Notes on Central Tendency, Arithmetic Mean, Partition Values | Grade 12 > Mathematics > Dispersion, correlation and regression | KULLABS.COM

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Basic Concepts of Central Tendency

The central value is the most important information of a data set. The central value of a dataset is known as its average. It is assumed that average depends on the whole data set as does the centre of gravity (CG). That represents the mass of a body. When we say that the per capita income of Nepal is 350 dollars or that the literacy rate is 54 percent they are average figures that represent the whole population of Nepal. The average is a value that represents the whole data set and it is assumed to lie at a centre of the frequency distribution. The tendency of average to be located at the centre of the data set is known as the central tendency.

Depending on the composition or nature of the data, the average is calculated differently by using the different formula. We have arithmetic average, geometric average, harmonic average, median and mode to represent the central value. Although arithmetic value is suitable for almost every type of data all other averages have their own speciality. Not all average are suitable for all type data. We should use an appropriate average by understanding the nature and type of data set.

Requirement of Good average

According to the renowned statistician Yule, an average should possess following properties to be called a good average so that it will represent the whole data set.

• It should be rigidly defined.( i.e. there should only be one average value for one data set)
• It should be readily comprehensible and easy to calculate.
• It should be based upon all observations.
• It should be suitable for further mathematical calculations ( For example, if we are given the average and size of two or more data sets, we should be able to calculate the average of combined data set)
• It should be least affected by sampling fluctuation (i.e. if we change very few values of the data set the average should be changed very little)
• It should be least affected by extreme values (i.e. if there are few very large or very small values in the whole data set it should be not affected by these values to produce a non-representative central value)

Important Formulae

Arithmetic Mean (A.M.)

When the sum of the set of observation of the same kind is divided by the number of observations, we get a unique value known as the arithmetic mean or simple mean of the given set data observations.

For Individual series

$$Arithmetic\;Mean\;(\overline{x})=\frac{\Sigma\;X}{n}$$

$$Arithmetic\;Mean\;(\overline{x})=A+\frac{\Sigma\;d}{n},\;d=X-A$$

$$A,\;is\;the\;assumed\;mean$$

For Discrete series

$$Arithmetic\;Mean\;(\overline{x})=\frac{\Sigma\;fX}{n},$$

$$Arithmetic\;Mean\;(\overline{x})=A+\frac{\Sigma\;fd}{n}$$

$$Arithmetic\;Mean\;(\overline{x})=A+\frac{\Sigma\;fd’}{n}\times\;h\;,where\;d’=\frac{X-A}{h}$$

For Continuous series

$$Arithmetic\;Mean\;(\overline{x})=\frac{\Sigma\;fX}{N},N=\sigma\;f\;and\;X-mid\;value$$

$$Arithmetic\;Mean\;(\overline{x})=A+\frac{\Sigma\;fd}{n},d=X-A$$

$$Arithmetic\;Mean\;(\overline{x})=A+\frac{\Sigma\;fd’}{n}\times\;h\;,where\;d’=\frac{X-A}{h}$$

Combined Mean

$$If\;x_1\;amd\;x_2\;be\;the\;sizes,\;\overline{X}_1\;and\;\overline{X}_2\;be\;the\;arithmetic\;means\;of\;two\;component\;series\;,$$

$$then\;their\;combined\;mean\;denoted\;by\;\overline{X}_{12}\;of\;the\;combined\;series\;N_1\;andN_2\;is\;given\;by\;$$

$$\overline{X}_{12}=\frac{N_1\overline{X}_1+ N_2\overline{X}_2}{N_1+N_2}$$

$$Weighted\;AM\;=\overline{X}_w=\frac{\sigma\;WX}{\sigma\;W}$$

Partition values:

The variate values dividing the total number of observations into equal number of parts is known as partition values. The equal parts may be two, four, ten or hundred.

Quartiles are the values dividing the whole observations into 4 equal parts. Deciles are the variate values dividing the whole observations into 10 equal parts. So there are 9 deciles.

In the same way, percentiles are the variate values dividing the entire observation into 100 equal parts and hence there are 99 percentiles.

Quartiles and Median

If the total number of observations be divided into 4 equal parts, then these variates values are known as quartile. The three variate values will divide the whole observation into four parts. Thus there are three quartiles first, second and third. The median is the second quartile dividing the whole data set into two equal parts. The variate that divides the lower half into two parts is known as the lower quartile of the first quartile and is denoted by Q­1­. The variate value dividing the upper half into two parts is known as the upper quartile or the third quartile denoted by Q­3­. The median is denoted by M­d­.

Taken reference from

( Basic mathematics Grade XII and A foundation of Mathematics Volume II and Wikipedia.com )

•  The tendency of average to be located at the centre of the data set is known as the central tendency.
• When the sum of the set of observation of the same kind is divided by the number of observations, we get a unique value known as the arithmetic mean or simple mean of the given set data observations.
• The variate values dividing the total number of observations into an  equal number of parts is known as partition values. The equal parts may be two, four, ten or hundred.

•

Quartiles are the values dividing the whole observations into 4 equal parts. Deciles are the variate values dividing the whole observations into 10 equal parts. So there are 9 deciles.

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