# Measure of Central Tendency and Dispersion Formulas

**Measure of Central Tendency and Dispersion Formulas:** In statistics, a central tendency or a measure of central tendency is a central or typical value for a probability distribution. We always use this Measure of Central Tendency and Dispersion concept to categorize the set of given data and find the average values along with the near and far values to average.

The most common measure of central tendency and dispersion concepts are mean, mode, median, standard deviation, etc. For easy calculation of all these concepts, we have gathered the list of Measure of Central Tendency and Dispersion Formulas here.

## Cheat Sheet for Measure of Central Tendency and Dispersion Formulas

Make the most out of this Measure of Central Tendency and Dispersion Formulas sheet and remember all concepts formulas in mind by practicing and studying them on a regular basis. You may also refer to the Measure of Central Tendency and Dispersion Formulas tables at any time from here and solve your complex problems in a fraction of seconds.

**1. Arithmetic Mean**

(i) Individual observation or unclassified data:

If x_{1}, x_{2},…….x_{n} be n observations, then their arithmetic mean is given by

\(\bar{x}=\frac{x_{1}+x_{2}+\ldots . .+x_{n}}{n} \text { or } \bar{x}=\frac{\sum_{i=1}^{n} x_{i}}{n}\)

(ii) Arithmetic Mean of discrete frequency distribution:

Let x_{1}, x_{2},…….x_{n} be n observation and let f_{1}, f_{2},…….f_{n} be their corresponding frequencies, then their mean

\(\bar{x}=\frac{f_{1} x_{1}+f_{2} x_{2}+\ldots \ldots+f_{n} x_{n}}{f_{1}+f_{2}+\ldots . .+f_{n}} \text { or } \bar{x}=\frac{\sum_{i=1}^{n} f_{i} x_{i}}{\sum_{i=1}^{n} f_{i}}\)

(iii) Short cut method:

If the values of x or (and) f are large the calculation of Arithmetic Mean by the previous formula used, is quite tedious and time consuming. In such case we take the deviation from an arbitrary point A.

\(\bar{x}=A+\frac{\Sigma f_{i} d_{i}}{\Sigma f_{i}}\)

where A = Assumed mean and d_{i} = x_{i} – A = deviation for each term

**2. Weighted Arithmetic Mean**

If w_{1}, w_{2}, w_{3},…….,w_{n} are the Weight assigned to the values x_{1}, x_{2}, x_{3},…….,x_{n} respectively, then the weighted average is defined as

weighted A.M. = \(\frac{w_{1} x_{1}+w_{2} x_{2}+\ldots . .+w_{n} x_{n}}{w_{1}+w_{2}+\ldots . . .+w_{n}} \text { or } \bar{x}=\frac{\sum_{i=1}^{n} w_{i} x_{i}}{\sum_{i=1}^{n} w_{i}}\)

**3. Combined mean**

If \(\bar{x}_{1}, \bar{x}_{2}, \ldots \ldots, \bar{x}_{k}\) are the mean of k series of sizes n_{1}, n_{2}, …….,n_{k} respectively then the mean \(\bar{x}\) if the composite series is given by

\(\bar{x}=\frac{n_{1} \bar{x}_{1}+n_{2} \bar{x}_{2}+\ldots \ldots+n_{k} \bar{x}_{k}}{n_{1}+n_{2}+\ldots \ldots \ldots+n_{k}}\)

**4. Properties of Arithmetic Mean**

- In a statistical data, the sum of the deviation of items from A.M. is always zero.
- If each of the n given observation be doubled, then their mean is doubled.
- If \(\bar{x}\) is the mean of x
_{1}, x_{2}, …….,x_{n}. The mean of ax_{1}, ax_{2}, …….,ax_{n}is a \(\bar{x}\) where a is any number different from zero. - Arithmetic Mean is independent of origin i.e. it is not effected by any change in origin.

**5. Geometric Mean**

(i) Individual data:

If x_{1}, x_{2}, x_{3},…….,x_{n} are n values of a variate x, none of them being zero, then the Geometric Mean G.M. is defined as

G.M. = (x_{1} x_{2} x_{3}…….x_{n})^{1/n}

or G.M. = antilog \(\left(\frac{\log x_{1}+\log x_{2}+\ldots \ldots+\log x_{n}}{n}\right)\)

or G.M. = antilog = \(\left(\frac{1}{n} \sum_{i=1}^{n} \log x_{i}\right)\)

(ii) Geometric Mean of grouped data:

Let x_{1}, x_{2},…….,x_{n} be n observation and let f_{1}, f_{2},…….,f_{n} be their corresponding frequency then their Geometric Mean is

G.M. = \(\left(x_{1}^{f_{1}} x_{2}^{f_{2}} \ldots x_{n}^{f_{n}}\right)^{1/ N}\) where N = \(\sum_{i=1}^{n} f_{i}\)

∴ G = antilog \(\left(\frac{\sum_{i=1}^{n} f_{i} \log x_{i}}{\sum_{i=1}^{n} f_{i}}\right)\)

**6. Harmonic Mean**

Harmonic Mean is reciprocal of arithmetic mean of reciprocals,

(i) Individual observation:

The H.M. of x_{1}, x_{2},…….,x_{n} of n observation is given by

H.M. = \(\frac{n}{\frac{1}{x_{1}}+\frac{1}{x_{2}}+\ldots .+\frac{1}{x_{n}}}\) or H.M. = \(\frac{n}{\sum_{i=1}^{n} \frac{1}{x_{i}}}\)

(ii) H.M. of grouped data:

Let x_{1}, x_{2},…….,x_{n} be n observation and let f_{1}, f_{2},…….,f_{n} be their corresponding frequency then H.M. \(=\frac{\sum_{i=1}^{n} f_{i}}{\sum_{i=1}^{n}\left(\frac{f_{i}}{x_{i}}\right)}\)

**7. Relation between A.M., G.M., and H.M.**

A.M. ≥ G.M. ≥ H.M.

Equality sign holds only when all the observations in the series are same.

8. Median of an Individual Series

Let n be the number of observations

(i) Arrange the data in ascending or descending order.

(ii) (a) If n is odd then

Median (M) = value of \(\left(\frac{n+1}{2}\right)^{t h}\) observation

(b) If n is even then

Median (M) = \(\frac{\left(\frac{\mathrm{n}}{2}\right)^{\mathrm{th}} \text { observation }+\left(\frac{\mathrm{n}}{2}+1\right)^{\mathrm{th}} \text { observation }}{2}\)

**9. Median of the discrete frequency distribution**

- Step-I: Find the cumulative frequency (C. F.)
- Step-II: Find \(\frac{N}{2}\), where N = \(\sum_{i=1}^{n} f_{i}\)
- Step-Ill: See the cumulative frequency (C.F.) just greater than \(\frac{N}{2}\) and determine the corresponding value of the variable.
- Step-IV: The value obtained in step III is the median

**10. Median of grouped data or continuous series**

Let the number of observations be n

(i) prepare the cumulative frequency table

(ii) find the median class i.e. the class in which the \(\frac{N}{2}\)^{th} observation lies

(iii) the median value is given by the formula

Median (M) = l + \(\left[\frac{\left(\frac{\mathrm{N}}{2}\right)-\mathrm{F}}{\mathrm{f}}\right] \times \mathrm{h}\)

N = total frequency = Σf_{i}

l = lower limit of median class

f = frequency of the median class

F = cumulative frequency of the class preceding the median class

h = class interval (width) of the median class

**11. Computation of Mode**

(i) Mode for individual series:

In the case of individual series, the value which is repeated maximum number of times is the mode of the series.

(ii) Mode for grouped data (discrete frequency distribution series):

In the case of discrete frequency distribution, mode is the value of the variate corresponding to the maximum frequency.

**12. Mode for continuous frequency distribution**

(i) First find the model class i.e. the class which has maximum frequency. The model class can be determined either by inspecting or with the help of grouping data.

(ii) The mode is given by the formula

Mode = l + \(\frac{f_{m}-f_{m-1}}{2 f_{m}-f_{m-1}-f_{m+1}} \times h\); where

l → lower limit of the model class

h → width of the model classj

f_{m} → frequency of the model class

f_{m-1} → frequency of the class preceding model class

f_{m+1} → frequency of the class succeeding model class

(iii) In case the model value lies in a class other than the one i containing maximum frequency (model class) then we use the j following formula

Mode = l + \(\frac{f_{m+1}}{f_{m-1}+f_{m+1}} \times h\)

**13. Relationship between Mean, Mode, and Median**

- In symmetrical distribution: Mean = Mode = Median
- In skew (moderately symmetrical) distribution: Mode = 3 Median – 2 Mean

**14. Mean Deviation of individual observations**

If x_{1}, x_{2},…….,x_{n} are n values of a variable x, then the Mean Deviation from an average A (median or AM) is given by M.D. = \(\frac{1}{n} \sum_{i=1}^{n}\left|x_{i}-A\right|\)

**15. Mean Deviation of a discrete frequency distribution**

If x_{1}, x_{2},…….,x_{n} are n observation with frequencies f_{1}, f_{2},…….,f_{n} then mean deviation from an average A is given by

Mean Deviation = \(\frac{1}{N}\)Σf_{i}|x_{i} – A| where N = \(\sum_{i=1}^{n} f_{i}\)

**16. Mean Deviation of a grouped or continuous frequency distribution**

For calculating mean deviation of a continuous frequency distribution the procedure is same as for a discrete frequency distribution. The only difference is that here we have to obtain the mid-point of the various classes and take the deviations of these mid points from the given central value (median or mean).

**17. Variance of Individual observations**

If x_{1}, x_{2},…….,x_{n} are n values of a variable x, then by definition

var (r) = \(\frac{1}{n}\left[\sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)^{2}\right]=\sigma^{2}\) ……..(i)

or var (x) = \(\frac{1}{n} \sum_{i=1}^{n} x_{1}^{2}-\bar{x}^{2}\) …….(ii)

**18. Variance of a discrete frequency distribution**

If x_{1}, x_{2},…….,x_{n} are n observations with frequencies f_{1}, f_{2},…….,f_{n} then

var (x) = \(\frac{1}{N}\left\{\sum_{i=1}^{n} f_{i}\left(x_{i}-\bar{x}\right)^{2}\right\}\) or var (x) =\(\frac{1}{N}\left(\frac{\beta}{\alpha}\right)\)

**19. Standard Deviation : Standard Deviation = + \(\sqrt{var}(\mathrm{x})\)**