INTRODUCTION:
Statistics
Statistics is one of the parts of mathematics in which we study about the collecting, organizing, analyzing, interpreting and presenting data.
Statistics is very helpful in real life situations as it is easy to understand if we represent a data in a particular number which represents all numbers. This number is called the measure of central tendency. Some of the central tendencies commonly in use are:
Mean:
It is the average of ‘n’ numbers, which is calculated by dividing the sum of all the numbers by n.
The mean x of n values x1, x2, x3, …… xn is given by
x = (x1+ x2,+ x3……xn)/n
Median:
If we arrange the numbers in an ascending or descending order then the middle number of the series will be median. If the number of series is even then the median will be the average of two middle numbers.
If n is odd then the median is the average of ( n/2 )th and [(n+1)/2]th
Mean of Grouped Data: (Without Class Interval)
If the data is organized in such a way that there is no class interval then we can calculate the mean by
x = (f_1 x1+ f2 x2+ f3 x3 ……fn xn)/(f1 + f2 + f3 ……fn )
= (∑_(i = 1)nf_(ix1) ) / (∑_(i = 1) nf1 )
where x1, x2, x1, …… xn are the observations.
f1, f2, f3, …… fn are the respective frequencies of the given observations.
1. The mean for grouped data can be found by :
( i ) The direct method x = (∑f1 x1 )/(∑f1)
(ii) the assumed mean method = x = a + (∑f1 x1 )/(∑f1)/ (∑f1 x1 )/(∑f1)
(iii) the step deviation method x = a + (∑f1 x1 )/(∑f1) × h
2. The mode for grouped data can be found by using the formula:
Mode = l + (f1 − f0) / (2f1 − f0 − f2 ) × h where symbols have their usual meanings.
3. The cumulative frequency of a class is the frequency obtained by adding the frequencies of all the classes preceding the given p class.
4.The median for grouped data is formed by using the formula:
Median = l + (n/2 − cf)/f ×h where symbols have their usual meanings.
5. Representing a cumulative frequency distribution graphically as a cumulative frequency curve, or an ogive of the less than type and of the more than type.
6. The median of grouped data can be obtained graphically as the x-coordinate of the point of intersection of the two ogives for this data.