Dear Dhritikesh

Arithmetic mean is not only the single measure to check the central tendency. there are other two measures median and mode which are important to check the central tendency. When all the three measures Mean, Median, Moe are almost alike we will assume that the data is normal. besides there are other measures to check the central tendency like kurtosis and Skewness and also the ( Z value shall be plus mines 1.96)

@ Mehraj Bhat ,

It is correct

(i) that arithmetic mean , median and mode are important to check the central tendency of data

and (ii) that when all the three measures namely mean , median & mode are equal then the distribution of the data is normal (symmetric).

However, the vital question , raised by me , is whether mean is the proper measure of the central tendency of data i.e. whether mean can give the actual value of the central tendency of data.

As almost all of the the data we dealed in real life is actually skewed, arithmetic mean is very rarely capable to represent central tendency;

Article TO DETERMINE SKEWNESS, MEAN AND DEVIATION WITH A NEW APPROAC...

Arithmetic mean is the ideal measure of central tendency when there are no extreme high and low values called outliers. With outliers, Median and mode are applicable. The arithmetic mean has further advantage of being used for additional analysis like standard deviation, variance and standard error of mean. Also useful in deriving confidence interval.

@ Mehmet Guven Gunver ,

This is true that as almost all of the the data we deal with real life is actually skewed, arithmetic mean is very rarely capable to represent central tendency.

However, I have not found any logic/ground on the arithmetic mean that it can yield the actual value of the central tendency of data even if the data are free from errors/fluctuations.

@ Taiwo Sogunle ,

This is true that arithmetic mean is the ideal measure of central tendency when there are no extreme high and low values called outliers and that the arithmetic mean has further advantage of being used for additional analysis like standard deviation, variance and standard error of mean.

But, can this be a logic/ground/cause that the arithmetic mean is a proper measure of central tendency of data. Moreover, I have not found any logic/ground that the arithmetic mean can be a proper measure of central tendency of data i.e. the arithmetic mean can yield the actual value of the central tendency of data.

There two issues here.

1. With normally distributed data the arithmetic mean provides ideal measure of central tendency

2. With non normal or skewed data, median and mode come into play

@ Taiwo Sogunle ,

It may be true that (i) with normally distributed data the arithmetic mean provides ideal measure of central tendency and (ii) with non normal or skewed data, median and mode come into play.

But, dose it imply that the arithmetic mean is a proper measure of central tendency of data. Of course, I am not convinced to accept that the arithmetic mean can be a proper measure of central tendency of data i.e. the arithmetic mean can yield the actual value of the central tendency of data.

The mean is usually the best measure of central tendency to use when your data distribution is continuous and symmetrical, such as when your data is normally distributed. However, it all depends on what you are trying to show from your data.

@ Zaffar Ahmed Nadaf ,

It is true that the arithmetic mean is the best (ideal) measure of central tendency of data among all the existing measures of central tendency.

However, this may not imply that arithmetic mean is the proper of of central tendency. The term "proper measure" has been used here to mean a measure which can give the actual (true) value of what we want to measure.

My thrust is to know whether arithmetic mean is such a measure (i.e. proper measure) of central tendency of data.

Thanks to Mehraj Bhat , Mehmet Guven Gunver , Taiwo Sogunle and Zaffar Ahmed Nadaf for their answers.

However, I have not yet got any hint / clue of my basic thrust on whether arithmetic mean is the proper measure of central tendency of data.

May I request any idea of this issue from the researchers / thinkers.

Dear Mercedes Orus-Lacort, Your view on the arithmetic mean as mentioned in the first paragraph in your answer make me more confident on my thinking of it.

The explanation provided in the second paragraph in your answer has enabled me to think of the practical application of arithmetic mean. Thank you very much.

Very relevant - Central limit theorem for normal distribution - https://www.investopedia.com/terms/c/central_limit_theorem.asp

@ Jaydip Datta ,

What happens if the the observations whose average is required to be determined do not follow normal distribution .

The problem or the objective is to measure the central tendency , but not the expectation , of the observations.