B.Com 1st Year Statistical Average Long Notes
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Long Question Answer
Q.1. Define the term ‘Average’ with and example. Discuss various objectives and measures of central tendency.
Ans. Concept of Average : An average is a technique which reduces mass complex data into a single figure and which exhibits most of the features of the data. By average (mean) we mean some central value (centroid) of a series which may be said to be most representative and which may be used as a single value for the whole series. Averages are also known as ‘measures of Central Tendency’. Measures of location is another name used for averages because it is the value that is located where larger part of total data are clustered.
Definitions of Average
The word ‘average’ has been defined differently by many authors. Some of the popular definition are given below:
‘average is a single value within the range of the data that is used to represent all the values in -Croxton and Cowden the series.
‘It means average is a single value which lies within that data and represent that data and that is why it is also called measure of central value.
‘A measure of central tendency is a typical value around which other figures congregate.
‘An average, therefore, may be defined as a typical single value, which is a fair representative of a group of values.
An average is sometimes called a measure of central tendency, because individual value of the variable usually cluster around it!
It is clear from the above definitions that an average is a single value which is a fair representano a group of values. This measure (average) play an important role because it depicts the characteristics of the whole set of data.
Objectives of Average
We know that statistical data is complex and difficult to understand. Thus, it is necessary to summarise the data. This may be achieved by the use of average. By an average, complexity of the data is removed and a single figure (average) is obtained. Thus, average serves the following objectives:
1. To Reduce the Data by a Single Value: It reduces the mass information into a single value. In other words, it presents a brief picture of the mass data. With the help of this single value, the mass information can be understood and grasped easily.
With the help of averages, complex and a large group of data can be presented in some important words or in a number!
2. Comparison of Two Series: Average provides facility of comparison. The average of one group may be compared with the average of other group. For example, if we want to compare the weight of persons of two groups, individual comparison will not be feasible. It is the average weight of the persons of two groups that alone serves the purpose.
3. Basis of Statistical Analysis: Average is very useful in statistical analysis. It is widely used in dispersion, skewness, regression, index number etc.
4. Useful in Sampling: Averages are also used in sampling. We can obtain a picture of a complete group or population with the help of a sample. The mean of a sample gives a good idea about the mean of the population.
Various Measures of Central Tendency
Basically there are two types of average:
1. Mathematical average,
2. Positional average.
Mathematical Averages: These are calculated values of a series. That is why they are classified under mathematical averages:
(i) Arithmetic mean,
(ii) Geometric mean,
(iii) Harmonic mean.
2. Positional Averages: They locate the position of the average in the series. That is why they are called positional averages. They are: (i) Median and (ii) Mode.
Merits and Demerits of Mean: Refer to Section-A, Q.3.
Merits and Demerits of Median: Refer to Section-B, Q.3.
Q.3. Definition of Mode
point of view “The mode of a set of observation is that which occurs most often or with the greatest frequency.’ Some of its definitions are as follows: “The value of variable which occurs most frequently in a distribution is called mode.
“The mode of a distribution is the value at the point around which the items tend to be most heavily concentrated. It may be regarded as the most typical value of a series.’
Mode may be defined as the predominant kind, type or size of item or the position of greatest tendency.’
For example: If maximum number of shoes sold in a shop is the size number 7, then mode is 7. Similarly, if the maximum number of readymade shirts sold is of size number 34, then mode size is 34.
Calculation of Mode
The rules and process for the calculation of mode in different series are as follows:
1. Calculation of Mode in Individual Series: In an individual series, mode is located generally by observation and the value occurring maximum number of items is the modal value. However, for the convenience of counting data may be placed in an array or may be converted into a discrete series.
2. Calculation of Mode in Discrete Series: Mode can be calculated with the following two methods:
(a) Inspection Method: If there is regularity and homogeneity in the series, mode can be located by inspection of the series. The size or value having the highest frequency will be identified as mode.
(b) Grouping Method: Under the following circumstances, mode is located by grouping method:
(i) When the maximum frequency is repeated or approximately equal concentration is found
in two or more neighbouring values.
(ii) When the maximum frequency occurs either in the very beginning or at the end of the distribution.
(iii) When there are irregularities in the distribution, i.e. the frequencies of the variable increase or decrease in a haphazard way. In grouping method, six columns are drawn in addition to the column of value (X) and frequencies are grouped in the following order:
(i) First Column: The frequencies given in the question are shown in this column.
(ii) Second Column: In this column, frequencies are grouped into twos, starting from the top.
(iii) Third Column: In this column, frequencies are again grouped into twos, but the first frequency is left out, i.e. in this column grouping starts from the second frequency
(iv) Fourth Column: In this column, frequencies are grouped into threes, starting from the top.
(v) fifth Column: In this column, frequencies are grouped into threes but this grouping starts from the second frequency.
(vi) Sixth Column: In this column, frequencies are grouped again in threes but this grouping starts from the third frequency.
After preparing the grouping table, tallies are marked against the values having highest frequency in first column and highest total in each of the other column. Finally, the value securing maximum tallies will be modal value.3. Calculation of Mode in Continuous Series: In continuous series, it should be checked before the calculation of mode that each class interval should be equal. If they are not equal, they should be equalised. After it there are two steps in the computation of mode:
6. All Techniques in a Continuous Series: In computation of mode in a continuous series following problems may arise:
(a) Intervals are unequal and they have to be equalised.
(b) Class intervals are inclusive and they have to be converted into exclusive type of intervals.
(c) Modal class is not clear by inspection and grouping is required in it.
(d) Modal class is not clear by grouping also and density method is require
om the original formula crosses the limits of modal class and alternate formula is required.
(e) The answer from the original formula crosses the limits of modal class and alternate formula is required. A problem including all these techniques has been solved.
7. Location of Mode by Graphic Method: In a frequency distribution, mode can be determined by graphic method also. For this, a histogram is drawn on the basis of frequency distribution. The highest rectangle will represent the modal class. Two lines are drawn diagonally inside this rectangle (modal class) in such a way that they touch corner of the modal bar and the upper corner of the adjacent bar. Then a perpendicular is drawn from the intersection of these lines to the X-axis. The point at w on perpendicular touches the X-axis gives the modal value.
Merits of Mode
Basic merits of mode are as follows:
1. Simplicity: It processes the merit of simplicity. In individual and discrete series, it can be located even by inspection.
2. Not Affected by Extreme Values: Mode is not affected by the values of extreme items. It can be calculated even when extremes are not known.
3. Graphical Determination: Mode can be determined and presented by graphical method also.
4. Most Representative Value: Mode is the most typical or representative value of a distribution because it is usually an actual value of an important part of the series, where there is maximuin concentration of frequencies.
5. Stability: The value of mode remains almost stable in sampling. It means that if an item is taken out randomly from the set of the data, the probability of a modal value being chosen will be more than any other values.
Demerits of Mode
Main demerits of mode are as follows:
1. Indeterminate and Ill-defined: In many cases, mode is ill-defined particularly if the maximum frequency is repeated or if the maximum frequency occurs either, in the very beginning or at the end of the distribution or if the distribution is irregular. In such a case, the value of mode is determined by of the distribution or if the distribution is irregular. In such a case, the valu grouping method. If the grouping method also gives two values of mode, then the distribution bi-modal distribution.
2. Not Based on all values: Since, mode is the value of X corresponding to the maximum frequency, it is not Not Based on all the Values of the series. Even in the case of continuous series mode depends only on the frequencies of modal class and the classes preceding and succeeding it.
3. Not Capable of Further Mathematical Treatment: Mode is not capable of two sets of data, we cannot compute the mode of combined data. treatment. For example, from the modes of two sets of data, we cannot compute the mode of combined data.
4. Not Rigidly Defined Measure: Mode is not a rigidly defined measure. There are several formulae for calculating the mode and all of which usually give somewhat different answers. Moreover, the value for calculating the mode and all of wh of the mode is effected significantly by the size of class intervals used in grouping of data. A change in the size of class interval will change the value of the mode.
So, among mean, median and mode, relative merits and demerits are properly analysed.
But even them, arithmetic mean is most sustactory where due weightage is given to the extreme items.
Limitations of Various Averages
There are also some limitations of the average which are as under:
1. It can give us a value that does not exist in the given series.
2. Measures of central value may fail to give any idea regarding the formation of the series. More than two or two or two series may have the central value but they differ widely in the composition.
4. It suffers lack of clarification of all the characteristics. Since an average is a single figure representing a series and not a single figure can condense all the properties of the items in itself.
5. There lies problem in proper selection of an average. The choice of average is an important and difficult problem. Inaccurate conclusions are likely to follow if any wrong average has been chosen.
So, limitations of average should always be kept in mind and should be used as and when desired.