Note on Concentration and important measure of central tendency

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Concentration.

Concentration is an important tool in data analysis. In this process after the collection of data , the next step is to analyse it. Since huge and widely masses of data are confusing and difficult to remember, so we need a widely value representing the whole data. Average are the typical values around which most of the data tend to cluster. These are the values which lie between two extreme observation of the entire data and gives us the idea about the distribution. Measurement of such single value is known as the measure of central tendency, these measures are also called as the statistical average. A measure of central tendency is valued around which all the observation have a tendency such a value is considered as the most representative figure of the entire data set. Concentration helps to make decision facilities comparison .

It is considered as an important measure in data analysis this is due to the following reason.

  1. Concentrations help to make the decision.
  2. This process helps in mathematical relation.
  3. Concentration halps to facilitates comparison.
  4. This process presents the salient feature of a mass of complex data.
  5. This process helps to know about the universe from a sample.

The most of the important tool for measurement of the concentration are of three types. They are

  1. Mean
  2. Median
  3. Mode

Mean.

This is the most common method of central tendency and may be defined as the value which we get divided the total of the values of various given items in a series by the total number of items. We can work in out as under.

$$Mean\,(or\,\overline{X})=\frac{\sum X_i }{n}$$

$$\frac{X_1+X_2+X_3......+X_n}{n}$$

$$Where\,,\overline{X})=The\,symbol\,we\,use\,for\,mean\,(Pronounced\,as\,\overline{X})$$

$$\sum=Symbol\,for\,summation$$

$$X_i=Value\,of\,i^th\,item\,X,I=1\,,2\,,3\,,...n$$

$$n=Total\,number\,of\,items$$

In case of frequency distribution when the whole data set of size n is summarized in k class several , mean is calculated as.

$$\overline{X} = \frac{\sum_{i=1}^{k}f_ix_i}{\sum_{i=1}^{k}f_i}$$

$$\frac{f_1X_1+f_2X_2+f_3X_3......+f_kX_k}{f_1+f_2+...+f_k}$$

$$Where\,X_i\,=Mid\,point\,of\,i^{th}\,class\,interval\,f_1\,f_2\,+...+f_k=n$$

Sometimes, instead of calculating the simple mean, as state above, we may workout the weighed mean for a realistic average. The weighted mean can be worked out as follows.

$$\overline{X_w})=\frac{\sum{w_i} X_i }{\sum{w_i}}$$

$$Where\,\overline{X_w}=Weight\,item$$

$$X_i=Value\,of\,the\,i^{th}\,item\,X$$

$$W_i=Weight\,of\,the\,i^{th}\,item\,X\,,assigned\,by\,the\,researcher\,using\,the\,some\,knowledge$$

This is the most common and simplest measure of central tendency. People use mean in daily life so much that it has become a synonym of average. For example, Ravi consumes 4 cigarette, on the average of daily; Suresh drinks approximately 2 litre of water, on an average daily; However, the mean suffers from some laminations. When the date set has one or more extreme values, of the magnitude of the mean is affected and it provides a wrong impression of the values in the data set when used to provide a wrong impression of the values of the date set when used to represent the whole date-set.

Median.

Mean divides the data set- into two equal parts. In this case, when the dataset has outlined, mean becomes flowed as a representative of the data-set. It is used as a measure of central tendency.

Half of the items are less than median and remaining half of the items are larger than the median. In order to obtain the median, we first arrange the data set into ascending or descending order. If a number of observation in the data set n, then the.

$$Median=(\frac{n+1 }{2})\,the\,observation\,,when\,n\,is\,odd$$

$$=\frac{1 }{2}[(\frac{n }{2})^{th}\,observation\,+(\frac{n }{2}+1^{th})\,observation]\,when\,n\,is\,even)$$

For example.

$$median\,of\,the\,data-set\,2\,,5\,,6\,,11\,,59\,,is\,6$$

$$Median\,of\,data\,set\,5\,,6\,,11\,,59\,,is\,\frac{1 }{2}\,[6\,+11\,]\,or\,8.5$$

This is a positional average and it is used only the context of a quantitative phenomenon. For example, in estimating intelligence, etc…. which are often encountered in sociological fields. The median is not useful where items need to be relatively important and weight. This process is not frequently used in sampling statistic.

Mode.

It is the most important data set and considered as the most frequently occurring observation. The mode is derived from France word which means fashion. The mode is particularly used usefully in the study of population size.

For example, a manufacture of shoes is usually interesting in finding out the size most in demand so that he may manufacture a large quantity of that size. Like median, the mode is amenable to algebraic treatment. A data set may have any mode or there may be more than one mode in a data set.

Another type of average used in the calculation.

There are two different average means used in the calculation.

They are .

  1. Harmonic mean.
  2. Geometric mean.

Harmonic mean.

The harmonic mean is defined as the reciprocal of the average of reciprocals of the values of items of a series. It can be expressed as

$$H.M=Rec.\frac{\sum RecX_i }{n}$$

$$=Rec.\frac{RecX_1.Rec.X_2+........+Rec.X_n }{n}$$

$$H.M=Harmonic\,Mean$$

$$Rec=Reciprocal$$

$$X_i=i^{th}\,value\,of\,the\,variable\,X$$

$$n=Number\,of\,items$$

For instance, the harmonic mean of the number 4, 5, and 10 is worked out as

$$H.M=Rec.\frac{1/4+1/5+1/10}{3}$$

$$=Rec.\frac{15+12+6/60}{3}$$

$$=REc.(\frac{33}{60}×\frac{1}{3})$$

$$=\frac{60}{11}$$

The harmonic mean gives the largest weight to the smallest item and smallest weight to the smallest weight of the largest item. As such it is used cases like time and motion study where time is variable and distance constant. The harmonic mean is the limited application, particularly in a case where time and rate are involved.

Geometric mean.

The geometric mean is defined as the nth root of the product of the values of nth times in a given series.

$$G.M=^n{\sqrt{πX_i}}$$

$$= ^n{\sqrt{X_1\,.X_2\,.X_3....X_n}}$$

$$G.M=Geometry\,mean$$

$$n=Number\,of\,items$$

$$X_i=i^{th}\,value\,of\,the\,variable\,X$$

$$π=Conventional\,Product\,notation$$

For instance, the geometric mean of the number 4, 6 and 9 worked out as

$$G.M=^3{\sqrt{4\,.6\,.9}}$$

The most frequently used applications of this average are in a determination of average per unit of change I,e it is often used in the preparation of index number or when deals in ratios.

Reference

Kerlinger, F.N. Foundation of Behavioural Research. New Delhi: Surjeet Publication, 2000.

Kothari, C.R. Research Methodology. India: Vishwa Prakashan, 1990.

Singh, M.L. and J.M Singh. Understanding Research Methodology. 1998.

Singh, Mrigendra Lal. Understanding Research Methodology. Nepal: National Book centre, 2013.

  1. Concentration is an important tool in data analysis. In this process after the collection of data , the next step is to analyse it.
  2. $$Mean\,(or\,\overline{X})=\frac{\sum X_i }{n}$$
  3. $$Median=(\frac{n+1 }{2})\,the\,observation\,,when\,n\,is\,odd$$
  4. $$=\frac{1 }{2}[(\frac{n }{2})^{th}\,observation\,+(\frac{n }{2}+1^{th})\,observation]\,when\,n\,is\,even)$$
  5. $$H.M=Rec.\frac{\sum RecX_i }{n}$$
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