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# Stat 101 formulae sheet

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Introduction to Statistics
Formulae sheet

Course Instructor--Iftekhar Mohammad Shafiqul Kalam (ISK)

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### Stat 101 formulae sheet

1. 1. Stat 101 Formulae Sheet MMM, Histogram, Central Tendency Name Sl.no. 1. Relative Frequency 3. l m Midpoint 2. Formula Angle of Pie s 2 RF= proportion= 0 f * 360 0 N U 4. Mean ( ) G n Me L0 F 2 Me * W Me f Me Me 5. Median (Me) Median L0 Lower Limit of the median class W Me Width of the median class CF of the pre median class n Total number G F F of the median class U 6. 7. Me f Me Mode (M0) 11304042 Classic way- count the middle value U The number with the most frequency Page 1
2. 2. 9. 10. Mode Lower Limit of Modal class f0 G Mo L0 8. F of the modal class f F of the pre modal class 1 F of post modal class f1 Width of the modal class W U Geometric Mean (GM) 1 G f GM f f x 1 1 . x 2 2 ... x k k n 1 ... Weighted Mean ( 14. fi HM Coefficient of Range (Co. R) 17. Quartile Deviation (QD) f1 f2 xi n n fk x2 ... xk W1 X 1 WX Xw Range (R) 16. 1 W2 X 2 fi i 1 ... xi Wn X n Quartile (Q) 15. i 1 i 1 x1 13. f i log x i i 1 k G 12. 1 x2 Harmonic Mean (HM) k n n xn 1 x1 11. 1 anti log n HM U GM n 11304042 ) W W1 R W2 A (Q 2 Q1 ) Xl X s Xl QD X s (Q 3 2 Wn Xs. Xl Coeff .. R ... Q2 ) R Q3 Q1 A 2 Page 2
3. 3. 18. 19. 20. 21. Coeff. Coefficient of QD U Mean Deviation (MD) Where A can be mean/ median/mode 1 MD Population i 1 i 2 i 1 N R 2 Sample 1 i k 1 i i 1 23. 1 N 1 2 2 k 1 s n 1 2 n f 1 x1 k 1 Ungroup data 22. i 2 fi xi N fi x i Sample n 2 xi 2 f 1 x1 xi N 1 2 fi Di N k i N s A = A 1 Population x Di N Group data x N 1 2 N 1 R Coeff. MD = Ungroup data N A = fi X i N Coefficient of MD 2 Q1 Xi N G Q1 Q3 1 MD Q3 QD 2 i i 1 1 n Group data Standard Deviation SD X [SD(X)] Coefficient of VAR X CV Variation (CV) VAR(X) = = SD ( X ) 2 / s2 A . R AM ( X ) Mean Skewness 11304042 Mode The distribution is positively skewed Mean Median Mode The distribution is negatively skewed Mean 24. Median Median Mode The distribution is symmetric Page 3
4. 4. 25. Pearson’s coefficient Sk of skewness Mean p 3 ( Mean Mode SD Median ) SD relatively higher peak is known as leptokurtic.  neither too peaked nor too flat topped is known as mesokurtic.  26.  Kurtosis more flat topped than the normal curve is called platykurtic. Key: U= Ungrouped Data G= Grouped Data A= Absolute measure of Dispersion R= Relative measure of Dispersion L2- Correlation and Regression Name Sl.no. Formula SP X , Y r xy SP XY 1. Correlation Coefficient ( r xy ) X i SS Y X Yi Yi Y rxy X Y 2. Regression Model 11304042 Y X Xi n V , X ,Y X V Y SS X 2 Y Xi 1 Xi X iYi 2 i Cov r xy SS X SS Y rxy X 2 1 Yi n 2 Yi Yi 2 2 n ˆ Yi ˆ ˆX X Independent variable Dependent variable Intercept Error term Y Regression coefficient of Y on X gradient (m) Page 4
5. 5. 3. 4. least square method & are the parameters of the model. Estimating the regression ˆ n X iYi n coefficient 2 Xi Xi Yi X iYi ˆ 2 Xi X 2 i nXY nX 2 Estimating the intercept ˆ : ˆX Y L3- Probability Name Sl.no. 3. 11304042 rule 2. Empirical or Frequency Probability ion Classic Probability Addit 1. Formula General Case P A m ; n P A m lim m n No. of times event A occurs. n P AorB P ( A) P(B) ; n Total no. of trials. P ( AandB ) Page 5
6. 6. ( NOT mutually exclusive) P A B P ( A) Special case ( mutually exclusive) 5. 6. Multiplication 4. P AorB P A P ( AB ) Independent events Events that are not independent P(B) P ( A) B P(A P(B) P ( A) B) P ( AB ) P(B) P ( A) P ( B ) conditional probability of A given B P ( AB ) P(A B) ; P(B) 0 P(B) conditional probability of B given A P ( AB ) P ( B A) ; P ( A) 0 P ( A) L4- Sampling Name 1. 2. 3. 4. 5. Perc Simple Random Sampling enta ge Standard stan error of dard erro r of Sl.no. 11304042 estimate of Population Mean Est Y estimate of population total Est the estimate of Population Mean SE ( y ) the estimate of Population Total Est SE ( N y ) Estimate of Population mean SE ( y ) % Formula n Y ˆ Y yi i 1 y n NY SE ( y ) ˆ NY Ny N n s 2 Nn SE ( N y ) SE ( y ) N * SE ( y ) SE ( y ) * 100 y Page 6
7. 7. 6. * 100 Ny s y n 1 2 n Sample Variance s2 7. SE ( N y ) SE ( N y ) Estimate of Population total % 2 yi 1 ny 2 i 1 Sample Mean, N = Population size 12. 13. 14. 15. 16. 17. 18. 19. 11304042 Standard error of Percentage standard error of 11. ˆ Y Y estimate of population Total Est the estimate of Population Mean SE ( y st ) the estimate of Population Total Est SE ( N y st ) Estimate of Population mean SE ( y st ) % Estimate of Population total % n NY N y st 2 the estimate of Population Mean SE ( y sys ) ni N the estimate of Population Total Est SE ( N y sys ) Estimate of Population mean SE ( y sys ) % Estimate of Population total % SE ( N y sys ) 2 W i si N * SE ( y st ) SE ( y st ) SE ( y st ) * 100 y st SE ( N y st ) SE ( N y st ) * 100 N y st ˆ Y 1 y sys NY SE ( y sys ) 2 1 SE ( N y st ) estimate of population Total Est Wi yi i 1 W i si SE ( y st ) Y k 1 y st SE ( N y st ) estimate of population Mean Est Standard error of 10. estimate of population Mean Est Percentage standard error of 9. Systematic Random Sampling 8. Stratified Random Sampling and n = Sample size N y sys 1 m (m SE ( N y sys ) SE ( y sys ) SE ( N y sys ) yj m 1) yj y sys N * SE ( y sys ) SE ( y sys ) * 100 y sys SE ( N y sys ) * 100 N y sys Page 7 2
8. 8. L5- Quality and Quality Control Name Sl.no. Formula x x x 1. Grand mean n k k n = No. of observation in each sample k = No. of samples taken x x 2. 3. 4. 5. Upper Control Limit (UCL) UCL d2 Sum of the sample means x 3R x d2 Lower Control Limit (LCL) Average of the sample ranges R central line of the control chart Sum of all observation and n Control chart factor from quality control chart LCL 3R x d2 n R R k C= the no. of defects counted in one unit of item C = mean of defects counted in several (usually 25 or more) such units he central line of the control chart for C is the C and the 3- sigma control limits are C 3 C Table: Quality Control Chart 11304042 Page 8
9. 9. Interpolation & Extrapolation yx 6. y0 u y0 u u 1 2 2! Newton’s Forward interpolation formulae y x u u y0 2 3 3! x u 1 u y0 ... ... ... x0 h x= the value of x for which the value of y is to be determined h=common intervals between x values 7. Newton’s Back ward interpolation formulae 11304042 yx yn u 1 yn u u 1 2 2! u u u yn 1 u 3! x 2 3 yn ... xn h Page 9 ... ...
10. 10. L5- Index Number Name Sl.no. 1. Un-weighted Index Numbers (Simple Aggregative Method) Formula 2. Total of base year prices for various commodities P 1 * 100 P 0 N P 01 Where N refers to no. of items log[ P 1 * 100 ] P 0 log P 01 log P or N Where 3. * 100 P0 Total of current year prices for various items P1 P0 Un-weighted Index Numbers (Simple Average Relative of Method) P1 P01 P01 Laspeyres Method N P1 P * 100 P2 PQ 1 0 * 100 P Q 0 0 Weight is Base year quantity 4. P 01 Paasche Method PQ 1 1 * 100 P Q 0 1 Weight is Current year quantity 5. 6. Dorbish and Bowley’s Method P1 Q 0 [Where L Fisher’s ‘Ideal’ Method 8. Marshall – Edgeworth Method Kelly’s Method P 01 P P0 Q 1 2 P1Q 0 L*P Paasche Index] P1Q1 * P0 Q 0 (Q 0 Q1 ) P1 (Q 0 Q1 ) P0 Paasche Index] P1Q 0 P1Q P1Q1 P0 Q 0 * 100 P 01 * 100 P0 Q1 Laspeyres Index and P [Where Q 11304042 * 100 2 Lasperes Index and P P 01 [Where L 7. L P 01 P1 Q 1 P0 Q 0 P0 Q1 * 100 * 100 P0 Q Q0 Q1 2 ] Page 10
11. 11. L5- Time Series Analysis & Forecasting Sl.no. 11304042 Name Formula Page 11