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Compute four different areas under the standard curve based on the z-value you enter.

The "standard score", is the statistical measurement of "how far is one particular observation away from the standard deviation".

The mathematical formula is: z = (x – m) / s, where:

- z is the standard score
- x is the "raw" score, to be standardized
- m is the mean of the population: the average value
- This is the sum of all observations, divided by the number of observations
- s is the standard deviation: the square root of the variance
- s = square root of ( ( S (xi – m )2 ) / n ) , for all i = 1 to n in the total population.

Statisticians like to point out that the "Student t-test" is more appropriate for incompletely-sampled populations. If you manufacture ball bearings, you probably don't measure each one. If you manufacture mirrors for Hubble-like telescopes, you would measure every one (several times).

Let's say that your company manufactures measuring cups for household kitchens. You and your customers expect them to be accurate.

Your quality assurance (QA) team tests each one and records what it really contains when filled to the "1 litre" mark.

You check on one of these cups and find that it actually contains 1.001 litres.

Clearly, your measuring cup is wrong by 1 millilitre. Is this a disaster? An anomaly? Acceptable?

Your QA team assures you that the average of all the measuring cups is 1.00003 litres, and the standard deviation is 0.005. Therefore the standard score e is (1.001 – 1.00003)/0.005 = 0.00097/0.005 = 0.194.

The QA team then reminds you that these measuring cups have a normal distribution. Therefore the probability, Q, that this standard score is due to chance is 42.3%. The QA team may consider that there is no problem at this time.

The ‘Q' probability expresses how likely an observation's value is to be a "random chance", rather than having a systemic cause. 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.

Let's say your company manufactures both measuring cups and oil drums. Each type of item should be its proper size. Each has its own QA team. Each team reports a deviation today. You need to decide where to deploy your scarce and valuable repair team.

The measuring cup is off by 1 millilitre; the drum by 40 millilitres. Clearly the drum is "more wrong", so the drum-making equipment is the higher priority. Or is it?

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%. The measuring cup's error is 0.1%. Now the measuring cup has the "larger" problem. Or does it?

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. Probably the z-score indicates that something has worsened recently. If the standard score were 0.9, for example, then the probability that it is just a chance occurrence is 18%. At worse than 1-in-5 odds of "just chance", you should check the drum manufacturing equipment for recent wear.

So: the standard score supports comparing the significance of deviations observed in different populations.

A larger standard score indicates that the observation is more unlikely – and therefore indicates a problem.

The standard score allows a comparison of quality control between apples and automobiles – how "deviant" is this apple's weight from the others in the orchard? How "deviant" is this automobile's paint thickness, compared to others from this factory?

*By Oskar Olofsson*

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z |
area |
z |
area |
z |
area |
z |
area |

-3.5 | 0.000233 | -1.5 | 0.066807 | 0.5 | 0.691462 | 2.5 | 0.99379 |

-3.45 | 0.00028 | -1.45 | 0.073529 | 0.55 | 0.70884 | 2.55 | 0.994614 |

-3.4 | 0.000337 | -1.4 | 0.080757 | 0.6 | 0.725747 | 2.6 | 0.995339 |

-3.35 | 0.000404 | -1.35 | 0.088508 | 0.65 | 0.742154 | 2.65 | 0.995975 |

-3.3 | 0.000483 | -1.3 | 0.0968 | 0.7 | 0.758036 | 2.7 | 0.996533 |

-3.25 | 0.000577 | -1.25 | 0.10565 | 0.75 | 0.773373 | 2.75 | 0.99702 |

-3.2 | 0.000687 | -1.2 | 0.11507 | 0.8 | 0.788145 | 2.8 | 0.997445 |

-3.15 | 0.000816 | -1.15 | 0.125072 | 0.85 | 0.802337 | 2.85 | 0.997814 |

-3.1 | 0.000968 | -1.1 | 0.135666 | 0.9 | 0.81594 | 2.9 | 0.998134 |

-3.05 | 0.001144 | -1.05 | 0.146859 | 0.95 | 0.828944 | 2.95 | 0.998411 |

-3 | 0.00135 | -1 | 0.158655 | 1 | 0.841345 | 3 | 0.99865 |

-2.95 | 0.001589 | -0.95 | 0.171056 | 1.05 | 0.853141 | 3.05 | 0.998856 |

-2.9 | 0.001866 | -0.9 | 0.18406 | 1.1 | 0.864334 | 3.1 | 0.999032 |

-2.85 | 0.002186 | -0.85 | 0.197663 | 1.15 | 0.874928 | 3.15 | 0.999184 |

-2.8 | 0.002555 | -0.8 | 0.211855 | 1.2 | 0.88493 | 3.2 | 0.999313 |

-2.75 | 0.00298 | -0.75 | 0.226627 | 1.25 | 0.89435 | 3.25 | 0.999423 |

-2.7 | 0.003467 | -0.7 | 0.241964 | 1.3 | 0.9032 | 3.3 | 0.999517 |

-2.65 | 0.004025 | -0.65 | 0.257846 | 1.35 | 0.911492 | 3.35 | 0.999596 |

-2.6 | 0.004661 | -0.6 | 0.274253 | 1.4 | 0.919243 | 3.4 | 0.999663 |

-2.55 | 0.005386 | -0.55 | 0.29116 | 1.45 | 0.926471 | 3.45 | 0.99972 |

-2.5 | 0.00621 | -0.5 | 0.308538 | 1.5 | 0.933193 | 3.5 | 0.999767 |

-2.45 | 0.007143 | -0.45 | 0.326355 | 1.55 | 0.939429 | 3.55 | 0.999807 |

-2.4 | 0.008198 | -0.4 | 0.344578 | 1.6 | 0.945201 | 3.6 | 0.999841 |

-2.35 | 0.009387 | -0.35 | 0.363169 | 1.65 | 0.950529 | 3.65 | 0.999869 |

-2.3 | 0.010724 | -0.3 | 0.382089 | 1.7 | 0.955435 | 3.7 | 0.999892 |

-2.25 | 0.012224 | -0.25 | 0.401294 | 1.75 | 0.959941 | 3.75 | 0.999912 |

-2.2 | 0.013903 | -0.2 | 0.42074 | 1.8 | 0.96407 | 3.8 | 0.999928 |

-2.15 | 0.015778 | -0.15 | 0.440382 | 1.85 | 0.967843 | 3.85 | 0.999941 |

-2.1 | 0.017864 | -0.1 | 0.460172 | 1.9 | 0.971283 | 3.9 | 0.999952 |

-2.05 | 0.020182 | -0.05 | 0.480061 | 1.95 | 0.974412 | 3.95 | 0.999961 |

-2 | 0.02275 | -1E-14 | 0.5 | 2 | 0.97725 | 4 | 0.999968 |

-1.95 | 0.025588 | 0.05 | 0.519939 | 2.05 | 0.979818 | 4.05 | 0.999974 |

-1.9 | 0.028717 | 0.1 | 0.539828 | 2.1 | 0.982136 | 4.1 | 0.999979 |

-1.85 | 0.032157 | 0.15 | 0.559618 | 2.15 | 0.984222 | 4.15 | 0.999983 |

-1.8 | 0.03593 | 0.2 | 0.57926 | 2.2 | 0.986097 | 4.2 | 0.999987 |

-1.75 | 0.040059 | 0.25 | 0.598706 | 2.25 | 0.987776 | 4.25 | 0.999989 |

-1.7 | 0.044565 | 0.3 | 0.617911 | 2.3 | 0.989276 | 4.3 | 0.999991 |

-1.65 | 0.049471 | 0.35 | 0.636831 | 2.35 | 0.990613 | 4.35 | 0.999993 |

-1.6 | 0.054799 | 0.4 | 0.655422 | 2.4 | 0.991802 | 4.4 | 0.999995 |

-1.55 | 0.060571 | 0.45 | 0.673645 | 2.45 | 0.992857 | 4.45 | 0.999996 |