11th Class Mathematics Statistics Notes - Mathematics Olympiads - Statistics

 

                                                                                                        Statistics

 

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

           

\[\overline{x}=\frac{x1+x2+x3+....xn}{n}=\frac{1}{n}.\sum\limits_{i=1}^{n}{xi}\]

           

\[=\frac{Sum\,\,o\text{f}\,\,\text{observations}}{no.\,\,o\text{f}\,\,\text{observations}}\]

 

• 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

 

\[\overline{x}=\frac{\sum{\text{f}\text{.x}}}{\sum{\text{f}}}\]

 

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

 

            \[A.M==\frac{{{w}_{1}}{{x}_{1}}+{{w}_{2}}{{x}_{2}}+{{w}_{3}}{{x}_{3}}+......{{w}_{n}}{{x}_{n}}}{{{w}_{1}}+{{w}_{2}}+{{w}_{3}}+.....{{w}_{n}}}\]

 

• 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.

           

\[{{\overline{x}}_{12}}=\frac{{{n}_{1}}{{\overline{x}}_{1}}+{{n}_{2}}{{\overline{x}}_{2}}}{{{n}_{1}}+{{n}_{2}}}\]

 

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

           

\[A.M\,\,=\overline{x}=\frac{{{x}_{1}}{{\text{f}}_{\text{1}}}+{{x}_{2}}{{\text{f}}_{2}}+...{{x}_{n}}{{\text{f}}_{n}}}{{{\text{f}}_{1}}+{{\text{f}}_{2}}+{{\text{f}}_{3}}+....{{\text{f}}_{n}}}=\frac{\sum\limits_{i=1}^{n}{{{\text{f}}_{i}}{{x}_{i}}}}{\sum\limits_{i=1}^{n}{\text{f}i}}\]

 

• 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,

\[G.M={{\{{{x}_{1}}.{{x}_{2}}.{{x}_{3}}....{{x}_{n}}\}}^{\frac{1}{n}}}\]

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

 

For classified/grouped data.

                       

\[G.M={{({{x}_{1}}.{{\text{f}}_{1}}+{{x}_{2}}.{{\text{f}}_{2}}+......{{x}_{n}}.{{f}_{n}})}^{\frac{1}{n}}}\]

                       

                        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

 

\[H.M=\frac{n}{\frac{1}{{{x}_{1}}}+\frac{1}{{{x}_{2}}}+\frac{1}{{{x}_{3}}}+....+\frac{1}{{{x}_{n}}}}\]

 

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.

Then

           

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

 

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)