Notes - Mathematics Olympiads - Statistics

Category : 11th Class




Statistics history is very old. Early statistics is considered as the imposed form of applied mathematics.


  • Statistics is used as singular and plural: Statistics used as singlular. It is the science in which we collect, analysis, interprete the data.


  • Statistics used as plural


(i)         Statistics are aggregate of facts.

(ii)        Statistics are affected by a number of factors.

(iii)       Statistics are collected in systematic manner.

(iv)       Statistics must be reasonable accurate.

It is both art and science.


  • Science: Systematised body of knowledge is said to be science.
  • Art: Handling of the fact of given information to skill up the knowledge about the matter is said to be art.


Note: Statistics without science has no fruit and science without statistics has no roots.


Measure of Central Tendency


  • Central Tendency: The properties of finding and the average value of the data is said to Central Tendency.


The commanly used measure of central tendency are:


(a) Arithmetic Mean                               

(b) Geometric Mean

(c) harmonic Mean                                

(d) Median

(e) Mode


  • Arithmetic Mean: Mean of unclassified/Raw data/Individual

Let \[{{x}_{1}},\,{{x}_{2}},\,{{x}_{3}}.....{{x}_{n}}\] are n observations. Then their arithmetic mean is written as






  • Mean of Classified Data: Let \[{{x}_{1}},\,{{x}_{2}},\,{{x}_{3}},\,{{x}_{4}},....{{x}_{n}}\] and let \[{{\text{f}}_{1}},\,{{\text{f}}_{2}},\,{{\text{f}}_{3}},....{{\text{f}}_{n}}\] are their corresponding frequencies. Then




Weighted Arithmetic Mean:

If \[{{w}_{1}},\,{{w}_{2}},\,{{w}_{3}},\,......{{w}_{n}}\] are the weights assigned to the values \[{{x}_{1}},\,{{x}_{2}},\,{{x}_{3}},\,{{x}_{4}}\,......{{x}_{n}}\] respectively. Then the weighted average, or weighted




  • Combined Mean: If we are given the A.M. of two data sets and their sizes, then the combined

A.M of two data sets can be obtained as.




Where, \[{{\overline{x}}_{12}}=\] combined mean of the two data sets 1 and 2

0 Mean of 1st data

\[{{\overline{x}}_{2}}=\]Mean of the 2nd data

\[{{n}_{1}}=\]size of the 1st data.

\[{{n}_{2}}=\]size of the 2nd data.

Some properties about A.M.

In statistical data, sum of the deviation of individual values from A.M. is always zero.


i.e.        \[\sum\limits_{i=1}^{n}{\text{f}i}({{x}_{1}}-\overline{x})=0\]


Where \[\text{f}i=\] frequencies of \[xi\,\,\{1\le i\le n\}\]

A.M is written s




  • Short-cut Method: For a given data, we suitably choose a term, usually the middle term and call it the assumed mean, to be denoted by A.

Then, we find deviation, \[{{d}_{i}}=({{x}_{i}}-A)\] for each term.

Thus \[A.M=\overline{x}=A+\frac{\sum{{{\text{f}}_{i}}{{d}_{i}}}}{{{\text{f}}_{i}}}\]


where A = Assumed Mean, f = frequency


  • Step-Deviation: \[A.M,\,\,\overline{x}=A+\frac{\Sigma {{\text{f}}_{i}}{{d}_{i}}}{N}\times h\]


Where A = Assumed mean


\[{{d}_{i}}=\frac{{{x}_{i}}-A}{h}=\]deviation of any variate from A


h = width of the class-interval and \[N=\Sigma {{\text{f}}_{i}}\]

In a statistical date, the sum of square of deviations of individual values from A.M. is least.


i.e.        \[\sum\limits_{i=1}^{n}{\text{f}i{{(x-\overline{x})}^{2}}=}\] least value          


If each of the given observation is doubled then their arithmetic mean is doubled

If \[\overline{x}\]then the mean of \[{{x}_{1}},\,{{x}_{2}},\,{{x}_{3}},.....{{x}_{n}}.\]then the mean of \[a{{x}_{1}},\,a{{x}_{2}},\,a{{x}_{3}},.....a{{x}_{n}}\]where a is any number different from zero, is \[\overline{ax}\]


  • Geometric Mean: if \[{{x}_{1}},\,{{x}_{2}},\,{{x}_{3}},.....{{x}_{n}}\] are n observations, none of them being zero, then their geometric mean is defined as,


\[G.M=Anti\log \left\{ \frac{\log {{x}_{1}}+\log {{x}_{2}}+\log {{x}_{n}}}{n} \right\}\]


For classified/grouped data.




                        Where \[N=\sum\limits_{i=1}^{n}{{{\text{f}}_{i}}}\]


                        \[G.M=Anti\log \left\{ \frac{\sum\limits_{i=1}^{n}{{{\text{f}}_{i}}lo{{x}_{i}}}}{n} \right\}\]


  • Harmonic Mean: The harmonic mean of n observation \[{{x}_{1}},\,{{x}_{2}},\,{{x}_{3}},.....{{x}_{n}}.\] is defined as




If \[{{x}_{1}},\,{{x}_{2}},\,{{x}_{3}},.....{{x}_{n}}\] are n observation which occur with frequencies \[{{\text{f}}_{1}},\,{{\text{f}}_{2}},\,{{\text{f}}_{3}},....{{\text{f}}_{n}}\] respectively.



\[H.M=\frac{\sum\limits_{i=1}^{n}{{{\text{f}}_{i}}}}{\sum\limits_{i=1}^{n}{\left( \frac{{{\text{f}}_{i}}}{{{x}_{i}}} \right)}}\]


Note: For Harmonic sequence be inverse sequence of arithmetic sequence (progression)


  • Relation between A.M., G.M. & H.M: To find single A.M., G.M & H.M between a & b be written as


\[A.M=\frac{a+b}{2},G.M=\sqrt{ab}\And H.M=\frac{2a.b}{a+b}\]


If \[a=b\]

the \[A.M=G.M=H.M.\]

\[A.M\ge G.M\ge H.M.\](for any observation)

Notes - Mathematics Olympiads - Statistics
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