Arithmetic, Geometric and Harmonic Mean (Statistics for Psychologists)

These notes are primarily posted for BSc Psychology (Research Methodology) and BSc Clinical Psychology (Statistics) students of Martin Luther Christian University, Shillong. Ideas are borrowed mainly from Gupta & Kapoor (link) (Textbook) and Sahoo (link).

Means

Arithmetic mean

Arithmetic mean is defined as the sum of all observations in a data series divided by the count of all observations in that series. Mathematically,

Arithmetic mean denoted by \bar{x}  of n observations x_1,x_2,\cdots,x_n is given by \bar{x}={x_1+x_2+\cdots+x_n\over n}={\sum_{i=1}^n x_i\over n}

The symbol ∑ (capital sigma) is a Greek letter; but here, it is used to simplify the way we write summations.
\sum_{i=1}^n x_i means x_1+x_2+\cdots+x_n\over n and is read as “summation i=1  to n, x_i”.

Example 1: Find the arithmetic mean of 1,3,4,3,5,6,1,1
Solution: Note that there are 8 numbers, hence, n=8. Then,
\bar{x}={1+3+4+3+5+6+1+1\over8}=3.


One of the ways we write the data more concisely is using the idea of ‘frequency’ of a particular number. Here’s the definition of arithmetic mean, in case of ‘frequency distribution’:

Arithmetic mean denoted by \bar{x}  of n observations x_i with frequencies f_i is given by
\bar{x}={f_1x_1+f_2x_2+\cdots+f_nx_n\over f_1+f_2+\cdots+f_n}={\sum_{i=1}^n f_ix_i\over \sum_{i=1}^n f_i}

Example 2:

x:1234567
f:59121714106
Find the arithmetic mean of the above frequency distribution.

Solution:

xffx
15 1 x 5 = 5
292 x 9 = 18
31236
41768
51470
61060
7642
 \sum f = 73\sum fx = 299

Thus, the arithmetic mean,

\bar{x}={\sum_{i=1}^n f_ix_i\over \sum_{i=1}^n f_i}={299\over73}=4.09

Example 3: Note Example 1. The given data can be re-written more neatly as:

x:13456
f:32111

Can you find the average using the second formula?

xffx
1
3
4
5
6
  f =_____fx = _____

Thus, arithmetic mean, (answer will be the same as Example 1)

\bar{x}={\sum_{i=1}^n f_ix_i\over \sum_{i=1}^n f_i}=_______

If the data is given in ranges, we take the midpoint of each range as our x for calculation.

Example 4: Calculate the arithmetic mean of the marks from the following table:

Marks:0-1010-2020-3030-4040-5050-60
No. of Students:12182720176

Solution:

MarksNo. of Students (f) Mid-point (x)fx
0-1012560
10-201815270
20-302725675
30-402035700
40-501745765
50-60655330
 f =__________ fx = _________

Thus, arithmetic mean,

\bar{x}={\sum_{i=1}^n f_ix_i\over \sum_{i=1}^n f_i}=_______

Example 5: Calculate the arithmetic mean from the following table:

Marks:0-2020-4040-6060-8080-100
No. of Students:131213122

Weighted Arithmetic Mean

Weighted Arithmetic mean denoted by \bar{x}  of n observations xi with weights wi is given by

\bar{x}={w_1x_1+w_2x_2+\cdots+w_nx_n\over w_1+w_2+\cdots+w_n}={\sum_{i=1}^n w_ix_i\over \sum_{i=1}^n w_i}

Example 6: Find the simple and weighted arithmetic mean of the first 6 natural numbers, the weight being the corresponding numbers.

Solution: The first 6 natural number are 1, 2, 3, 4, 5, 6.

xwwx
11 
22 
33 
44 
55 
66 
Sum∑w =   ∑wx =
Fill in the details and finish the problem!

Weighted arithmetic mean =

Properties of Arithmetic Mean

  1. Sum of the  x_i - \bar{x} is zero. i.e., \sum_{i=1}^n{f_i [x_i-\bar x]}=0
  2. The sum of the squared of the deviations of a set of values is minimum when taken about mean.
  3. (Mean of Composite Series) If \bar{x}_i (i=1,2,3,…k) are the means of -series of sizes ni (i=1,2,3,…k) respectively, then the mean  of the composite series obtained on combining these series is given by the formula

\bar x = {n_1 \bar x_1+ n_2 \bar x_2 +\cdots+ n_k \bar x_k \over n_1+\cdots+n_k}={\sum_i n_i\bar x_i\over\sum_i n_i}

The last property might be useful in future, in your research and entrance examinations for higher studies!

Merits and Demerits of Arithmetic Mean

Remember the discussion about characteristics of an ideal central measure? Let’s see how ‘good’ arithmetic mean is.

RequisiteMeritDemeritNotes
It is rigidly defined. Has a mathematical definition
It is easy to understand and calculate. It is in fact the easiest central tendency to calculate and understand.
It is based on all the observations.  
It is suitable for further mathematical treatment. We already saw in property 3 how it is fit for further mathematical treatment.
It is affected as little as possible by fluctuations of sampling.  Among central tendencies, arithmetic mean is the one which is least affected by fluctuations in sampling.
It is affected by extreme observations. In case of extreme items, arithmetic mean might give misleading results. Say, consider the distribution 1, 2, 1, 100 – mean is not near any of the observations.
Cannot be used for Nominal Scale observations  
Cannot be determined by inspection or graphically  
If the values are too distributed, mean might not give useful insights Consider a set of observation with 4 with frequency 10 and 99 with frequency 15. Mean will be between 50-60, which is less insightful.

Remember requisites for ideal central tendency we discussed earlier? Turns our arithmetic mean is not ‘ideal’ – it has its own merits and demerits. That’s why we have different central tendencies, which will be discussed below.

There are two more mathematical means which are useful in a few cases, called Geometric Mean and Harmonic mean. But Arithmetic mean is the easiest to understand, easiest to calculate and most widely used. Hence, if we simply say ‘mean’ of a distribution, we are referring to Arithmetic Mean.

Geometric and Harmonic Mean

Consider a distribution with n observations, x1, x2, x3, …, xn.

Geometric mean is the nth root of their product.

G = [x_1\cdot x_2 \times \cdots \cdot x_n]^\frac1n

and Harmonic mean, H = \frac1{\frac1n(\frac1{x_1}+\frac1{x_2}+\cdots+\frac1{x_n})}

Both are rigidly defined, but mathematically hard to calculate and understand. Also, is one of the observations is zero, then G becomes zero and H becomes impossible to calculate!

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