Statistics
Category : 10th Class
Statistics
(i) Find the adjustment factor. i.e. \[\frac{1}{2}\][lower limit of second class – upper limit of first class]
(ii) Subtract the adjustment factor from the lower limit of a class and add it to the upper limit of each class.
(iii) The series obtained is the exclusive series
\[(\overline{x})=\frac{{{x}_{1}}+{{x}_{2}}+.....+{{x}_{n}}}{n}=\frac{1}{n}\sum\limits_{i=1}^{n}{{{x}_{i}}}\]
(a) The algebraic sum of deviations from mean is zero; i.e. if \[{{x}_{1}},{{x}_{2}},.....{{x}_{n}}\]are ’n’ observations, then \[\sum\limits_{i=1}^{n}{({{x}_{i}}-\bar{x})}=0\]
(b) If each observation of a data is increased by 'a', then the mean is increased by 'a'; i.e., if \[\bar{x}\]is the mean of \[{{x}_{1}},{{x}_{2}},....,{{x}_{n}}\], then the mean of \[{{x}_{1}}+a,{{x}_{2}}+a,....,{{x}_{n}}+a\]is \[\bar{x}+a\].
(c) If each observation of a data is decreased by 'a', then the mean is decreased by 'a'; i.e., if the mean of \[{{x}_{1}},{{x}_{2}},....,{{x}_{n}}\]is \[\bar{x}\], then the mean of \[{{x}_{1}}-a,{{x}_{2}}-a,....,{{x}_{n}}-a\]is \[\bar{x}-a\].
(d) If each observation of a data is multiplied by a non-zero number, 'a', then the mean of the new observations is the product of 'a' and\[\bar{x}\], i.e., if \[{{x}_{1}},{{x}_{2}},....,{{x}_{n}}\]are 'n' observations whose mean is\[\bar{x}\], then the mean of \[{{x}_{1}}a,{{x}_{2}}a,....,{{x}_{n}}a\]is \[\bar{x}a\].
(e) If each observations is \[\frac{{\bar{x}}}{a}\]; i.e., if the mean of ‘n’ observations \[{{x}_{1}},{{x}_{2}},....,{{x}_{n}}\]is \[\bar{x}\]then the mean of the observations\[\frac{{{x}_{1}}}{a},\frac{{{x}_{2}}}{a},....,\frac{{{x}_{n}}}{a}\]is \[\frac{{\bar{x}}}{a}\].
Note: Class mark of mid-value of a class \[\mathbf{=}\frac{\mathbf{Upper}\,\mathbf{limit+Lower}\,\mathbf{limit}}{\mathbf{2}}\]
Note: It is time-consuming and tedious to find the product \[{{f}_{i}}{{x}_{i}}\]
Arithmetic mean\[=a+\frac{\sum\limits_{i=1}^{n}{{{f}_{i}}{{d}_{i}}}}{\sum\limits_{i=1}^{n}{{{f}_{i}}}}\]
Note: (i) This is also called the short cut method.
(ii) The assumed mean is chosen in such a way that it is one of the central values, the deviations are small and one of the deviations is zero.
Note: The mode of the data lies in the modal class.
\[l=\]lower limit of the modal class,
h = size of the class interval (assuming all class sizes to be equal)
\[{{f}_{1}}=\]frequency of the modal class
\[{{f}_{0}}=\]frequency of the class preceding the modal class and\[{{f}_{2}}=\]frequency of the class succeeding the modal class.
(a) Unimodal series: The series of observations containing a single mode.
(b) Bimodal series: The series of observations containing two modes.
(c) Trimodal series: The series of observations containing three modes.
Note: Ill-defined mode: If a series has more than one mode, then the mode is said to be ill-defined.
If 'n' is odd, median = value of\[{{\left( \frac{n+1}{2} \right)}^{th}}\]term.
If 'n' is even, median = value of \[\frac{1}{2}\left[ {{\left( \frac{n}{2} \right)}^{th}}term+{{\left( \frac{n+1}{2} \right)}^{th}}term \right]\]
\[l=\]lower limit of median class interval
\[C=\]cumulative frequency preceding the median class frequency
\[f=\]frequency of the class interval to which median belongs
\[h=\]width of the class interval
\[\text{N}={{\text{f}}_{1}}+{{\text{f}}_{2}}+{{\text{f}}_{3}}+......+{{\text{f}}_{n}}\]
(i) Lesser than cumulative frequency
(ii) Greater than cumulative frequency
(a)The freehand smooth curve on a graph paper, obtained by plotting the upper limits of the class intervals on the X-axis and their corresponding cumulative frequencies on the Y-axis choosing some convenient scale is called a cumulative frequency curve (of lesser than type).
Note: The other name for cumulative frequency curve is 'ogive'.
Method I:
(i) Compute\[\frac{n}{2}\], where 'n' is the sum of frequencies.
(ii) From \[\frac{n}{2}\]on the Y-axis, draw a line onto the curve, parallel to the X-axis.
(iii) From this point, drop a perpendicular onto the X-axis.
(iv) The point of intersection of the perpendicular and the X-axis is the median of the data.
(i) Draw both the ogives for the given data, on the same coordinate plane.
(ii) Locate the point of intersection of the two ogives.
(iii) From the point of intersection of the ogives, draw a perpendicular onto the X-axis.
(iv) The point of intersection of the perpendicular and the X-axis is the median of the given data.
Note: To draw ogives, the class-intervals must be continuous.
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