A measure of central tendency is a typical value around which other figures congregate. A measure of central tendency is a single value which represent the entire data.

The various averages in the measure of central tendency are:

- Mathematical averages - Arithmetic mean, Geometric mean and Harmonic mean.
- Averages of position - Median and Mode.

**Mathematical averages**

**a. Arithmetic mean**

- If x
_{1},_{ }x_{2}, x_{3},........, x_{n}are sizes of n items, their arithmetic mean x = (x_{1}+ x_{2 }+ x_{3 }+........+ x_{n})_{}/n - If the data is a frequency distribution where x
_{1},_{ }x_{2}, x_{3},........, x_{n}are the mid-values of the class intervals and f_{1},_{ }f_{2}, f_{3},........, f_{n }are corresponding frequencies, then x = (x_{1}f_{1 }+_{ }x_{2}f_{2 }+ x_{3}f_{3}+........+ x_{n}f_{n})/(f_{1}+_{ }f_{2 }+ f_{3 }+........+ f_{n}) - When the variables x
_{1},_{ }x_{2}, x_{3},........, x_{n}may not have same importance, the weights w_{1}, w_{2}, w_{3}, ........, w_{n }are given to each of the variables, then

Weighted arithmetic mean = (x_{1}w_{1} + _{ }x_{2}w_{2 }+ x_{3}w_{3} + ........ + x_{n}w_{n})/(w_{1 }+ w_{2 }+ w_{3 }+ ........ + w_{n})

- If x
_{1}and x_{2}be the means of two series n_{1}and n_{2}observations respectively, then the

combined arithmetic mean = (n_{1}x_{1} + n_{2}x_{2})/(n_{1 }+ n_{2})

b. **Geometric mean**

**For raw data:**

If x_{1},_{ }x_{2}, x_{3}, ........, x_{n} are n observations, then their Geometric mean = `root(n)(x_(1)* x_(2)*x_(3)....x_(n))`

**For frequencies distribution:**

Geometric mean = `root(N)'(`x_{1}^{} f_{1} *_{ }x_{2}^{} f_{2 }* x_{3}^{} f_{3 }*_{.}....... * x_{n} ^{}f_{n}), where N = f_{1 }+ _{ }f_{2 }+ f_{3 }+ ........ + f_{n}

c. **Harmonic mean**

**For raw data:**

If x_{1},_{ }x_{2}, x_{3}, ........, x_{n} are n observations, then their 1 /Harmonic mean = (1/x_{1}+ 1/_{ }x_{2}+ 1/ x_{3} +_{.}.......+ 1/x_{n} )/(1/n) .

**For frequencies distribution:**

Harmonic mean = (f_{1}/x_{1}+ f_{2}/_{ }x_{2}+ f_{3}/ x_{3} +_{.}.......+ f_{n}/x_{n} )/(1/N) where N = f_{1 }+ _{ }f_{2 }+ f_{3 }+ ........ + f_{n} .

**Median and Mode**

**Median**

Median is the size of the middle most term.

**For raw data:**

If x_{1}, x_{2}, x_{3}, ......... , x_{n} are arranged in ascending order of magnitude, then the median is the size of (n + 1)/2 th item**.**

**For frequencies distribution:**

Median = l + `((N/2-m)c)/(f)`

where

l is the lower boundary of the median class

m is the cumulative frequency upto median class

c is the class interval

f is the frequency of median class

N is the total frequency

**Mode**

Mode is the value which occurs most frequently.

**For raw data:**

Mode is the item that has highest frequency.

**For frequencies distribution:**

Mode = l_{} + ( `Delta` _{1}/(`Delta`_{1} + `Delta` _{2})) * c

where

l is the lower boundary of the mode class

c is the class interval

`Delta` _{1 }= f - f_{1}

`Delta` _{2 }= f - f_{2}

f is the frequency of mode class

f_{1} is the frequency of the class preceding the model class

f_{2} is the frequency of the class succeeding the model class.

More topics in Central Tendency | |

Dispersion | Median |

Mean | Mode |