The ancient Phoenicians were great traders and sea farers, but they tended to
overload their boats. In stormy weather, goods had to be thrown overboard in
order to save the ship. Owners of lost goods were then compensated by those who
did not lose their goods, and the amounts involved depended on the total value
of the goods. This arrangement was named havara, and this term evolved over the
centuries to become average, and from average the mathematical term mean developed.

The astronomer, Gauss, measured distances between stars. He noticed that it
was difficult to reproduce his measurements exactly. However, the measurements
clustered around a central value, more common near the mean, and becoming less
common as they are further away from the mean (fig. left).

He concluded that any set of
measurements would normally distribute to this pattern and called it the
Normal Distribution.

De Moivre derived the formula for the Normal Distribution curve in
mathematical terms (fig. right).

Once this was done, the features of Normal Distribution could be
mathematically handled. Using simple calculus, the area under the curve (or any
part of it) can be estimated. Fisher developed the concept further and used the
area under the curve as a measure of probability. He argues that if the area
under the whole curve be consider totality, then any portion represents the
probability of the events that describe the portion. From this, he was able to
calculate the probability of obtaining a measurement that exceeds a deviation
from the mean value. He standardized the measurement of this deviation and
called it the Normal Standard Deviate (z), later abbreviated to Standard
Deviation, and derived the relationships between z and probability.

The other panels of this page are

**Calculations: **Javascript program to calculate the probability of z
**Table: ** Table of probability of z
**Codes: ** R and Python codes for calculation of z

**References**

Calculation are presented in

maroon and results produced presented in

navy
### R Code

**Probability of z**

ZToP<-function(z) #function to calculate probability from z
{
return (1-pnorm(z))
}
ZToP(1.65) # testing

0.04947147

PToZ<-function(p) #function to calculate z from probability
{
return (-qnorm(p))
}
PToZ(0.05) # testing

1.644854

### Python Code

**Header and Library**

import scipy.stats as st

**Probability of z**
def ZToP(z):
""" Calculate probability from z """
return 1 - st.norm.cdf(z)
print(ZToP(1.64)) # testing

0.050502583474103746

def PToZ(p):
""" Calculate z from probability """
return st.norm.ppf(1 - p)
print(PToZ(0.05)) # testing

1.6448536269514722

The z value is the sum of the first row and first column, and probability is the cell of that row and column.
for example, probability=0.40905 when z=0.23

**Please be reminded **that, although probabilities of z are presented
to 5 decimal points of precision in the table, only a 2-4 decimal point precisions
are more commonly used.

| 0.00 | 0.01 | 0.02 | 0.03 | 0.04 | 0.05 | 0.06 | 0.07 | 0.08 | 0.09 |

0.0 | 0.50000 | 0.49601 | 0.49202 | 0.48803 | 0.48405 | 0.48006 | 0.47608 | 0.47210 | 0.46812 | 0.46414 |

0.1 | 0.46017 | 0.45620 | 0.45224 | 0.44828 | 0.44433 | 0.44038 | 0.43644 | 0.43251 | 0.42858 | 0.42465 |

0.2 | 0.42074 | 0.41683 | 0.41294 | 0.40905 | 0.40517 | 0.40129 | 0.39743 | 0.39358 | 0.38974 | 0.38591 |

0.3 | 0.38209 | 0.37828 | 0.37448 | 0.37070 | 0.36693 | 0.36317 | 0.35942 | 0.35569 | 0.35197 | 0.34827 |

0.4 | 0.34458 | 0.34090 | 0.33724 | 0.33360 | 0.32997 | 0.32636 | 0.32276 | 0.31918 | 0.31561 | 0.31207 |

0.5 | 0.30854 | 0.30503 | 0.30153 | 0.29806 | 0.29460 | 0.29116 | 0.28774 | 0.28434 | 0.28096 | 0.27760 |

0.6 | 0.27425 | 0.27093 | 0.26763 | 0.26435 | 0.26109 | 0.25785 | 0.25463 | 0.25143 | 0.24825 | 0.24510 |

0.7 | 0.24196 | 0.23885 | 0.23576 | 0.23270 | 0.22965 | 0.22663 | 0.22363 | 0.22065 | 0.21770 | 0.21476 |

0.8 | 0.21186 | 0.20897 | 0.20611 | 0.20327 | 0.20045 | 0.19766 | 0.19489 | 0.19215 | 0.18943 | 0.18673 |

0.9 | 0.18406 | 0.18141 | 0.17879 | 0.17619 | 0.17361 | 0.17106 | 0.16853 | 0.16602 | 0.16354 | 0.16109 |

| 0.00 | 0.01 | 0.02 | 0.03 | 0.04 | 0.05 | 0.06 | 0.07 | 0.08 | 0.09 |

1.0 | 0.15866 | 0.15625 | 0.15386 | 0.15151 | 0.14917 | 0.14686 | 0.14457 | 0.14231 | 0.14007 | 0.13786 |

1.1 | 0.13567 | 0.13350 | 0.13136 | 0.12924 | 0.12714 | 0.12507 | 0.12302 | 0.12100 | 0.11900 | 0.11702 |

1.2 | 0.11507 | 0.11314 | 0.11123 | 0.10935 | 0.10749 | 0.10565 | 0.10383 | 0.10204 | 0.10027 | **0.09853** |

1.3 | 0.09680 | 0.09510 | 0.09342 | 0.09176 | 0.09012 | 0.08851 | 0.08691 | 0.08534 | 0.08379 | 0.08226 |

1.4 | 0.08076 | 0.07927 | 0.07780 | 0.07636 | 0.07493 | 0.07353 | 0.07214 | 0.07078 | 0.06944 | 0.06811 |

1.5 | 0.06681 | 0.06552 | 0.06426 | 0.06301 | 0.06178 | 0.06057 | 0.05938 | 0.05821 | 0.05705 | 0.05592 |

1.6 | 0.05480 | 0.05370 | 0.05262 | 0.05155 | 0.05050 | **0.04947** | 0.04846 | 0.04746 | 0.04648 | 0.04551 |

1.7 | 0.04457 | 0.04363 | 0.04272 | 0.04182 | 0.04093 | 0.04006 | 0.03920 | 0.03836 | 0.03754 | 0.03673 |

1.8 | 0.03593 | 0.03515 | 0.03438 | 0.03363 | 0.03288 | 0.03216 | 0.03144 | 0.03074 | 0.03005 | 0.02938 |

1.9 | 0.02872 | 0.02807 | 0.02743 | 0.02680 | 0.02619 | 0.02559 | **0.02500** | 0.02442 | 0.02385 | 0.02330 |

| 0.00 | 0.01 | 0.02 | 0.03 | 0.04 | 0.05 | 0.06 | 0.07 | 0.08 | 0.09 |

2.0 | 0.02275 | 0.02222 | 0.02169 | 0.02118 | 0.02068 | 0.02018 | 0.01970 | 0.01923 | 0.01876 | 0.01831 |

2.1 | 0.01786 | 0.01743 | 0.01700 | 0.01659 | 0.01618 | 0.01578 | 0.01539 | 0.01500 | 0.01463 | 0.01426 |

2.2 | 0.01390 | 0.01355 | 0.01321 | 0.01287 | 0.01255 | 0.01222 | 0.01191 | 0.01160 | 0.01130 | 0.01101 |

2.3 | 0.01072 | 0.01044 | 0.01017 | **0.00990** | 0.00964 | 0.00939 | 0.00914 | 0.00889 | 0.00866 | 0.00842 |

2.4 | 0.00820 | 0.00798 | 0.00776 | 0.00755 | 0.00734 | 0.00714 | 0.00695 | 0.00676 | 0.00657 | 0.00639 |

2.5 | 0.00621 | 0.00604 | 0.00587 | 0.00570 | 0.00554 | 0.00539 | 0.00523 | 0.00508 | **0.00494** | 0.00480 |

2.6 | 0.00466 | 0.00453 | 0.00440 | 0.00427 | 0.00415 | 0.00402 | 0.00391 | 0.00379 | 0.00368 | 0.00357 |

2.7 | 0.00347 | 0.00336 | 0.00326 | 0.00317 | 0.00307 | 0.00298 | 0.00289 | 0.00280 | 0.00272 | 0.00264 |

2.8 | 0.00255 | **0.00248** | 0.00240 | 0.00233 | 0.00226 | 0.00219 | 0.00212 | 0.00205 | 0.00199 | 0.00193 |

2.9 | 0.00187 | 0.00181 | 0.00175 | 0.00169 | 0.00164 | 0.00159 | 0.00154 | 0.00149 | 0.00144 | 0.00139 |

| 0.00 | 0.01 | 0.02 | 0.03 | 0.04 | 0.05 | 0.06 | 0.07 | 0.08 | 0.09 |

3.0 | 0.00135 | 0.00131 | 0.00126 | 0.00122 | 0.00118 | 0.00114 | 0.00111 | 0.00107 | 0.00103 | 0.00100 |

3.1 | **0.00097** | 0.00094 | 0.00090 | 0.00087 | 0.00084 | 0.00082 | 0.00079 | 0.00076 | 0.00074 | 0.00071 |

3.2 | 0.00069 | 0.00066 | 0.00064 | 0.00062 | 0.00060 | 0.00058 | 0.00056 | 0.00054 | 0.00052 | 0.00050 |

3.3 | 0.00048 | 0.00047 | 0.00045 | 0.00043 | 0.00042 | 0.00040 | 0.00039 | 0.00038 | 0.00036 | 0.00035 |

3.4 | 0.00034 | 0.00032 | 0.00031 | 0.00030 | 0.00029 | 0.00028 | 0.00027 | 0.00026 | 0.00025 | 0.00024 |

3.5 | 0.00023 | 0.00022 | 0.00022 | 0.00021 | 0.00020 | 0.00019 | 0.00019 | 0.00018 | 0.00017 | 0.00017 |

3.6 | 0.00016 | 0.00015 | 0.00015 | 0.00014 | 0.00014 | 0.00013 | 0.00013 | 0.00012 | 0.00012 | 0.00011 |

3.7 | 0.00011 | 0.00010 | 0.00010 | 0.00010 | 0.00009 | 0.00009 | 0.00009 | 0.00008 | 0.00008 | 0.00008 |

3.8 | 0.00007 | 0.00007 | 0.00007 | 0.00006 | 0.00006 | 0.00006 | 0.00006 | 0.00005 | 0.00005 | 0.00005 |

3.9 | 0.00005 | 0.00005 | 0.00004 | 0.00004 | 0.00004 | 0.00004 | 0.00004 | 0.00004 | 0.00003 | 0.00003 |

Probability | z value |

0.20 | 0.85 | power=0.8, β=0.2 |

0.10 | 1.29 |

0.05 | 1.65 | statistical significane p<0.05 |

0.025 | 1.96 | 95% confidence interval (2.5% each side) |

0.01 | 2.33 |

0.005 | 2.58 |

0.0025 | 2.81 |

0.001 | 3.10 |