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# Central Tendency

A measure of central tendency is a typical value around which other figures congregate. A measure of central tendency is a single value which represent the entire data.

The various averages in the measure of central tendency are:

1. Mathematical averages - Arithmetic mean, Geometric mean and Harmonic mean.
2. Averages of position - Median and Mode.

Mathematical averages

a.  Arithmetic mean

• If x1, x2, x3,........, xn are sizes of n items, their arithmetic mean x = (x1 + x2 + x3 +........+ xn)/n
• If the data is a frequency distribution where x1, x2, x3,........, xn are the mid-values of the class intervals and f1, f2, f3,........, fn are corresponding frequencies, then x = (x1f1 + x2f2 +  x3f3 +........+ xnfn)/(f1 f2 + f3 +........+ fn)
• When the variables x1, x2, x3,........, xn may not have same importance, the weights w1, w2, w3, ........, wn are given to each of the variables, then

Weighted arithmetic mean = (x1w1 + x2w2 +  x3w3 + ........ + xnwn)/(w1 + w2 +  w3 + ........ + wn)

• If x1 and x2 be the means of two series n1 and n2 observations respectively, then the

combined arithmetic mean = (n1x1 + n2x2)/(n1 + n2)

b.  Geometric mean

For raw data:

If x1, x2, x3, ........, xn are n observations, then their Geometric mean =  root(n)(x_(1)* x_(2)*x_(3)....x_(n))

For frequencies distribution:

Geometric mean = root(N)'(x1 f1 * x2 f2 * x3 f3 *........ * xn fn), where N = f1 f2 +  f3 + ........ + fn

c.  Harmonic mean

For raw data:

If x1, x2, x3, ........, xn are n observations, then their 1 /Harmonic mean = (1/x1+ 1/ x2+ 1/ x3 +........+ 1/xn )/(1/n) .

For frequencies distribution:

Harmonic mean = (f1/x1+ f2/ x2+ f3/ x3 +........+ fn/xn )/(1/N) where N = f1 f2 +  f3 + ........ + fn .

Median and Mode

Median

Median is the size of the middle most term.

For raw data:

If x1, x2, x3, ......... , xn are arranged in ascending order of magnitude, then the median is the size of (n + 1)/2 th item.

For frequencies distribution:

Median = l +  ((N/2-m)c)/(f)

where

l is the lower boundary of the median class

m is the cumulative frequency upto median class

c is the class interval

f is the frequency of median class

N is the total frequency

Mode

Mode is the value which occurs most frequently.

For raw data:

Mode is the item that has highest frequency.

For frequencies distribution:

Mode = l + ( Delta 1/(Delta1 + Delta 2)) * c

where

l is the lower boundary of the mode class

c is the class interval

Delta 1 = f - f1

Delta 2 = f - f2

f is the frequency of mode class

f1 is the frequency of the class preceding the model class

f2 is the frequency of the class succeeding the model class.

 More topics in  Central Tendency Dispersion Median Mean Mode
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