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 meandenoted by of observations is given by

The symbol ∑ (capital sigma) is a Greek letter; but here, it is used to simplify the way we write summations.

means and is read as “summation i=1 to n, ”.

**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,

.

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 meandenoted by of observations with frequencies is given by

**Example 2:**

x: | 1 | 2 | 3 | 4 | 5 | 6 | 7 |

f: | 5 | 9 | 12 | 17 | 14 | 10 | 6 |

**Solution**:

x | f | fx |

1 | 5 | 1 x 5 = 5 |

2 | 9 | 2 x 9 = 18 |

3 | 12 | 36 |

4 | 17 | 68 |

5 | 14 | 70 |

6 | 10 | 60 |

7 | 6 | 42 |

Thus, the arithmetic mean,

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

:x | 1 | 3 | 4 | 5 | 6 |

f: | 3 | 2 | 1 | 1 | 1 |

Can you find the average using the second formula?

x | f | fx |

1 | ||

3 | ||

4 | ||

5 | ||

6 | ||

∑ f =_____ | ∑ fx = _____ |

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

_______

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: | 0-10 | 10-20 | 20-30 | 30-40 | 40-50 | 50-60 |

No. of Students: | 12 | 18 | 27 | 20 | 17 | 6 |

**Solution:**

Marks | No. of Students () f | Mid-point ()x | fx |

0-10 | 12 | 5 | 60 |

10-20 | 18 | 15 | 270 |

20-30 | 27 | 25 | 675 |

30-40 | 20 | 35 | 700 |

40-50 | 17 | 45 | 765 |

50-60 | 6 | 55 | 330 |

∑ f =__________ | ∑ fx = _________ |

Thus, arithmetic mean,

_______

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

Marks: | 0-20 | 20-40 | 40-60 | 60-80 | 80-100 |

No. of Students: | 13 | 12 | 13 | 12 | 2 |

#### Weighted Arithmetic Mean

Weighted Arithmetic meandenoted by of observations x_{i}with weights w_{i}is given by

**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.

x | w | wx |

1 | 1 | |

2 | 2 | |

3 | 3 | |

4 | 4 | |

5 | 5 | |

6 | 6 | |

Sum | ∑w = | ∑wx = |

Weighted arithmetic mean =

#### Properties of Arithmetic Mean

- Sum of the is zero. i.e.,
- The sum of the squared of the deviations of a set of values is minimum when taken about mean.
- (Mean of Composite Series) If (i=1,2,3,…k) are the means of -series of sizes
*n*(i=1,2,3,…k) respectively, then the mean of the composite series obtained on combining these series is given by the formula_{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.

Requisite | Merit | Demerit | Notes |

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, *x _{1}, x_{2}, x_{3}, …, x_{n}*.

Geometric mean is the nth root of their product.

and Harmonic mean,

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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