42. The Central Limit Theorem The Central Limit Theorem states that the mean values of samples taken from ANY distribution tend towards a normal distribution as the sample size increases . This computer demonstration provides convincing evidence of this surprising fact. Thus, taking samples from a distribution and averaging the observations within the samples effectively eliminates the effect of the underlying distribution, however 'non-normal' it may be. This demonstration works with two symmetrical distributions: one is triangular and has some features in common with the normal distribution while the other is a 'V'-shaped notch - almost the total opposite of the type of distributions we see in applied statistics. Both distributions have a mean of 50.00. We can model these distributions by supposing we have two packs containing cards numbered 1 - 99. The first pack would have: One 1, two 2s .... fifty 50s, forty nine 51s, .... two 98s, and one 99 While the second would have: Fifty 1s, forty nine 2s ... two 48s, one 50, two 51s, ..... fifty 99s The computer draws cards according to these distributions for sample sizes of 1 (to verify the concept of 'distribution'), 2, 5 and 10. When the sample size is 1, we are really confirming that the data 'in the long run' will behave like the distribution - which is in itself an important statistical lesson. The case {Sample Size = 2} is particularly interesting. It is not easy to to 'outguess' the computer and predict the shape of the lower curve; however, once the curve is seen, it can be readily explained in terms of basic probability. Although the second case is very extreme (literally!) compared with the first, it eventually falls into a 'normal' shape although it takes longer to do so. LEANRING 9
43. The Arithmetic mean : Most of the time when we refer to the average of something we are talking about arithmetic mean only. To find out the arithmetic mean , we sum the values and divide by the number of observation. Advantages : it's a good measure of central tendency.It easily understood by most people Disadvantages :- Although the mean is reliable in that it reflects all the values in the data set, it may also be affected by extreme values that are not representative of the rest of the data.
44. The Median : The median is a single value from the data set that measures the central item in the set of numbers.Half of the item lie above this point and the other half lie below it. We can find median even when our data are qualitative descriptions. For example we have five runs of the printing press the results of which must be rated according to the sharpness of the image. Extremely sharp, very sharp, sharp slightly blurred, and very blurred. Mode :- The mode is a value that is repeated most often in the data set. Infect it is the value with highest frequency.
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51. Control Limits for Average and Range Chart R = R+R+R+…R 1 2 3 n n UCL = X + A 2 R CL = X LCL = X - A 2 R UCL = D 4 R CL = R LCL = D 3 R LEANRING 10 X = X+X+X+…X 1 2 3 n n
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58. LEANRING 11 Cp = Tol band / 6 sigma Cpk = Min of (Avg - LSL) or (USL - Avg) / 3 sigma (R abr) R d 2 = (n-1) = √ (x-x ) + (x-x ) + … (x-x ) (n - 1) _ _ _ 1 2 n 2 2 2
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