pyyeti.stats.kdouble

pyyeti.stats.kdouble(p, c, n, tol=1e-12)

Compute statistical k-factor for a double-sided tolerance interval.

Parameters:

• p (scalar or array_like; real) – Portion of population to bound; 0 < p < 1

• c (scalar or array_like; real) – Probability level (confidence); 0 < c < 1

• n (scalar or array_like; integer) – Number of observations in sample; n > 1

• tol (scalar; optional) – Error tolerance to pass to the _getr() routine

Returns:

k (scalar or ndarray; real) – The statistical k-factor for a double-sided tolerance interval.

Notes

The inputs p, c, and n must be broadcast-compatible.

The k-factor allows the computation of a tolerance interval that has the probability c of containing at least the proportion p of the population. The statistics are based on having a sample of a normally distributed population; n is the number of observations in the sample.

The bounds of the tolerance interval are calculated by:

lower = m - k std
upper = m + k std

where m is the sample mean and std is the sample standard deviation.

See references [1], [2], and [3] for the mathematical background on these tolerance limits.

References

[1]

Churchill Eisenhart; Millard Hastay; and W. Allen Wallis, ‘Selected Techniques of Statistical Analysis for Scientific and Industrial Research and Production and Management Engineering,’ by the Statistical Research Group, Columbia University, First Edition, New York and London, McGraw-Hill Book Company, Inc, 1947.

[2]

Albert H. Bowker, ‘Computation of Factors for Tolerance Limits on a Normal Distribution when the Sample Size is Large,’ Annals of Mathematical Statistics, Vol. 17, 1946, pp 238-240.

[3]

A. Wald and J. Wolfowitz, ‘Tolerance Limits for a Normal Distribution,’ Annals of Mathematical Statistics, Vol. 17, 1946, pp 208-215.

See also

ksingle()

Examples

Assume we have 21 samples. Determine the k-factor to have a 90% probability of the interval containing 99% of the population. In other words, we need the ‘P99/90’ double-sided k-factor for N = 21. (From table: N = 21, k = 3.340)

>>> from pyyeti.stats import kdouble
>>> kdouble(.99, .90, 21)
3.3404115111514...

Make a table of double-sided k-factors using 50% confidence. The probabilities p will be: 95%, 97.725%, 99% and 99.865%. Number of samples n will be: 2-10, 1000000. Have n define the rows and p define the columns:

>>> import numpy as np
>>> from pandas import DataFrame
>>> n = [[i] for i in range(2, 11)]  # create list of lists
>>> n.append([1000000])
>>> p = [.95, .9545, .99, .9973]
>>> table = kdouble(p, .50, n)
>>> DataFrame(table, index=[i[0] for i in n], columns=p)
           0.9500    0.9545    0.9900    0.9973
2        3.502564  3.568593  4.502465  5.175585
3        2.697379  2.750060  3.498689  4.041051
4        2.456441  2.505211  3.200478  3.706120
5        2.336939  2.383750  3.052519  3.540237
6        2.264675  2.310279  2.962781  3.439587
7        2.215998  2.260775  2.902111  3.371448
8        2.180884  2.225054  2.858183  3.322030
9        2.154316  2.198021  2.824834  3.284445
10       2.133494  2.176829  2.798616  3.254847
1000000  1.959966  2.000004  2.575831  2.999979