Module stikpetP.helper.help_quartileIndexing

Functions

def he_quartileIndexing(data, method='sas1')

Quartile Indexing

Helper function for me_quartiles() and he_quartileIndexing() to return the index number of the first and third quartile for different methods of determining this index.

Parameters

data : pandas series with numeric values

 

method : optional which method to use to calculate quartiles

 

Returns

q1Index : the index of the first (lower) quartile

 

q3Index : the index of the third (upper/higher) quartile

 

Notes

The inclusive method divides the data into two, and then includes the median in each half (if the sample size is odd). The first and third quarter are then the median of each of these two halves (Tukey, 1977, p. 32).

For the inclusive method, the index of the first quartile can be found using: $$iQ_1 = \begin{cases} \frac{n+2}{4} & \text{ if } n \text{ mod } 2 = 0 \\ \frac{n+3}{4} & \text{ else } \end{cases}$$

And the third quartile: $$iQ_3 = \begin{cases} \frac{3\times n +2}{4} & \text{ if } n \text{ mod } 2 = 0 \\ \frac{3\times n+1}{4} & \text{ else } \end{cases}$$

The exclusive method does the same as the inclusive method, but excludes the median in each half (if the sample size is odd) (Moore & McCabe, 1989, p. 33; Joarder & Firozzaman, 2001, p. 88).

For the exclusive method, the index of the first quartile can be found using: $$iQ_1 = \begin{cases} \frac{n+2}{4} & \text{ if } n \text{ mod } 2 = 0 \\ \frac{n + 1}{4} & \text{ else } \end{cases}$$

And the third quartile: $$iQ_3 = \begin{cases} \frac{3\times n +2}{4} & \text{ if } n \text{ mod } 2 = 0 \\ \frac{3\times n + 3}{4} & \text{ else } \end{cases}$$

Other methods use a different indexing. Six alternatives for the indexing is:

Most basic (SAS1): $$iQ_i = n\times p_i$$

SAS4 method uses for indexing (SAS, 1990, p. 626; Snedecor, 1940, p. 43): $$iQ_i = \left(n + 1\right)\times p_i$$

Hog and Ledolter use for their indexing (Hogg & Ledolter, 1992, p. 21; Hazen, 1914, p. ?):

$$iQ_i = n\times p_i + \frac{1}{2}$$

MS Excel uses for indexing (Gumbel, 1939, p. ?; Hyndman & Fan, 1996, p. 363): $$iQ_i = \left(n - 1\right)\times p_i + 1$$

Hyndman and Fan use for their 8th version (Hyndman & Fan, 1996, p. 363): $$iQ_i = \left(n + \frac{1}{3}\right)\times p_i + \frac{1}{3}$$

Hyndman and Fan use for their 9th version (Hyndman & Fan, 1996, p. 364): $$iQ_i = \left(n + \frac{1}{4}\right)\times p_i + \frac{3}{8}$$

References

Gumbel, E. J. (1939). La Probabilité des Hypothèses. Compes Rendus de l’ Académie des Sciences, 209, 645–647.

Hazen, A. (1914). Storage to be provided in impounding municipal water supply. Transactions of the American Society of Civil Engineers, 77(1), 1539–1640. https://doi.org/10.1061/taceat.0002563

Hogg, R. V., & Ledolter, J. (1992). Applied statistics for engineers and physical scientists (2nd int.). Macmillan.

Hyndman, R. J., & Fan, Y. (1996). Sample quantiles in statistical packages. The American Statistician, 50(4), 361–365. https://doi.org/10.2307/2684934

Joarder, A. H., & Firozzaman, M. (2001). Quartiles for discrete data. Teaching Statistics, 23(3), 86–89. https://doi.org/10.1111/1467-9639.00063

Moore, D. S., & McCabe, G. P. (1989). Introduction to the practice of statistics. W.H. Freeman.

SAS. (1990). SAS procedures guide: Version 6 (3rd ed.). SAS Institute.

Snedecor, G. W. (1940). Statistical methods applied to experiments in agriculture and biology (3rd ed.). The Iowa State College Press.

Tukey, J. W. (1977). Exploratory data analysis. Addison-Wesley Pub. Co.

Author

Made by P. Stikker

Please visit: https://PeterStatistics.com

YouTube channel: https://www.youtube.com/stikpet

Expand source code

def he_quartileIndexing(data, method="sas1"):
    '''
    Quartile Indexing
    
    Helper function for **me_quartiles()** and **he_quartileIndexing()** to return the index number of the first and third quartile for different methods of determining this index.
    
    Parameters
    ----------
    data : pandas series with numeric values
    method : optional which method to use to calculate quartiles
    
    Returns
    -------
    q1Index : the index of the first (lower) quartile
    q3Index : the index of the third (upper/higher) quartile
    
    Notes
    -----
    The **inclusive** method divides the data into two, and then includes the median in each half (if the sample size is odd). The first and third quarter are then the median of each of these two halves (Tukey, 1977, p. 32).
    
    For the **inclusive** method, the index of the first quartile can be found using:
    $$iQ_1 = \\begin{cases} \\frac{n+2}{4} & \\text{ if } n \\text{ mod } 2 = 0 \\\\ \\frac{n+3}{4} & \\text{ else } \\end{cases}$$
    
    And the third quartile:
    $$iQ_3 = \\begin{cases} \\frac{3\\times n +2}{4} & \\text{ if } n \\text{ mod } 2 = 0 \\\\ \\frac{3\\times n+1}{4} & \\text{ else } \\end{cases}$$
    
    The **exclusive** method does the same as the inclusive method, but excludes the median in each half (if the sample size is odd) (Moore & McCabe, 1989, p. 33; Joarder & Firozzaman, 2001, p. 88).
    
    For the **exclusive** method, the index of the first quartile can be found using:
    $$iQ_1 = \\begin{cases} \\frac{n+2}{4} & \\text{ if } n \\text{ mod } 2 = 0 \\\\ \\frac{n + 1}{4} & \\text{ else } \\end{cases}$$

    And the third quartile:
    $$iQ_3 = \\begin{cases} \\frac{3\\times n +2}{4} & \\text{ if } n \\text{ mod } 2 = 0 \\\\ \\frac{3\\times n + 3}{4} & \\text{ else } \\end{cases}$$
    
    Other methods use a different indexing. Six alternatives for the indexing is:
    
    Most basic (**SAS1**):
    $$iQ_i = n\\times p_i$$

    **SAS4** method uses for indexing (SAS, 1990, p. 626; Snedecor, 1940, p. 43):
    $$iQ_i = \\left(n + 1\\right)\\times p_i$$

    **Hog and Ledolter** use for their indexing (Hogg & Ledolter, 1992, p. 21; Hazen, 1914, p. ?):

    $$iQ_i = n\\times p_i + \\frac{1}{2}$$

    **MS Excel** uses for indexing (Gumbel, 1939, p. ?; Hyndman & Fan, 1996, p. 363):
    $$iQ_i = \\left(n - 1\\right)\\times p_i + 1$$

    **Hyndman and Fan** use for their 8th version (Hyndman & Fan, 1996, p. 363):
    $$iQ_i = \\left(n + \\frac{1}{3}\\right)\\times p_i + \\frac{1}{3}$$

    **Hyndman and Fan** use for their 9th version (Hyndman & Fan, 1996, p. 364):
    $$iQ_i = \\left(n + \\frac{1}{4}\\right)\\times p_i + \\frac{3}{8}$$
    
    References
    ----------
    Gumbel, E. J. (1939). La Probabilité des Hypothèses. Compes Rendus de l’ Académie des Sciences, 209, 645–647.
    
    Hazen, A. (1914). Storage to be provided in impounding municipal water supply. Transactions of the American Society of Civil Engineers, 77(1), 1539–1640. https://doi.org/10.1061/taceat.0002563
    
    Hogg, R. V., & Ledolter, J. (1992). Applied statistics for engineers and physical scientists (2nd int.). Macmillan.
    
    Hyndman, R. J., & Fan, Y. (1996). Sample quantiles in statistical packages. The American Statistician, 50(4), 361–365. https://doi.org/10.2307/2684934
    
    Joarder, A. H., & Firozzaman, M. (2001). Quartiles for discrete data. Teaching Statistics, 23(3), 86–89. https://doi.org/10.1111/1467-9639.00063
    
    Moore, D. S., & McCabe, G. P. (1989). Introduction to the practice of statistics. W.H. Freeman.
    
    SAS. (1990). SAS procedures guide: Version 6 (3rd ed.). SAS Institute.
    
    Snedecor, G. W. (1940). Statistical methods applied to experiments in agriculture and biology (3rd ed.). The Iowa State College Press.
    
    Tukey, J. W. (1977). Exploratory data analysis. Addison-Wesley Pub. Co.
    
    Author
    ------
    Made by P. Stikker
    
    Please visit: https://PeterStatistics.com
    
    YouTube channel: https://www.youtube.com/stikpet
    
    '''
    
    
    
    n = len(data)
    if method=="inclusive":
        if (n % 2) == 0:
            q1Index = (n + 2)/ 4
            q3Index = (3*n + 1)/4
        else:
            q1Index = (n + 3)/ 4
            q3Index = (3*n + 1)/4
    elif method=="exclusive":
        if (n % 2) == 0:
            q1Index = (n + 2)/ 4
            q3Index = (3*n + 1)/4
        else:
            q1Index = (n + 1)/ 4
            q3Index = (3*n + 3)/4
            
    elif method=="sas1":        
        q1Index = n*1/4
        q3Index = n*3/4
    elif method=="sas4":
        q1Index = (n + 1)*1/4
        q3Index = (n + 1)*3/4
    elif method=="hl":  
        q1Index = n*1/4 + 1/2
        q3Index = n*3/4 + 1/2
    elif method=="excel":
        q1Index = (n - 1)*1/4 + 1
        q3Index = (n - 1)*3/4 + 1
    elif method=="hf8":
        q1Index = (n + 1/3)*1/4 + 1/3
        q3Index = (n + 1/3)*3/4 + 1/3
    elif method=="hf9":
        q1Index = (n + 1/4)*1/4 + 3/8
        q3Index = (n + 1/4)*3/4 + 3/8
        
    return q1Index, q3Index