If the oil drum's standard score is higher than the measuring cup's, however, then the drum's equipment has the more serious problem – it is farther from average than the measuring cup's equipment. Now the measuring cup has the "larger" problem. What if the percentage error is smaller for the drum than the measuring cup? The 55-gallon drum (US gallons) holds 208.20 litres, so the percentage error is 0.019%. Clearly the drum is "more wrong", so the drum-making equipment is the higher priority. The measuring cup is off by 1 millilitre the drum by 40 millilitres. You need to decide where to deploy your scarce and valuable repair team. Each type of item should be its proper size. Let's say your company manufactures both measuring cups and oil drums. On the assumption that the population of data follows a normal "bell-shaped" curve, statisticians can show a relationship between the z-score and the Q probability. The ‘Q' probability expresses how likely an observation's value is to be a "random chance", rather than having a systemic cause. The QA team may consider that there is no problem at this time. Therefore the probability, Q, that this standard score is due to chance is 42.3%. The QA team then reminds you that these measuring cups have a normal distribution. Your QA team assures you that the average of all the measuring cups is 1.00003 litres, and the standard deviation is 0.005. Is this a disaster? An anomaly? Acceptable? You check on one of these cups and find that it actually contains 1.001 litres.Ĭlearly, your measuring cup is wrong by 1 millilitre. Your quality assurance (QA) team tests each one and records what it really contains when filled to the "1 litre" mark. You and your customers expect them to be accurate. Let's say that your company manufactures measuring cups for household kitchens. What does the score mean for me? I mean, really? If you manufacture mirrors for Hubble-like telescopes, you would measure every one (several times). If you manufacture ball bearings, you probably don't measure each one. Statisticians like to point out that the "Student t-test" is more appropriate for incompletely-sampled populations. s = square root of ( ( S (xi – m )2 ) / n ), for all i = 1 to n in the total population.s is the standard deviation: the square root of the variance.This is the sum of all observations, divided by the number of observations.m is the mean of the population: the average value.x is the "raw" score, to be standardized.The mathematical formula is: z = (x – m) / s, where: The "standard score", is the statistical measurement of "how far is one particular observation away from the standard deviation". Compute four different areas under the standard curve based on the z-value you enter.
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