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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:
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_{d} | M_{0} | ||||||||
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
Types of Measures of Dispersion:
Ways of Measuring Dispersion
Taken reference from
( Basic mathematics Grade XII and A foundation of Mathematics Volume II and Wikipedia.com )
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