Module stikpetP.measures.meas_median

Functions

def me_median(data, levels=None, tieBreaker='between')

Median

Function to determine the median of a set of data. The median can be defined as "the middle value in a distribution, below and above which lie values with equal total frequencies or probabilities" (Porkess, 1991, p. 134). This means that 50% of the respondents scored equal or higher to the median, and also 50% of the respondents scored lower or equal.

This function is shown in this YouTube video and the measure is also described at PeterStatistics.com

Parameters

data : list or pandas series

 

levels : dictionary, optional

indicate what values represent

tieBreaker : {"between", "low", "high"}, optional

which to return if median falls between two values. Default is "between"

Returns

medNum : float

numeric value of the median

medText : string

string value of the median

Notes

The formula that is used, assuming the data has been sorted, is: $$\tilde{x} = \begin{cases} x_{MI} & \text{ if } MI= \left \lfloor MI \right \rfloor \\ \frac{x_{MI-0.5} + x_{MI+0.5}}{2} & \text{ if } MI\neq \left \lfloor MI \right \rfloor \end{cases}$$

With: $$MI = \frac{n + 1}{2}$$

Symbols used:

• $n$ the sample size
• $x_i$ the i-th score of X, assuming X has been sorted.
• $MI$ the index of the median
• $\tilde{x}$ the median

If the number of scores is an odd number, and the median falls between two categories, the tieBreaker can be used. If this is set to "between", the function will return the average of the two values, or "between x and y" if levels are used. If it is set to "tieBreaker="low"", the lower value is returned, and if set to "tiebreaker="high"" the upper value is returned.

Some old references to the median are Pacioli (1523) in Italian, Cournot (1843, p. 120) in French, and Galton (1881, p. 246) in English.

Before, After and Alternatives

Before this measure you might want an impression using a frequency table or a visualisation: * tab_frequency for a frequency table * vi_bar_stacked_single for Single Stacked Bar-Chart * vi_bar_dual_axis for Dual-Axis Bar Chart

After this you might want some other descriptive measures: * me_consensus for the Consensus * me_hodges_lehmann_os for the Hodges-Lehmann Estimate (One-Sample) * me_quantiles for Quantiles * me_quartiles for Quartiles / Hinges * me_quartile_range for Interquartile Range, Semi-Interquartile Range and Mid-Quartile Range

or perform a test: * ts_sign_os for One-Sample Sign Test * ts_trinomial_os for One-Sample Trinomial Test * ts_wilcoxon_os for Wilcoxon Signed Rank Test (One-Sample)

References

Cournot, A. A. (1843). Exposition de la théorie des chances et des probabilités. L. Hachette.

Galton, F. (1881). Report of the anthropometric committee. Report of the British Association for the Advancement of Science, 51, 225–272.

Pacioli, L. (1523). Summa de arithmetica geometria proportioni: Et proportionalita. Paganino de Paganini.

Porkess, R. (1991). The HarperCollins dictionary of statistics. HarperPerennial.

Author

Made by P. Stikker

Companion website: https://PeterStatistics.com
YouTube channel: https://www.youtube.com/stikpet
Donations: https://www.patreon.com/bePatron?u=19398076

Examples

Example 1: Text Pandas Series

>>> import pandas as pd
>>> df2 = pd.read_csv('https://peterstatistics.com/Packages/ExampleData/StudentStatistics.csv', sep=';', low_memory=False, storage_options={'User-Agent': 'Mozilla/5.0'})
>>> ex1 = df2['Teach_Motivate']
>>> order = {"Fully Disagree":1, "Disagree":2, "Neither disagree nor agree":3, "Agree":4, "Fully agree":5}
>>> me_median(ex1, levels=order)
(np.float64(2.0), 'Disagree')

Example 2: Numeric data

>>> ex2 = [1, 1, 1, 2, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5]
>>> me_median(ex2)
(np.float64(4.0), '4.0')

Example 3: Text data with between median

>>> ex3 = ["a", "b", "f", "d", "e", "c"]
>>> order = {"a":1, "b":2, "c":3, "d":4, "e":5, "f":6}
>>> me_median(ex3, levels=order)
(np.float64(3.5), 'between c and d')
>>> me_median(ex3, levels=order, tieBreaker="low")
(np.float64(3.5), 'c')
>>> me_median(ex3, levels=order, tieBreaker="high")
(np.float64(3.5), 'd')

Example 4: Numeric data with between median

>>> ex4 = [1, 2, 3, 4, 5, 6]
>>> me_median(ex4)
(np.float64(3.5), '3.5')
>>> me_median(ex4, tieBreaker="low")
(3, '3')
>>> me_median(ex4, tieBreaker="high")
(4, '4')

Expand source code

def me_median(data, levels=None, tieBreaker="between"):
    '''
    Median
    ------
    
    Function to determine the median of a set of data. The median can be defined as "the middle value in a distribution, below and above which lie values with equal total frequencies or probabilities" (Porkess, 1991, p. 134). This means that 50% of the respondents scored equal or higher to the median, and also 50% of the respondents scored lower or equal.

    This function is shown in this [YouTube video](https://youtu.be/iI07nJ3wlOQ) and the measure is also described at [PeterStatistics.com](https://peterstatistics.com/Terms/Measures/Quantiles.html)
    
    Parameters
    ----------
    data : list or pandas series
    levels : dictionary, optional 
        indicate what values represent
    tieBreaker : {"between", "low", "high"}, optional 
        which to return if median falls between two values. Default is "between"
    
    Returns
    -------
    medNum : float
        numeric value of the median
    medText : string 
        string value of the median
    
    Notes
    -----
    The formula that is used, assuming the data has been sorted, is:
    $$\\tilde{x} = \\begin{cases} x_{MI} & \\text{ if } MI= \\left \\lfloor MI \\right \\rfloor \\\\ \\frac{x_{MI-0.5} + x_{MI+0.5}}{2} & \\text{ if } MI\\neq \\left \\lfloor MI \\right \\rfloor \\end{cases}$$
    
    With:
    $$MI = \\frac{n + 1}{2}$$
    
    *Symbols used:*
    
    * $n$ the sample size
    * $x_i$ the i-th score of X, assuming X has been sorted.
    * $MI$ the index of the median
    * $\\tilde{x}$ the median
    
    If the number of scores is an odd number, and the median falls between two categories, the *tieBreaker* can be used. If this is set to *"between"*, the function will return the average of the two values, or "between x and y" if levels are used. If it is set to "tieBreaker="low"", the lower value is returned, and if set to "tiebreaker="high"" the upper value is returned.
    
    Some old references to the median are Pacioli (1523) in Italian, Cournot (1843, p. 120) in French, and Galton (1881, p. 246) in English.

    Before, After and Alternatives
    ------------------------------
    Before this measure you might want an impression using a frequency table or a visualisation:
    * [tab_frequency](../other/table_frequency.html#tab_frequency) for a frequency table
    * [vi_bar_stacked_single](../visualisations/vis_bar_stacked_single.html#vi_bar_stacked_single) for Single Stacked Bar-Chart
    * [vi_bar_dual_axis](../visualisations/vis_bar_dual_axis.html#vi_bar_dual_axis) for Dual-Axis Bar Chart

    After this you might want some other descriptive measures:
    * [me_consensus](../measures/meas_consensus.html#me_consensus) for the Consensus
    * [me_hodges_lehmann_os](../measures/meas_hodges_lehmann_os.html#me_hodges_lehmann_os) for the Hodges-Lehmann Estimate (One-Sample)
    * [me_quantiles](../measures/meas_quantiles.html#me_quantiles) for Quantiles
    * [me_quartiles](../measures/meas_quartiles.html#me_quantiles) for Quartiles / Hinges
    * [me_quartile_range](../measures/meas_quartile_range.html#me_quartile_range) for Interquartile Range, Semi-Interquartile Range and Mid-Quartile Range
    
    or perform a test:
    * [ts_sign_os](../tests/test_sign_os.html#ts_sign_os) for One-Sample Sign Test
    * [ts_trinomial_os](../tests/test_trinomial_os.html#ts_trinomial_os) for One-Sample Trinomial Test
    * [ts_wilcoxon_os](../tests/test_wilcoxon_os.html#ts_wilcoxon_os) for Wilcoxon Signed Rank Test (One-Sample)
    
    References
    ----------
    Cournot, A. A. (1843). *Exposition de la théorie des chances et des probabilités*. L. Hachette.
    
    Galton, F. (1881). Report of the anthropometric committee. *Report of the British Association for the Advancement of Science, 51*, 225–272.
    
    Pacioli, L. (1523). *Summa de arithmetica geometria proportioni: Et proportionalita*. Paganino de Paganini.
    
    Porkess, R. (1991). *The HarperCollins dictionary of statistics*. HarperPerennial.
    
    Author
    ------
    Made by P. Stikker
    
    Companion website: https://PeterStatistics.com  
    YouTube channel: https://www.youtube.com/stikpet  
    Donations: https://www.patreon.com/bePatron?u=19398076
    
    Examples
    --------
    Example 1: Text Pandas Series
    >>> import pandas as pd
    >>> df2 = pd.read_csv('https://peterstatistics.com/Packages/ExampleData/StudentStatistics.csv', sep=';', low_memory=False, storage_options={'User-Agent': 'Mozilla/5.0'})
    >>> ex1 = df2['Teach_Motivate']
    >>> order = {"Fully Disagree":1, "Disagree":2, "Neither disagree nor agree":3, "Agree":4, "Fully agree":5}
    >>> me_median(ex1, levels=order)
    (np.float64(2.0), 'Disagree')
    
    Example 2: Numeric data
    >>> ex2 = [1, 1, 1, 2, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5]
    >>> me_median(ex2)
    (np.float64(4.0), '4.0')
    
    Example 3: Text data with between median
    >>> ex3 = ["a", "b", "f", "d", "e", "c"]
    >>> order = {"a":1, "b":2, "c":3, "d":4, "e":5, "f":6}
    >>> me_median(ex3, levels=order)
    (np.float64(3.5), 'between c and d')
    >>> me_median(ex3, levels=order, tieBreaker="low")
    (np.float64(3.5), 'c')
    >>> me_median(ex3, levels=order, tieBreaker="high")
    (np.float64(3.5), 'd')
    
    Example 4: Numeric data with between median
    >>> ex4 = [1, 2, 3, 4, 5, 6]
    >>> me_median(ex4)
    (np.float64(3.5), '3.5')
    >>> me_median(ex4, tieBreaker="low")
    (3, '3')
    >>> me_median(ex4, tieBreaker="high")
    (4, '4')
    
    '''
    
    if type(data) is list:
        data = pd.Series(data)
        
    # set myField
    data = data.dropna()
    
    # if no coding is used, all values must be numeric, so we can use the median functions:
    if levels is None:
        
        if tieBreaker=="low":
            medNum = median_low(data)
        elif tieBreaker=="high":
            medNum = median_high(data)
        else:
            medNum = data.median()
                
        medText = str(medNum)
        
    else:
        # make sure we get a full coding
        uniqueVals = data.unique()
        fullCoding = levels
        for i in uniqueVals:
            if i not in fullCoding:
                fullCoding[i] = float(i) 
        
        # replace the values in the field with the numeric codes
        pd.set_option('future.no_silent_downcasting', True)
        data3 = data.map(fullCoding).astype('Int8')
        
        # now find the numeric value of the median
        medNum = data3.median()
                
        keys = list(fullCoding.keys())
        values = list(fullCoding.values())
        
        # in case the numeric median is not in the coding
        if values.count(medNum) == 0:
            adf = lambda list_value : abs(list_value - medNum)
            nearest_value = min(values, key=adf)
            if nearest_value < medNum:
                lower_value = nearest_value
                lower_index = values.index(lower_value)
            else:
                higher_value = nearest_value
                lower_index = values.index(higher_value)-1
                
            if tieBreaker=="low":
                medText = str(keys[lower_index])
            elif tieBreaker=="high":
                medText = str(keys[lower_index+1])
            else:
                medText = ('between ' + str(keys[lower_index]) + ' and ' + str(keys[lower_index+1]))
                
        # if it is in the coding
        else:
            medText = str(keys[values.index(medNum)])
    
    return medNum, medText