Subject: Mathematics

**Median**

Arrange the data either in ascending or descending order and use the given or following formula for the respective series to find the median

$$Individual\;Series:\;Median=M_d=\bigg(\frac{N+1}{2}\bigg)^{th}\;item$$

$$Discrete\;series\;Median=M_d=\bigg(\frac{N+1}{2}\bigg)^{th}\;item$$

Continuous Series

To find the median class, the median lies in

$$the\;\bigg(\frac{N+}{2}\bigg)^{th}\;item\;then\;$$

$$Median=M_d=L+\frac{\frac{N}{2}-c.f.}{f}\times\;h$$

Here, L = lower limit of the modal class

c.f = cumulative frequency

f = frequency of the corresponding class

h = class interval

**Quartiles:**

The variate values which divides the total observation into four equal parts are called quartiles. They are denoted by Q_{1}, Q_{2}, Q_{3} as the three quartiles. Q_{2} is the median.

For Individual and discrete series:

Arrange the data in either ascending or descending order and then use the formula,

$$Quartiles=Q_i=\bigg[\frac{i(N+1)}{4}\bigg]^{th}\;item,\;i=1\;for\;first\;quartiles\;2\;for\;median,\;3\;for\;upper\;quartile\;$$

For continuous series,

$$Quartiles=Q_i=L+\frac{\frac{iN}{4}-c.f.}{f}\times\;h\;where\;i=1\;for\;first\;quartiles\;2\;for\;median,\;3\;for\;upper\;quartile\;$$

**Mode**

Mode is the value of variates which occurs most frequently in the series.

Continuous series

$$Mode=M_0=L+\frac{f_1-f_0}{2f_1-f_0-f_2}\times\;h$$

Here, L = lower limit of the modal class

f_{1}= maximum frequency

f_{0}= frequency of preceding modal class

f_{1}= frequency of Succeeding modal class

h = Size of modal class

**Relation between various measures of central tendency:**

- For symmetrical distribution , mean = median – mode
- If the distribution is not symmetrical, then

Mean – Mode = 3(Mean - Mode)

Or Mode = 3 Median – 2 Mode

This relation is also known as the empirical relation.

**Measures and Meaning of Dispersion **

Measures of central tendency reveal only one characteristic of the data, i.e. the point of central tendency. It fails to give us the idea, about the extent to which the item of the distribution deviates from a central value. The averages only cannot give the clear information or picture about the distribution because two distributions having the same averages may have different variations in the items from the central value. We obtain this information by the study of dispersion.

A measure of dispersion is a descriptive statistical device used to measure the variation or spread or the scattering of the data in the distribution. It is also known as the measure of variation.

For this we can see the following two series:

$$\overline{X}$$ | M | M | ||||||||

A | 25 | 26 | 27 | 27 | 27 | 28 | 29 | 27 | 27 | 27 |

B | 5 | 10 | 18 | 27 | 27 | 27 | 75 | 27 | 27 | 27 |

From the above table, we can that mean median and mode of the two series A and series B are same. Only with these results, we cannot say that the two series A and B are the same. Because the two series A and B may differently be constituted. That is, the differences between the least and greatest items from central value may be different. That is, the scattering of the extreme data from the central value may differ.

**From Series A: **The difference of the least value from the central value = 27 – 25 = 2 and the difference of the greatest value from the central value = 29 – 27 = 2 . Since the differences are so small, so the items are close to the central value. That is, the items are not so scattered from the central value.

**From Series B: **The difference of the least value from the central value = 27 – 5 = 22 and the difference of the greatest value from the central value = 75 – 27 = 48 . Since the differences are so large, so the items are far to the central value. That is, the items are more scattered from the central value.

Thus the two series A and B have the same averages but the scattering of the items from the central values are different. These results show that the two series are differently formed.

Thus the meaning of dispersion is the scattering of the items from the central value. Hence, Dispersion is defined as the measure of scattering or variation of the items from the central value.

**Requirements of a good measure of Dispersion:**

For measure of dispersion to be classified as a good measure of dispersion, It must have the following characteristics

- The measure should be rigidly defined.
- The measure should be simple to understand and easy to calculate.
- All items must be included in the measure.
- The measure must be suitable for further mathematical treatment.
- It should be least fluctuated from sample to sample drawn from the same population.
- Extreme values should not unduly affect the measure.

**Types of Measures of Dispersion:**

- Absolute Dispersion
- Relative Dispersion

**Ways of Measuring Dispersion**

- Range
- Quartile Deviation [ or Semi-Interquartile Range ]
- Mean Deviation
- Standard Deviation

Taken reference from

( Basic mathematics Grade XII and A foundation of Mathematics Volume II and Wikipedia.com )

- Measures of central tendency reveal only one characteristic of the data, i.e. the point of central tendency. It fails to give us the idea, about the extent to which the item of the distribution deviates from a central value. The averages only cannot give the clear information or picture about the distribution because two distributions having the same averages may have different variations in the items from the central value. We obtain this information by the study of dispersion.
- The formulae of mean, median and mode are very important during problem solving of this chapter.

- It includes every relationship which established among the people.
- There can be more than one community in a society. Community smaller than society.
- It is a network of social relationships which cannot see or touched.
- common interests and common objectives are not necessary for society.

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