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Types of Averages

Measures of central tendency are divided into two categories:
  1. Three types of mathematical averages are
    1. arithmetic mean,
    2. geometric mean, and
    3. harmonic mean.
  2. Five types of positional averages are
    1. median,
    2. quartiles,
    3. deciles,
    4. percentiles, and
    5. mode.

Arithmetic mean

Individual observations: 94768.png = sum of items, N = number of observations
Discrete series: 94779.pngf = frequency, N = Σf
Short-cut method: 94786.pngd = X - AN = ΣfA = assumed mean
Continuous series: 94792.pngm = mid-values of various classes, N = Σf
Short-cut method: 94798.pngA = assumed mean, d = x - AN = Σf
Combined mean: If a series of N observation consists of two components, with means 94807.png and number of items N1 and N2, then combined mean is


Median is defined as the central value of set of observations. In order to calculate median, first of all, arrange the data in ascending or descending order of magnitude of the observations.
Individual observation: If N is odd, then median = size of (N + 1)/2th item. If N is even, median = average of N/2th and 94819.pngitems.
Discrete series: First, arrange the data in ascending or descending order, find cumulative frequencies, then median is the size of the observation which lies in the class having cumulative frequency just greater than N/2.
Continuous series: Median class is the class corresponding to cumulative frequency just greater than N/2 and median is given by the formula
Median = 94839.png
where l = lower limit of median class, c.f. = cumulative frequency of class preceding to the median class, f = frequency of median class, and i = class interval of median class.


It is that value of the variable, which occurs greatest number of times, i.e., variable with maximum frequency. In case of a discrete frequency distribution of value of mode is determined by the method of grouping. In case of a grouped or continuous frequency distribution, mode is given by the formula
Mode = l + 94850.png × h
where l = lower limit of the modal class, h = width of modal class, f1 = frequency of the class preceding the modal class, f2 = frequency of the class following the modal class, and f = frequency of the modal class.


Note: If there are two observations (or modal classes) with the same maximum frequency, then the mode can be found by using the formula (known as empirical formula). Mode = 3 Median - 2 Mean

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