Statistics - Revision Notes

 CBSE Class 11 Mathematics

Revision Notes
Chapter-15
STATISTICS

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1. Mean: x¯¯¯=x1+x2+........+xnn$\overline{x}=\frac{x_1+x_2+........+x_n}{n}$
2. Median: If the number of observations n$n$ is odd, then median is (n+12)th${\left(\frac{n+1}{2}\right)}^{th}$observation and if the number of observations n$n$ is even, then median is the mean of (n2)th${\left(\frac{n}{2}\right)}^{th}$ and (n+12)th${\left(\frac{n+1}{2}\right)}^{th}$ observations.
3. Measures of Dispersion, Range  and Mean Deviation
4. Variance and Standard Deviation
5. Analysis of Frequency Distributions

• Measures of dispersion Range, Quartile deviation, mean deviation, variance, standard deviation are measures of dispersion.
• Range = Maximum Value – Minimum Value
• Mean deviation for ungrouped data
  M.D. (x¯)=∑|xi−x¯|n$M.D.(\overline{x})=\frac{\sum |x_i-\overline{x}{|}_{}}{n}$
• Mean Deviation from Median for ungrouped data
  M.D. (M)=∑|xi−M|n$M.D.(M)=\frac{\sum |x_i-M{|}_{}}{n}$
• Mean deviation for grouped data
  M.D. (x¯)=∑fi|xi−x¯|N$M.D.(\overline{x})=\frac{\sum f_i|x_i-\overline{x}{|}_{}}{N}$
• Mean Deviation from Median for grouped data
  M.D. (M)=∑fi|xi−M|N$M.D.(M)=\frac{\sum f_i|x_i-M{|}_{}}{N}$ where N  = N=∑fi$N=\sum _{}^{}{f}_i$
• Variance and standard deviation for ungrouped data
  Variance: σ2=1n∑(xi−x¯)2${\sigma }^2=\frac{1}{n}\sum _{}^{}{(x_i-\overline{x})}^2$
  Standard deviation: σ2=1n∑(x1−x¯)2−−−−−−−−−−−√${\sigma }^2=\sqrt{\frac{1}{n}\sum _{}^{}{(x_1-\overline{x})}^2}$
• Variance and standard deviation of a discrete frequency distribution
  Variation: σ2=1N∑(xi−x¯)2${\sigma }^2=\frac{1}{N}\sum _{}^{}{(x_i-\overline{x})}^2$
  Standard deviation: σ2=1N∑fi(x1−x¯)2−−−−−−−−−−−−−√${\sigma }^2=\sqrt{\frac{1}{N}\sum _{}^{}{f}_i{(x_1-\overline{x})}^2}$
• Variance and standard deviation of a continuous frequency distribution

          (i)    If xif1;$\frac{x_i}{f_1};$i$i$ = 1, 2, 3, .........., n$n$ is a continuous frequency distribution of a variate X,

                  then  σ2=1N∑fi(xi−x¯)2${\sigma }^2=\frac{1}{N}\sum _{}^{}{f}_i(x_i-\overline{x}{)}^2$

         (ii)   If x1,x2,.......,xn$x_1,x_2,.......,x_n$ be the n$n$ given observations with respective frequencies

                f1,f2,.......,fn$f_1,f_2,.......,f_n$, then    σ=1NN∑fix2i−(∑fix1)2−−−−−−−−−−−−−−−−−√$\sigma =\frac{1}{N}\sqrt{N\sum _{}^{}{f}_i{x}_i^2-(\sum _{}^{}{f}_i{x}_1{)}^2}$, where N = ∑f1$\sum f_1$

         (iii)  If di=xi−A,$d_i=x_i-\mathrm{A},$ where A is assumed mean, then σ2=1N∑fid2i−(∑fidiN)2${\sigma }^2=\frac{1}{\mathrm{N}}\sum f_i{d}_i^2-{\left(\frac{\sum f_i{d}_i}{\mathrm{N}}\right)}^2$

         (iv)  If ui=xi−Ah,$u_i=\frac{x_i-\mathrm{A}}{h},$ where h$h$ is the common difference of values of x,$x,$ then

                 σ2=1N[∑fiu2i−(∑fiuiN)2]${\sigma }^2=\frac{1}{\mathrm{N}}\left[\sum f_i{u}_i^2-{\left(\frac{\sum f_i{u}_i}{\mathrm{N}}\right)}^2\right]$

• Analysis of frequency distribution with equal means but different variances: If the S.D. of group A < the S.D. of group B, then group A is considered more consistent or uniform.
• Ananlysis of frequency distribution with unequal means: In this case we compare the coefficient of variation [Coefficient of variation (C.V. = 100×S.D.Mean$\frac{100\times \mathrm{S}.\mathrm{D}.}{\mathrm{M}\mathrm{e}\mathrm{a}\mathrm{n}}$. The series having greater coefficient of variation is said to be more variable than the other.
• Variance of the combined two series: σ2=1n1+n2[n1(σ21+d21)+n2(σ22+d22)]${\sigma }^2=\frac{1}{n_1+n_2}\left[n_1\left({\sigma }_1^2+d_1^2\right)+n_2\left({\sigma }_2^2+d_2^2\right)\right]$
  where n1$n_1$ and n2$n_2$ are the sizes of two groups, σ1${\sigma }_1$ and σ2${\sigma }_2$ are the S.D. of two groups, d1=a¯¯¯−x¯¯¯$d_1=\overline{a}-\overline{x}$, d2=b¯¯−x¯¯¯$d_2=\overline{b}-\overline{x}$ and x¯¯¯=n1a¯¯¯+n2b¯n1+n2