tallest and the smallest person can be interchanged = 2 =12
75. How many words can be formed from MOBILE so that consonent always occupies odd place ? There are 3 odd and 3 even places. We have 3! *3! =36 answer
76. In how many ways can we arrange 6 + and 4 – signs so that no two – signs are together? + + + + + + there are 5 places between 2 +. one on extreme left and one on extreme right. We have 7 positions for – sign 7c4 we have 6 places for 6 + sign, so we have 6c6 total = 35 answer
77. There are 10 buses between Bikaner and Jaipur. In how many ways can Gajendra go to Jaipur and come back without using the same bus in return journey? There are 10 options while going there are 9 options while returning (one bus used earlier will not be used) 10*9 = 90 answer
78. In how many ways can yamini distribute 8 sweets to 8 persons provided the largest sweet is served to Jigyasha? 1 sweet is fixed so we have 7! = 5040 answer
79. Yamini & Jigyasha go to a train and they find 6 vacant seats. In how many ways can they sit? Yamini has 6 options but Jigyasha has only 5 options left = 6*5 = 30 answer
81. Gajendra has 12 friends out of whom 8 are relatives. In how many ways can he invite 7 in such a way that 5 are relatives? 8c5 * 4c2 =56*6 =336 answer
82. There are 8 points on a plane. No 3 points are on a straight line. How many traiangles can be made out of these ? 8c3 = 56 answer
83. In how many ways can you form a committee of 3 persons out of 12 persons ? 12c3 =220 answer
84. How many different factors are possible from 75600 ? The factors are : 2^4* 3^3*5^2 *7 formula = (number of factors +1) (number of factors +1) .... - 1 (4+1)(3+1)(2+1)(1+1) -1 =119 answer
85. A box contains 7 red 5 white and 4 blue balls. How many selections can be made that we pick up 3 balls and all are red? It is a question of combination. Total possibilities = 7c3 7c3 = 7*6*5 / 3*2*1 = 35 thus there are 35 chances of getting
86. A box contains 7 red 5 white and 4 blue balls. What is the probability that in our selections we pick up 3 balls and all are red? Total possibilities for red = 7c3 7c3 = 7*6*5 / 3*2*1 = 35 total possibility of 3 balls : 16c3 =(16*15*14/3*2*1) =560 probability - thus there are 35/560 chances of getting red in all the three selections
87. What is the probability of getting 3 heads when I toss a coin 5 times? This is a case of binomial probability (where there are only 2 outcomes possible, we can use this theory) Here we can use this formula : Ncr (p)^r * (q)^(n-r) =n =5, p = ½ q = (1-p) = ½ , r = 3 5c3 (1/2)^3*(1/2)^2 =5/48 answer
88. In how many ways can Gajendra invite some or all of his 5 friends in party hosted by him? (at least 1) Frmula of combination of 1 to all = 2^n – 1 = 2^5 - 1 = 32-1 =31 answer
89. How many words can be formed by using all the letters of the word DRAUGHT so that a. vowels always come together & b. vowels are never together? A There are 2 vowels. We treat them as 1. solution : 6!*2! = 1440 answer b. total possibilities = 7! = 5040 number of cases when vowels are not together = 5040-1440 = 3600 answer
90. In how many ways can a cricket eleven be chosen out of a batch of 15 players. 15c11 =15! / ((15-11)!*11!) =15!/(4!*11!) =(15*14*13*12)/(4*3*2*1) 1365 answer
91. In how many a committee of 5 members can be selected from 6 men 5 ladies consisting of 3 men and 2 ladies 6c3 *5c2 =[(6*5*4)/(3*2*1)] [(5*4)/(2*1)] =20*10 =200 answer
92. How many 4-letter word with or without meaning can be formed out of the letters of the word 'LOGARITHMS' if repetition of letters is not allowed 10p4 =(10*9*8*7) =5040 answer
93. how many ways can the letter of word 'LEADER' be arranged We have two e, so divide 6p6 by 2 6!/2! =720 / 2 =360 answer
94. How many arrangements can be made out of the letters of the word 'MATHEMATICS' be arranged so that the vowels always come together Let us treat all 4 vowels as 1 total digits are 11 we we take 11 – 4+1 = 8 digits vowels can be arranged among themselves = 4!/2! =8!/ (2!*2!) * 4!/2! = 120960 answer
95. In how many different ways can the letter of the word 'DETAIL' be arranged in such a way that the vowels occupy only the odd positions We have 3 odd and 3 even positions =3! *3! =36 answer
96. How many 3 digit numbers can be formed from the digits 2,3,5,6,7 and 9 which are divisible by 5 and none of the digits is repeated? Last digit must be 5 now we have 5 options for 1 st and 4 options for 2 nd digit =5*4 = 20 answer
97. In how many ways can 21 books on English and 19 books on Hindi be placed in a row on a self so that two books on Hindi may not be together? We have 22 places for Hindi books. 22p19 *21!
98. Out of 7 constants and 4 vowels how many words of 3 consonants and 2 vowels can be formed? Selection of 5 digits =7c3 *4c2 =35*6 = 210 5 digits can be arranged in 5! ways =120 total options : 210*120 = 25200 answer
99. What is effective rate of interest ? In the case of compound interest questions, the effective rate is generally higher than the rate. For example: if rate is 20% compounded quarterly , (4 times in a year) it will be equal to : (1+20/400)^4 =1.2155 so effectiveinterest here is 21.55% answer
100. What is present value ? When you are trying to find the present worth of some money which is due after some time, it is called present value. Due to factors like inflation, risk, uncertainity, present value is always less. Suppose you have to get 1100 after 1 year, at a discount rate of 10% its present value is 1000. (you can see here that there is a discount of 100) Money due – discount for time factor = present value
101. What is future value ? Future value takes up interest and therefore it is more than the sum invested. If I invest 1000 today, with an interest rate of 10%, it will become 1100 after 1 year.
102. Formula for present value ? Amount / (1+rate) ^ number of years suppose 1221 is due after 3 years and rate of interest is 10%, present value is : 1221 / (1+10/100)^3 =917.35 answer
103. What is the formula for future value ? Amount *(1+rate) ^ number of years suppose 1000 is invested for 3 years and rate of interest is 10% annually compounding, future value is : 1000 * (1+10/100)^3 =1331 answer
104. How to calculate EMI? You may use the formula for present value of annuity. Here you need a factor formula = ((1+rate)^n -1) / (rate(1+rate)^n) here n= number of instalments rate = rate % / number of instalments in a year*100 EMI = amout to pay / factor of annuity(calculated from above formula)
105. What will be EMI for Rs. 5 lakh rate of interest = 10%, payable in 20 annual instalments = ((1+rate)^n -1) / (rate(1+rate)^n) ((1+10/100)^20 - 1)/(10/100 (1+10/100)^20) =5.73/.67 =8.55 EMI=500000/8.55 =58479 ANSWER
106. What will be EMI for Rs. 5 lakh rate of interest = 10%, payable in Monthly instalments in 20 years. = ((1+rate)^n -1) / (rate(1+rate)^n) ((1+10/1200)^240 - 1)/(10/1200* (1+10/1200)^240) 6.328 / .061 =103.624 EMI = 500000 / 103.624 =4825 ANSWER
107. What is sinking fund ? If you deposit a sum of money every year and you are able to have a lot of money after some time this is sinking fund you create sinking fund to purchase a new machinary / building etc it is just reverse of the EMI (where you were looking at present value of annuities), because here you are taking future value of annuities.
108. How to calculate sinking fund contribution? For calculation of sinking fund contribution, we have to use the following formula : = ((1+rate)^n -1 )/(rate) here n = number of instalment rate = rate / number of instalments in a year*100.
109. Jigyasa has to collect 1 ml. After 5 years to start a new factory. How should she save every month? Rate = 12% = ((1+rate)^n -1 )/(rate) =((1+12/1200)^60 -1) / (12/1200) =.8167 / .01 dividing factor =81.669 monthly savings = 1000000/81.669 =12244.44 per month answer
110. What is a sample ? Instead of contacting every person, we may contact only a few persons, this is called sample. Suppose we go to check the quality of wheat to purchase. Instead of checking all the bags, we pick up one bag randomly and pick out a few grains, this is also a sample.
111. What are the methods of sampling ? 1. random sampling = purely by chance – just like a lottery 2. judgement sampling – here we are using some basis for judgement – the basis of judgement is related to our purpose of research. 3. quota sampling – taking some number of persons from each group 4. cluster sampling – here we divide populationin clusters (based on their geography / demography / location / etc.) and then pick up a few clusters (groups) of people and study them all
112. contd... Stratified ramdom sampling : here we divide population in different stratas (strata = population divided on some logical criteria) then we randomly take a few % of persons from each strata. Convenience sampling = taking sample on the basis of your convenience
113. What is confidence level ? It is the confidence created / associated with an interval estimate If we are using a confidence level of 95%, it means that there are 95% chances that our estimate will be close to population parameter (mean).
114. What is the difference between population parameters and sample statistics ? Population = actual population – but it is not possible to collect all the information about population due to our own resource constraints we dont have time or resources to collect data about population. Therefore we go for sample. When we use sample, we are using sample statistics. We try to estimate population parameters from sample statistics.
115. What is population parameter? If you go for census study (you contact each element in the population and take their data), you can calculate population parameter. There are different parameters which are of use like : mean, mode, median, standard deviation, etc. But we actually take sample so we estimate population parameters from sample statistics.
116. What is sample statistics? Sample characteristics like mean, mode, median, standard deviation etc. Which are used to estimate population parameter
117. What is sampling error? The difference in the value identified by sample and the population parameter is called sampling error. For example, population mean is 20 but sample mean is 18, so sampling error = 2
118. What is quantitative data and qualitative data ? quantitative data = data which tell about what and how much qualitative data=data which only contain nominal scale – just name / labels etc.
119. What are the various types of scales of data ? 1 nominal scale = only names are there – like ram, shyam 2. ordinal scale - they give order or ranks 3.interval scale: they have identifiable gaps, but they dont have zero 4. ratio scale – they can be used to calculate ratio – they have a zero and ratio can also be calculated, they are the best in numerical analysis
120. What are the various methods to present data ? Scatter chart / diagrams bar chart Histogram Ogive Dot plot etc.
121. What is statistical inference ? When we try to estimate or test hypothesis using sample data, it is called statistical inference (here we use sample data, not the population parameters).
122. What is a variable ? It is a characteristic of some interest relating to some element. It can take different values. Variables are denoted by X,Y,Z etc. Examples of variables are : for people = their education, for car=their car, fuel efficiency etc.
123. What is cross sectional data ? Data collected at the same point of time from different segments
124. What is cross tabulation? There are two variables, their data are presented in one table – one variable as X axis and other variable as Y axis for example : Age and Height or Marks and Attendance
125. Can we take up same element again in sampling ? Yes, it is possible (by chance) there are two types of sampling : 1. sampling with replacement 2. sampling without replacement in sampling with replacement, it is possible that by chance we may pick up same element again (we should avoid).
126. What is normal distribution ? There are many types of probabilty distributions, normal distribution is used most widely. It assumes that the data are bell shaped and mean=mode=median. Normal distribution assumes that most of the data are near mean and extreme data are very few.
127. How do you calculate mode ? Mode is that element, which has highest frequency if there is continuous data,you may use the following formula : Mode = L1 + (D1 / (D1+D2) * class interval) L1 = lower limit of the modal class D1=higest frequency – frequency in preceding class D2=higest frequency – frequency in succeeding class
128. Example of mode : 2,3,5,6,7,8,9,11,13,13,14,14,14,15,17,21,22,34,43 out of these mode is 14 (because its frequency is 3)
129. Example of mode ? Class frequency 10 to 20 4 20 to 30 8 30 to 40 12 40 to 50 4 apply the formula : modal class = 30 to 40 = 30 + ((12-8) / ((12-8)+(12-4)) * class interval = 30 + 4/12 * 10 = 30+3.3 = 33.3 answer
130. What is median ? Median = exact mid point in the data formula = n/2 or (n+1) / 2 example : 1,3,5,7,9 thre are 5 values, so n = 5 (5+1)/2 = 3 so 3 rd value is median. Median = 5 answer
131. Formula for median ? L1 + ((M-C) / F)* class interval L1 = lower limit F = frequency M=median = n/2 C = cumulative frequency of the previous class
132. Example of median ? Class frequency C.F 10 to 20 4 4 20 to 30 8 12 30 to 40 12 24 40 to 50 4 28 L1 + ((M-C) / F)* class interval M=28/2 = 14, so median class is 30 to 40 30 + (( 14-12)/12) * 10 =30+1.6 = 31.6 answer
133. What is cumulative frequency ? When you add up frequencies, it is called cumulative frequencies in the previous example , 10to 20 is 4, but 20 to 30 is shown as 16 (4 of 10 to 20 is added in it) cumulative frequency
141. What is formula of coefficient of mean deviation ? Mean deviation / Median or mean deviation / mean
142. calculate combined standard deviation. Means A=8 B = 3, std. Deviation A = 2 B = 1 n1 of a = 20 n2 =30 Formula = sqrt ((n1s1 +n2s2 +n1d1+n2d2)/(n1+n2)) d1 = mean of a – combined mean d2 = mean of b -combined mean combined mean = (160+90)/50 = 5 d1=3 d2 =-2 sqrt ((20*2 +30*1 +20*3+30*(-2))/(20+30)) =1.18 answer
143. FORMULA OF RANK CORRELATION = 1- (6 ∑ D^2) / (N^3 -N) = 1 – (6*20)/(64 -4) =1 - 120/60 =1-2 =-1 Thus two series have perfectly negative correlation
144. What is sample space? A set of all experimental outcomes is called sample space
145. What is experiment ? In research, we manipulate some data, we change some variables that is called experiment,
146. What is experimental group? There are generally two types of groups – one on which you undertake experiment (experimental group) and one on which you dont do any experiment, just do observation.(control group) Example – if you have two plants, on one plant you pour fertilisers and on the other you dont put any fertilizer, then the former is experimental group and 2 nd is control group.
147. What is standard deviation? Deviation = difference here we find the difference of each value with mean and this will create standard deviation. Formula = square root of (sum of squares of difference of each element from mean)
148. Example : of standard deviation.. X dx^2 2 4 3 1 5 1 6 4 average = 16/4 = 4, dx = x-average = 2-4 = -2 average of dx^2 = variance = 10 / 4 = 2.5 standard deviation = square root of variance = sqrt(2.5) =1.58 answer
149. Steps in calculation of standard deviation ? 1. calculate average. For this total all the values of X and then divide it by n (in our example, we have divided 16/4, where 16 is total of all values and 4 is number of elements. 2. find dx (difference of x from mean) 3. square the dx to get dx^2 4 . find average of dx^2 this is called variance. 5. find square root of variance. This is called standard deviation.