Some definitions for Measures of central tendency:
" An average may be thought of as a measure of central value " - John I. Griffin
" The inherit inability of human to group in its entirely a large body of numerical data compels us to seek relatively few constants that will adequately show the data " - R. A. Fisher
" Averages are statistical constants which enable us to comprehend in a single effort the significance of the whole " - A. L. Bowley
Properties of central tendency:
- It should be rigidly defined.
- It should be simple and easy to calculate.
- It depends on all the observations.
- It should be suitable for further mathematical treatment.
- It should be easily located from the graph.
- It should not be much affected by the extreme observations.
- Arithmetic mean
- Geometric mean
- Harmonic mean
Positional averages:
- Median
- Mode
- Quartiles
- Quintiles
- Octiles
- Deciles
- Percentiles.
Commercial averages:
- Moving average
- Progressive average
- Composite average.
The five measures of central tendency that uses very commonly.
- Arithmetic mean
- Median
- Mode
- Geometric mean
- Harmonic mean
Arithmetic Mean: It is set of observations is their sum divided by the number of observations.
Merits for mean:
- It is rigidly defined.
- It is easy to calculate.
- It is based upon all observations.
- It is amenable to algebraic treatment.
- Of all averages, mean is affected least by fluctuations of sampling. This property is sometimes described by saying that mean is a stable average
Demerits for mean:
- It cannot be determined by inspection nor it can be located graphically.
- Mean cannot be used if we are dealing with qualitative characteristics which cannot be measured quantitatively.
- Mean cannot be obtained if a single observation is missing or lost or illegible unless we drop it out and compute the mean of the remaining values.
- Mean cannot be calculated if the extreme class is open.
- In extremely asymmetrical distribution, usually mean is not a suitable measure of location.
Median: Median of a distribution is the value of the variable which divides it into two equal parts. It is the value which exceeds and is exceeded by the same number of observations. The median is a positional average.
- It is rigidly defined.
- It is simple and easy to calculate.
- It can be located from the graph.
- It does not depend on all the observations.
- It is not suitable for further mathematical treatment.
- In some situations, median is affected by extreme observations.
Mode: Mode is the value which occurs most frequently in a set of observations and around which the other items of the set cluster densely. In the other words, mode is the value of the variable which is predominantly in the series.
- It is rigidly defined.
- It is simple and easy to calculate.
- It is based on all the observations.
- It can be located from graph.
- It is not affected by the extreme observations.
- It is not rigidly defined.
- It does not depend on all the observations.
- It is not suitable for further mathematical treatment.
Geometric mean: It is a set of n observations is the nth root of their product.
- It is rigidly defined.
- It depends on all the observations.
- It is not affected by the extreme observations.
- The calculation of GM is not simple and easy.
- It cannot be located from the graph
Harmonic mean: Harmonic mean of a number of observations none of which is zero, is the reciprocal of the AM of the reciprocals of the given values.
- It is rigidly defined.
- It depends on all the observations.
- It is not affected by the extreme observations.
- It is not simple and easy.
- It cannot be located from the graph.
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