Category: Big Data & Analytics

Statistics – Ti 83 Exponential Regression ”; Previous Next Ti 83 Exponential Regression is used to compute an equation which best fits the co-relation between sets of indisciriminate variables. Formula ${ y = a times b^x}$ Where − ${a, b}$ = coefficients for the exponential. Example Problem Statement: Calculate Exponential Regression Equation(y) for the following data points. Time (min), Ti 0 5 10 15 Temperature (°F), Te 140 129 119 112 Solution: Let consider a and b as coefficients for the exponential Regression. Step 1 ${ b = e^{ frac{n times sum Ti log(Te) – sum (Ti) times sum log(Te) } {n times sum (Ti)^2 – times (Ti) times sum (Ti) }} } $ Where − ${n}$ = total number of items. ${ sum Ti log(Te) = 0 times log(140) + 5 times log(129) + 10 times log(119) + 15 times log(112) = 62.0466 \[7pt] sum log(L2) = log(140) + log(129) + log(119) + log(112) = 8.3814 \[7pt] sum Ti = (0 + 5 + 10 + 15) = 30 \[7pt] sum Ti^2 = (0^2 + 5^2 + 10^2 + 15^2) = 350 \[7pt] implies b = e^{frac {4 times 62.0466 – 30 times 8.3814} {4 times 350 – 30 times 30}} \[7pt] = e^{-0.0065112} \[7pt] = 0.9935 } $ Step 2 ${ a = e^{ frac{sum log(Te) – sum (Ti) times log(b)}{n} } \[7pt] = e^{frac{8.3814 – 30 times log(0.9935)}{4}} \[7pt] = e^2.116590964 \[7pt] = 8.3028 } $ Step 3 Putting the value of a and b in Exponential Regression Equation(y), we get. ${ y = a times b^x \[7pt] = 8.3028 times 0.9935^x } $ Print Page Previous Next Advertisements ”;

Statistics – Formulas ”; Previous Next Following is the list of statistics formulas used in the Tutorialspoint statistics tutorials. Each formula is linked to a web page that describe how to use the formula. A Adjusted R-Squared – $ {R_{adj}^2 = 1 – [frac{(1-R^2)(n-1)}{n-k-1}]} $ Arithmetic Mean – $ bar{x} = frac{_{sum {x}}}{N} $ Arithmetic Median – Median = Value of $ frac{N+1}{2})^{th} item $ Arithmetic Range – $ {Coefficient of Range = frac{L-S}{L+S}} $ B Best Point Estimation – $ {MLE = frac{S}{T}} $ Binomial Distribution – $ {P(X-x)} = ^{n}{C_x}{Q^{n-x}}.{p^x} $ C Chebyshev”s Theorem – $ {1-frac{1}{k^2}} $ Circular Permutation – $ {P_n = (n-1)!} $ Cohen”s kappa coefficient – $ {k = frac{p_0 – p_e}{1-p_e} = 1 – frac{1-p_o}{1-p_e}} $ Combination – $ {C(n,r) = frac{n!}{r!(n-r)!}} $ Combination with replacement – $ {^nC_r = frac{(n+r-1)!}{r!(n-1)!} } $ Continuous Uniform Distribution – f(x) = $ begin{cases} 1/(b-a), & text{when $ a le x le b $} \ 0, & text{when $x lt a$ or $x gt b$} end{cases} $ Coefficient of Variation – $ {CV = frac{sigma}{X} times 100 } $ Correlation Co-efficient – $ {r = frac{N sum xy – (sum x)(sum y)}{sqrt{[Nsum x^2 – (sum x)^2][Nsum y^2 – (sum y)^2]}} } $ Cumulative Poisson Distribution – $ {F(x,lambda) = sum_{k=0}^x frac{e^{- lambda} lambda ^x}{k!}} $ D Deciles Statistics – $ {D_i = l + frac{h}{f}(frac{iN}{10} – c); i = 1,2,3…,9} $ Deciles Statistics – $ {D_i = l + frac{h}{f}(frac{iN}{10} – c); i = 1,2,3…,9} $ F Factorial – $ {n! = 1 times 2 times 3 … times n} $ G Geometric Mean – $ G.M. = sqrt[n]{x_1x_2x_3…x_n} $ Geometric Probability Distribution – $ {P(X=x) = p times q^{x-1} } $ Grand Mean – $ {X_{GM} = frac{sum x}{N}} $ H Harmonic Mean – $ H.M. = frac{W}{sum (frac{W}{X})} $ Harmonic Mean – $ H.M. = frac{W}{sum (frac{W}{X})} $ Hypergeometric Distribution – $ {h(x;N,n,K) = frac{[C(k,x)][C(N-k,n-x)]}{C(N,n)}} $ I Interval Estimation – $ {mu = bar x pm Z_{frac{alpha}{2}}frac{sigma}{sqrt n}} $ L Logistic Regression – $ {pi(x) = frac{e^{alpha + beta x}}{1 + e^{alpha + beta x}}} $ M Mean Deviation – $ {MD} =frac{1}{N} sum{|X-A|} = frac{sum{|D|}}{N} $ Mean Difference – $ {Mean Difference= frac{sum x_1}{n} – frac{sum x_2}{n}} $ Multinomial Distribution – $ {P_r = frac{n!}{(n_1!)(n_2!)…(n_x!)} {P_1}^{n_1}{P_2}^{n_2}…{P_x}^{n_x}} $ N Negative Binomial Distribution – $ {f(x) = P(X=x) = (x-1r-1)(1-p)x-rpr} $ Normal Distribution – $ {y = frac{1}{sqrt {2 pi}}e^{frac{-(x – mu)^2}{2 sigma}} } $ O One Proportion Z Test – $ { z = frac {hat p -p_o}{sqrt{frac{p_o(1-p_o)}{n}}} } $ P Permutation – $ { {^nP_r = frac{n!}{(n-r)!} } $ Permutation with Replacement – $ {^nP_r = n^r } $ Poisson Distribution – $ {P(X-x)} = {e^{-m}}.frac{m^x}{x!} $ probability – $ {P(A) = frac{Number of favourable cases}{Total number of equally likely cases} = frac{m}{n}} $ Probability Additive Theorem – $ {P(A or B) = P(A) + P(B) \[7pt] P (A cup B) = P(A) + P(B)} $ Probability Multiplicative Theorem – $ {P(A and B) = P(A) times P(B) \[7pt] P (AB) = P(A) times P(B)} $ Probability Bayes Theorem – $ {P(A_i/B) = frac{P(A_i) times P (B/A_i)}{sum_{i=1}^k P(A_i) times P (B/A_i)}} $ Probability Density Function – $ {P(a le X le b) = int_a^b f(x) d_x} $ R Reliability Coefficient – $ {Reliability Coefficient, RC = (frac{N}{(N-1)}) times (frac{(Total Variance – Sum of Variance)}{Total Variance})} $ Residual Sum of Squares – $ {RSS = sum_{i=0}^n(epsilon_i)^2 = sum_{i=0}^n(y_i – (alpha + beta x_i))^2} $ S Shannon Wiener Diversity Index – $ { H = sum[(p_i) times ln(p_i)] } $ Standard Deviation – $ sigma = sqrt{frac{sum_{i=1}^n{(x-bar x)^2}}{N-1}} $ Standard Error ( SE ) – $ SE_bar{x} = frac{s}{sqrt{n}} $ Sum of Square – $ {Sum of Squares = sum(x_i – bar x)^2 } $ T Trimmed Mean – $ mu = frac{sum {X_i}}{n} $ Print Page Previous Next Advertisements ”;

Statistics – T-Distribution Table ”; Previous Next The critical values of t distribution are calculated according to the probabilities of two alpha values and the degrees of freedom. The Alpha (a) values 0.05 one tailed and 0.1 two tailed are the two columns to be compared with the degrees of freedom in the row of the table. One Tail 0.05 0.025 0.01 0.005 0.0025 0.001 0.0005 Two Tails 0.1 0.05 0.02 0.01 0.005 0.002 0.001 df   1 6.3138 12.7065 31.8193 63.6551 127.3447 318.4930 636.0450 2 2.9200 4.3026 6.9646 9.9247 14.0887 22.3276 31.5989 3 2.3534 3.1824 4.5407 5.8408 7.4534 10.2145 12.9242 4 2.1319 2.7764 3.7470 4.6041 5.5976 7.1732 8.6103 5 2.0150 2.5706 3.3650 4.0322 4.7734 5.8934 6.8688 6 1.9432 2.4469 3.1426 3.7074 4.3168 5.2076 5.9589 7 1.8946 2.3646 2.9980 3.4995 4.0294 4.7852 5.4079 8 1.8595 2.3060 2.8965 3.3554 3.8325 4.5008 5.0414 9 1.8331 2.2621 2.8214 3.2498 3.6896 4.2969 4.7809 10 1.8124 2.2282 2.7638 3.1693 3.5814 4.1437 4.5869 11 1.7959 2.2010 2.7181 3.1058 3.4966 4.0247 4.4369 12 1.7823 2.1788 2.6810 3.0545 3.4284 3.9296 4.3178 13 1.7709 2.1604 2.6503 3.0123 3.3725 3.8520 4.2208 14 1.7613 2.1448 2.6245 2.9768 3.3257 3.7874 4.1404 15 1.7530 2.1314 2.6025 2.9467 3.2860 3.7328 4.0728 16 1.7459 2.1199 2.5835 2.9208 3.2520 3.6861 4.0150 17 1.7396 2.1098 2.5669 2.8983 3.2224 3.6458 3.9651 18 1.7341 2.1009 2.5524 2.8784 3.1966 3.6105 3.9216 19 1.7291 2.0930 2.5395 2.8609 3.1737 3.5794 3.8834 20 1.7247 2.0860 2.5280 2.8454 3.1534 3.5518 3.8495 21 1.7207 2.0796 2.5176 2.8314 3.1352 3.5272 3.8193 22 1.7172 2.0739 2.5083 2.8188 3.1188 3.5050 3.7921 23 1.7139 2.0686 2.4998 2.8073 3.1040 3.4850 3.7676 24 1.7109 2.0639 2.4922 2.7970 3.0905 3.4668 3.7454 25 1.7081 2.0596 2.4851 2.7874 3.0782 3.4502 3.7251 26 1.7056 2.0555 2.4786 2.7787 3.0669 3.4350 3.7067 27 1.7033 2.0518 2.4727 2.7707 3.0565 3.4211 3.6896 28 1.7011 2.0484 2.4671 2.7633 3.0469 3.4082 3.6739 29 1.6991 2.0452 2.4620 2.7564 3.0380 3.3962 3.6594 30 1.6973 2.0423 2.4572 2.7500 3.0298 3.3852 3.6459 31 1.6955 2.0395 2.4528 2.7440 3.0221 3.3749 3.6334 32 1.6939 2.0369 2.4487 2.7385 3.0150 3.3653 3.6218 33 1.6924 2.0345 2.4448 2.7333 3.0082 3.3563 3.6109 34 1.6909 2.0322 2.4411 2.7284 3.0019 3.3479 3.6008 35 1.6896 2.0301 2.4377 2.7238 2.9961 3.3400 3.5912 36 1.6883 2.0281 2.4345 2.7195 2.9905 3.3326 3.5822 37 1.6871 2.0262 2.4315 2.7154 2.9853 3.3256 3.5737 38 1.6859 2.0244 2.4286 2.7115 2.9803 3.3190 3.5657 39 1.6849 2.0227 2.4258 2.7079 2.9756 3.3128 3.5581 40 1.6839 2.0211 2.4233 2.7045 2.9712 3.3069 3.5510 41 1.6829 2.0196 2.4208 2.7012 2.9670 3.3013 3.5442 42 1.6820 2.0181 2.4185 2.6981 2.9630 3.2959 3.5378 43 1.6811 2.0167 2.4162 2.6951 2.9591 3.2909 3.5316 44 1.6802 2.0154 2.4142 2.6923 2.9555 3.2861 3.5258 45 1.6794 2.0141 2.4121 2.6896 2.9521 3.2815 3.5202 46 1.6787 2.0129 2.4102 2.6870 2.9488 3.2771 3.5149 47 1.6779 2.0117 2.4083 2.6846 2.9456 3.2729 3.5099 48 1.6772 2.0106 2.4066 2.6822 2.9426 3.2689 3.5051 49 1.6766 2.0096 2.4049 2.6800 2.9397 3.2651 3.5004 50 1.6759 2.0086 2.4033 2.6778 2.9370 3.2614 3.4960 51 1.6753 2.0076 2.4017 2.6757 2.9343 3.2579 3.4917 52 1.6747 2.0066 2.4002 2.6737 2.9318 3.2545 3.4877 53 1.6741 2.0057 2.3988 2.6718 2.9293 3.2513 3.4838 54 1.6736 2.0049 2.3974 2.6700 2.9270 3.2482 3.4800 55 1.6730 2.0041 2.3961 2.6682 2.9247 3.2451 3.4764 56 1.6725 2.0032 2.3948 2.6665 2.9225 3.2423 3.4730 57 1.6720 2.0025 2.3936 2.6649 2.9204 3.2394 3.4696 58 1.6715 2.0017 2.3924 2.6633 2.9184 3.2368 3.4663 59 1.6711 2.0010 2.3912 2.6618 2.9164 3.2342 3.4632 60 1.6706 2.0003 2.3901 2.6603 2.9146 3.2317 3.4602 61 1.6702 1.9996 2.3890 2.6589 2.9127 3.2293 3.4573 62 1.6698 1.9990 2.3880 2.6575 2.9110 3.2269 3.4545 63 1.6694 1.9983 2.3870 2.6561 2.9092 3.2247 3.4518 64 1.6690 1.9977 2.3860 2.6549 2.9076 3.2225 3.4491 65 1.6686 1.9971 2.3851 2.6536 2.9060 3.2204 3.4466 66 1.6683 1.9966 2.3842 2.6524 2.9045 3.2184 3.4441 67 1.6679 1.9960 2.3833 2.6512 2.9030 3.2164 3.4417 68 1.6676 1.9955 2.3824 2.6501 2.9015 3.2144 3.4395 69 1.6673 1.9950 2.3816 2.6490 2.9001 3.2126 3.4372 70 1.6669 1.9944 2.3808 2.6479 2.8987 3.2108 3.4350 71 1.6666 1.9939 2.3800 2.6468 2.8974 3.2090 3.4329 72 1.6663 1.9935 2.3793 2.6459 2.8961 3.2073 3.4308 73 1.6660 1.9930 2.3785 2.6449 2.8948 3.2056 3.4288 74 1.6657 1.9925 2.3778 2.6439 2.8936 3.2040 3.4269 75 1.6654 1.9921 2.3771 2.6430 2.8925 3.2025 3.4250 76 1.6652 1.9917 2.3764 2.6421 2.8913 3.2010 3.4232 77 1.6649 1.9913 2.3758 2.6412 2.8902 3.1995 3.4214 78 1.6646 1.9909 2.3751 2.6404 2.8891 3.1980 3.4197 79 1.6644 1.9904 2.3745 2.6395 2.8880 3.1966 3.4180 80 1.6641 1.9901 2.3739 2.6387 2.8870 3.1953 3.4164 81 1.6639 1.9897 2.3733 2.6379 2.8859 3.1939 3.4147 82 1.6636 1.9893 2.3727 2.6371 2.8850 3.1926 3.4132 83 1.6634 1.9889 2.3721 2.6364 2.8840 3.1913 3.4117 84 1.6632 1.9886 2.3716 2.6356 2.8831 3.1901 3.4101 85 1.6630 1.9883 2.3710 2.6349 2.8821 3.1889 3.4087 86 1.6628 1.9879 2.3705 2.6342 2.8813 3.1877 3.4073 87 1.6626 1.9876 2.3700 2.6335 2.8804 3.1866 3.4059 88 1.6623 1.9873 2.3695 2.6328 2.8795 3.1854 3.4046 89 1.6622 1.9870 2.3690 2.6322 2.8787 3.1844 3.4032 90 1.6620 1.9867 2.3685 2.6316 2.8779 3.1833 3.4020 91 1.6618 1.9864 2.3680 2.6309 2.8771 3.1822 3.4006 92 1.6616 1.9861 2.3676 2.6303 2.8763 3.1812 3.3995 93 1.6614 1.9858 2.3671 2.6297 2.8755 3.1802 3.3982 94 1.6612 1.9855 2.3667 2.6292 2.8748 3.1792 3.3970 95 1.6610 1.9852 2.3662 2.6286 2.8741 3.1782 3.3959 96 1.6609 1.9850 2.3658 2.6280 2.8734 3.1773 3.3947 97 1.6607 1.9847 2.3654 2.6275 2.8727 3.1764 3.3936 98 1.6606 1.9845 2.3650 2.6269 2.8720 3.1755 3.3926 99 1.6604 1.9842 2.3646 2.6264 2.8713 3.1746 3.3915 100 1.6602 1.9840 2.3642 2.6259 2.8706 3.1738 3.3905 101 1.6601 1.9837 2.3638 2.6254 2.8700 3.1729 3.3894 102 1.6599 1.9835 2.3635 2.6249 2.8694 3.1720 3.3885 103 1.6598 1.9833 2.3631 2.6244 2.8687 3.1712 3.3875 104 1.6596 1.9830 2.3627 2.6240 2.8682 3.1704 3.3866 105 1.6595 1.9828 2.3624 2.6235 2.8675 3.1697 3.3856 106 1.6593 1.9826 2.3620 2.6230 2.8670 3.1689 3.3847 107 1.6592 1.9824 2.3617 2.6225 2.8664 3.1681 3.3838 108 1.6591 1.9822 2.3614 2.6221 2.8658 3.1674 3.3829 109 1.6589 1.9820 2.3611 2.6217 2.8653 3.1667 3.3820 110 1.6588 1.9818 2.3607 2.6212 2.8647 3.1660 3.3812 111 1.6587 1.9816 2.3604 2.6208 2.8642 3.1653 3.3803 112 1.6586 1.9814 2.3601 2.6204 2.8637 3.1646 3.3795 113 1.6585 1.9812 2.3598 2.6200 2.8632 3.1640 3.3787 114 1.6583 1.9810 2.3595 2.6196 2.8627 3.1633 3.3779 115 1.6582 1.9808 2.3592 2.6192 2.8622 3.1626

Statistics – Residual Sum of Squares ”; Previous Next In statistics, the residual sum of squares (RSS), also known as the sum of squared residuals (SSR) or the sum of squared errors of prediction (SSE), is the sum of the squares of residuals (deviations of predicted from actual empirical values of data). Residual Sum of Squares (RSS) is defined and given by the following function: Formula ${RSS = sum_{i=0}^n(epsilon_i)^2 = sum_{i=0}^n(y_i – (alpha + beta x_i))^2}$ Where − ${X, Y}$ = set of values. ${alpha, beta}$ = constant of values. ${n}$ = set value of count Example Problem Statement: Consider two populace bunches, where X = 1,2,3,4 and Y = 4, 5, 6, 7, consistent worth ${alpha}$ = 1, ${beta}$ = 2. Locate the Residual Sum of Square (RSS) values of the two populace bunch. Solution: Given, ${X = 1,2,3,4 Y = 4,5,6,7 alpha = 1 beta = 2 }$ Arrangement: Substitute the given qualities in the recipe, Remaining Sum of Squares Formula ${RSS = sum_{i=0}^n(epsilon_i)^2 = sum_{i=0}^n(y_i – (alpha + beta x_i))^2, \[7pt] = sum(4-(1+(2x_1)))^2 + (5-(1+(2x_2)))^2 + (6-(1+(2x_3))^2 + (7-(1+(2x_4))^2, \[7pt] = sum(1)^2 + (0)^2 + (-1)^2 + (-2)^2, \[7pt] = 6 }$ Print Page Previous Next Advertisements ”;

Statistics – Regression Intercept Confidence Interval ”; Previous Next Regression Intercept Confidence Interval, is a way to determine closeness of two factors and is used to check the reliability of estimation. Formula ${R = beta_0 pm t(1 – frac{alpha}{2}, n-k-1) times SE_{beta_0} }$ Where − ${beta_0}$ = Regression intercept. ${k}$ = Number of Predictors. ${n}$ = sample size. ${SE_{beta_0}}$ = Standard Error. ${alpha}$ = Percentage of Confidence Interval. ${t}$ = t-value. Example Problem Statement: Compute the Regression Intercept Confidence Interval of following data. Total number of predictors (k) are 1, regression intercept ${beta_0}$ as 5, sample size (n) as 10 and standard error ${SE_{beta_0}}$ as 0.15. Solution: Let us consider the case of 99% Confidence Interval. Step 1: Compute t-value where ${ alpha = 0.99}$. ${ = t(1 – frac{alpha}{2}, n-k-1) \[7pt] = t(1 – frac{0.99}{2}, 10-1-1) \[7pt] = t(0.005,8) \[7pt] = 3.3554 }$ Step 2: ${ge} $Regression intercept: ${ = beta_0 + t(1 – frac{alpha}{2}, n-k-1) times SE_{beta_0} \[7pt] = 5 – (3.3554 times 0.15) \[7pt] = 5 – 0.50331 \[7pt] = 4.49669 }$ Step 3: ${le} $Regression intercept: ${ = beta_0 – t(1 – frac{alpha}{2}, n-k-1) times SE_{beta_0} \[7pt] = 5 + (3.3554 times 0.15) \[7pt] = 5 + 0.50331 \[7pt] = 5.50331 }$ As a result, Regression Intercept Confidence Interval is ${4.49669}$ or ${5.50331}$ for 99% Confidence Interval. Print Page Previous Next Advertisements ”;

Statistics – Probability ”; Previous Next Probability Probability implies ”likelihood” or ”chance”. When an event is certain to happen then the probability of occurrence of that event is 1 and when it is certain that the event cannot happen then the probability of that event is 0. Hence the value of probability ranges from 0 to 1. Probability has been defined in a varied manner by various schools of thought. Some of which are discussed below. Classical Definition of Probability As the name suggests the classical approach to defining probability is the oldest approach. It states that if there are n exhaustive, mutually exclusive andequally likely cases out of which m cases are favourable to the happening ofevent A, Then the probabilities of event A is defined as given by the following probability function: Formula ${P(A) = frac{Number of favourable cases}{Total number of equally likely cases} = frac{m}{n}}$ Thus to calculate the probability we need information on number of favorable cases and total number of equally likely cases. This can he explained using following example. Example Problem Statement: A coin is tossed. What is the probability of getting a head? Solution: Total number of equally likely outcomes (n) = 2 (i.e. head or tail) Number of outcomes favorable to head (m) = 1 ${P(head) = frac{1}{2}}$ Print Page Previous Next Advertisements ”;