Cumulative frequency distribution with example

Cumulative frequency distribution introduction:

From a grouped frequency distribution, we can simply read how many observations fall within different classes. E.g. consider the following frequency distance.

Mark:                      0 – 20             20 – 40          40 – 60          60 – 80

No. of students:         5                      25                  45                   19

The number of students who secured marks 40 or more but less than 60 are 45, those who secured marks 20 or more but less than 40 are 25. But if one wants to know the number of students who obtained 40 or more marks or those who obtained less than 60 marks, then the given frequency distribution does not serve our purpose. To answer such type of questions we construct a frequency distribution known as cumulative frequency distribution.

Definition:

The total frequency of all the classes less than the upper class boundary or move than or equal to the lower class boundary of a particular class is called the cumulative frequency of that class. A table showing the cumulative frequencies of different classes is called the cumulative frequency distribution.

There are two methods of constructing the cumulative frequency distributions.

i.                        Less than type cumulative frequency distance:

A table showing total frequencies of all the classes less than the upper class boundaries is called less than type cumulative frequency distribution. In this type the frequencies are serially added from top to bottom. The cumulative frequency of last class should equal the total frequency.

ii.                        More than type cumulative frequency distance:

A table showing total frequencies of all the classes more than equal to the lower class boundaries is called the more than type cumulative frequency distribution. In this type the frequencies are serially added from bottom to top. The cumulative frequency of first class should equal the total frequency.

Question:

From the following frequency distribution construct less than type and more than type cumulative frequency distribution.

Marks:    0 – 10     10 – 20       20 – 30         30 – 40    40 – 50      50 - 60

Frequency:   5              25                37                  43            13                7

Sol:

a)                 Less type cumulative frequency distribution

Marks	No. of students	Cumulative frequency
Below 10	5	5
Below 20	5 + 25	30
Below 30	5 + 25 +37	67
Below 40	5 + 25 + 37 + 43	110
Below 50	5 + 25 + 37 + 43 + 13	123
Below 60	5 + 25 + 37 + 43 + 13 +7	130

b)                More than type cumulative frequency distribution

Marks	No. of students	Cumulative frequency
Above 0	7 + 13 + 43 + 37 + 25 + 5	130
Above 10	7 + 13 + 43 + 37 + 25	125
Above 20	7 + 13 + 43 + 37	100
Above 30	7 + 13 + 43	63
Above 40	7 + 13	20
Above 50	7	7

What is Frequency distribution and construction of frequency distribution?

Frequency distribution:

The arrangement of data into groups or classes together with the number of observations in each group or class is called frequency distribution.

The number of observations falling (lying) in a particular class is called frequency and is usually denoted by (f). Data presented in the form of frequency distribution is also known as grouped data, while the data in original form is called ungrouped data while constructing a grouped frequency distribution. The following terms are associated with its construction. i.e. class limits, class boundaries, class mark, or midpoint of a class, width of a class or class interval size.

i.                        Class limits:

The pair of number of a variable which describe a class are called class limits. The smaller number is called the lower class limit while the larger number is called the upper class limit. The class limits are constructed in such a way that the upper limit of one class do not coincide with the lower limit of next higher class. Thus there is a gap between successive classes e.g. 10-19, 20-29, 30-39 etc.

ii.                        Class boundaries:

When class are constructed in such a way that the upper limit of one class coincides with the lower limit of next higher class, then such limits are called class boundaries. Thus there will be no gap between the successive classes. The class boundaries are exclusive.

 e.g. 10-20, 20-30, 30-40, 40-50, etc hence 20 will be included in the class 20-30 instead of 10-20.

iii.                    Midpoint of a class or class mark:

The midpoint of a class is obtained by dividing the sum of upper and lower class limits/boundaries by 2. Since individual identity of the observation is lost in grouping process, hence for convenience of computation midpoint are compute, about midpoint of a class, we assume that each value in a class is equal to its midpoint e.g. if frequency of a class is 9 and its midpoint is 24, it means that all 9 values of a class are equal to 24.

iv.                        Width of a class or class interval size:

The different between successive lower limits or between successive upper limits is called width of class or class interval size. It may also be obtained by finding the difference between successive midpoints. The width of a class is usually denoted by (h).

Construction of frequency distribution:

While constructing a grouped frequency distribution the following steps should be taken into consideration.

i.      For convenience arrange the data in an array using stem and leaf display.

ii.    Determine the largest and smallest number from an arrayed data in order to find range i.e. the difference between largest and smallest number.

iii.  Decide upon the number of classes. There is no hand and fast rule for deciding the number of classes, but a reasonable number of classes between 5 to 20 may be included depending upon the size of the data. Sturges also suggested a formula for deciding the number of classes i.e.

          

      K = 1+3.3 log N

iv.  To find the class interval size (h) divide the range by the desired number of classes e.g. if range of values is 87 and number of classes are 9, then class interval size will be   87/ 9 = 9.67 or  approximately 10. Similarly if range is 43 and number of classes are 8, then class interval size will be  43/ 8 = 5.375 or 6 approximately (round off to next higher integer).

v.   Decide what should be the starting value of the first class. The starting value is usually taken as the lowest value of the given data or less than that which is a multiple of 2 5, 10, and such other figures. The upper limit is obtained by adding the width of a class with the lower class limit. The remaining class limits are determined similarly.

vi.  Distribute the values in to appropriate classes either by listing actual values In their proper classes or by using tally bars. The number of tallies is then written infrequency column.

Types of Classification in Statistic?

Definition:

The process of arranging data into various groups or classes according to some common characteristics is called classification. 

Types of classification:

There are four types of classification.

i.                        Quantitative classification:

When the data are classified according to quantitative variable, then it is known quantitative classification e.g. when population of a city is classified by income, age, weight, height etc.

Height (inches): 54-56  57-59  60-62  63-65  65-67.

No. of person :   289  356  589  297 240.

ii.                        Qualitative classification:

When the data are classified according to qualitative characteristics like sex, literacy, religion education etc. then it is called qualitative classifications. E.g. classification of population according to sex (i.e. male and female), according to education (i.e. literate and illiterate), according to wealth (i.e. rich and poor) etc.

iii.                        Geographic classification:

When the data are classified according to places or geographic location. Then it is called geographic  classification. E.g. population of Khyber Pakhtunkhwa recorded in 1990 district wise, literacy rate in Pakistan province wise etc. the following example illustrate geographic classification.

Country:	Canada	U.S.A	Germany	France
National income	7930	7880	7510	6730

Series which are obtained by arranging the data on the basis of places are called “spatial series”.

iv.                        Chronological OR temporal classification:

When the data are classified on the basis of time, then it is known as chronological classification and the series so obtained is called time series. The following table would give an idea of chronological classification.

Year :                           1930  1940  1950  1960  1970.

Population(crores) :     2311  1785  3135  3688  3940.

What is Classification? Objectives and Basic principle of classification in statistic?

Classification:

Introduction:

It is difficult to draw any inferences form the data that have been originally collected. There are chances of making wrong decisions about the nature of the data, because human mind is not so capable of memorizing all the figures. Thus there arises a need to reduce and simplify the raw data (primary data) into such a form that is easily understood. One such form of reducing the data is classification.

Definition:

The process of arranging data into various groups or classes according to some common characteristics is called classification.

Aims or objectives of classification:

The main objectives of classifying the data are:

1. Since human mind is not so fertile to remember all the figures. Therefore classification is the only way to reduce the large mass of data.
2. Classification facilitate comparison i.e. when data are classified it becomes easy to know how many students source marks between 20-40, 40-60 etc
3. Classification simplifies calculation of statistical measure like mean, median, standard deviation etc.
4. When data are originally collected, there is repetition of values which consumes too much space and time. The classification technique saves time and space because data are presented in a compact form in comparison to lose form.

Basic principle of classification:

While studying the larger set of data, the following points should be taken into Consideration.
The classes into which data are to be distribute should be mutually exclusive i.e. successive classes should not overlap.

The classification procedure should be exhaustive i.e. classes should completely cover the whole data. For the proper analysis no item should be left classified.

Classification should be clear and simple. Ambiguities and doubtful entries must be removed.
The classification procedure should not be so slab orate  to lead trivial classes nor it should be so crude as to accommodate whole  data in one or two classes.

Secondary data and Methods for collection of secondary data in statistic?

Secondary data:

The data that have already been collected by someone and the statistic techniques are applied at least once on such data, are called secondary data.

When statistical methods are applied on primary data, then they lose their original shape and become secondary e.g. if the data in different census years are again used to measure the changes in the population growth, sex ratio, mortality rate etc.

Methods for collection of secondary data:

Secondary data are those, which have already been collected by someone for their own use and now he same data is used by different persons for another purpose. Such data can be collected from the following sources.

1.                 Official sources:

E.g. publication of statistical divisions, reports of ministries of finance, food and agriculture, planning and development etc.

2.                 Semi official sources:

E.g. publication of state bank, wapda, P.I.A local bodies etc.

3.                 Private sources:

E.g. publication of state association, chambers of commerce and industry, private commercial and financial institutions etc.

4.                 Research organizations:

E.g. publication of research organizations like universities, institute of education and research etc.