# Central Tendency

Central tendency may be defined as the tendency of a given set of observations to cluster around a single, central or a middle value. It means that a single or central value can give adequate idea about the data or the set of observations. This value, which represents the given set of observations, is described as the average value or the measure of central tendency.# Measures of Central Tendency

Following are the different measures of central tendency:- Arithmetic Mean (A.M.)
- Median (Me)
- Mode (Mo or
*Z*) - Geometric Mean (G.M.)
- Harmonic Mean (H.M.)

# Criteria for an Ideal Measure of Central Tendency

Following are the criteria for a measure of central tendency:- It should be clearly defined.
- It should be easy to understand and interpret.
- Its computation should be simple.
- It should be based on all observations of the data.
- It should not be affected by extreme observations.

# Arithmetic Mean (A.M.)

If*x*

_{1, }

*x*

_{2, }

*x*

_{3 }â€¦

*x*are the observations of an ungrouped data, then the sum of all observations divided by the number of observations is called the

_{n}*arithmetic mean*and is denoted by.

- Arithmetic Mean of the given observations can be found using the relation
ExampleFind the arithmetic mean of 15, 20 and 25.Solution
*x*_{1}= 15,*x*_{2}= 20,*x*_{3}= 25,*n*= 3 - When there is a simple frequency distribution relating to an attribute, we have

**Note:** The above formula can be used in case of grouped frequency distribution with x_{i} as the mid value of the *i*th class interval.

Example

What is the arithmetic mean for the given data?

Marks |
5â€“14 | 15â€“24 | 25â€“34 | 35â€“44 | 45â€“54 | 55â€“64 |

No. of students |
10 | 18 | 32 | 26 | 14 | 10 |

Solution

Let us add the columns for

*x*_{i}and*f*_{i }*x*_{i}values in the table.Marks |
No. of students |
Mid value (x_{i}) |
f_{i} x_{i} |

14 | 10 | 9.5 | 95 |

15â€“24 | 18 | 19.5 | 351 |

25â€“34 | 32 | 29.5 | 944 |

35â€“44 | 26 | 39.5 | 1027 |

45â€“54 | 14 | 49.5 | 693 |

55â€“64 | 10 | 59.5 | 595 |

Total |

- We can use the following equation to find the A.M. for a grouped frequency distribution in order to simplify the calculations when the values of observations are large or tedious.
*x*value corresponding to the middle-most class interval. In case of even number of intervals, any of the two middle values can be taken as the assumed mean._{i}

ExampleFind the mean for the following frequency distribution.SolutionFirst, let us write the given inclusive table as exclusive table.*C*= 369.5 - 349.5 = 20*A*= 419.5

# Properties of Arithmetic Mean

- The sum of deviations taken from the arithmetic mean is zero.
- Arithmetic mean is affected due to the change of origin and scale.
*y*=*a*+*bx*be the relation between the original variable*x*and the variable*y*obtained when there is a change of origin â€˜*a*â€™ and a change of scale â€˜*b*â€™; then the A.M. of*y*is given by - If there are three groups containing
*n*_{1,}*n*_{2}and*n*_{3}observations and their respective arithmetic means are*then the combined mean can be expressed as*ExampleThe arithmetic mean of marks scored by 30 girls of a class is 44%. The arithmetic mean for 50 boys is 42%. Find the arithmetic mean for the whole class.SolutionHere, - If each observation of the variable
*x*is increased or decreased by a constant*c*,*c.***Example:**Consider 3 numbers 5, 10 and 15.*The A.M. of the above numbers is**Let us increase the given numbers by a quantity 3. The new set of numbers will be 5 + 3, 10 + 3 and 15 + 3,**i.e.*, 8, 13 and 18.*The A.M. of 8, 13 and 18 is**We can see that the A.M. is also increased by the same quantity as the numbers.* - If each observation of the variable
*x*is multiplied or divided by a constant*k*, then the resulting A.M. is obtained by multiplying or dividing the original A.M. by the same constant.**Example:**Consider 4 numbers: 2, 7, 12 and 15.*i.e.*, 4, 14, 24 and 30.The A.M. of 4, 14, 24 and 30 is which is 9 Ã— 2.

# Merits

- It can be easily calculated.
- Its calculations are based on all the observations.
- It is easy to understand.
- It is rigidly defined by the mathematical formula.
- It is least affected by sampling fluctuations.
- It is best measure to compare two or more series (data).
- It is the average obtained by calculations and it does not depend upon any position.

# Demerits

- It may not be represented in actual data and so it is theoretical.
- The extreme values have greater effect on mean.
- It cannot be calculated if all the values are not known.
- It cannot be determined for the qualitative data such as love, beauty, honesty, etc.
- Mean may lead to fallacious conditions in the absence of original observations.