Statistics

**Category : **10th Class

**Statistics**

**Statistics:**The science of collection, organization, presentation, analysis and interpretation of numerical data is called statistics.

**Raw data:**The collection of actual information used to make logical inferences is called raw data.

**Frequency table:**The table in which raw data is condensed and presented is called the frequency table.

**Measure of central tendency:**the tendency in the data to be concentrated around a certain single value that represents the whole set of data is called a measure of central tendency or average.

**Series:**While 'preparing a frequency distribution table, classes can be considered in two ways- (i) inclusive or discrete series and (ii) Exclusive or continuous series.

**Discrete series:**The series in which the class intervals are so fixed that the upper limit of the class is included in the same class interval is called the continuous series.

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**Rule to convert a discrete (or discontinuous) series into a continuous series.**

(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

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**Arithmetic mean of raw data (when frequencies are not given):**The arithmetic mean of raw data is obtained by adding all the values of the variables and dividing the sum by the total number of values that are added.

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- If \[{{x}_{1}},{{x}_{2}},.....{{x}_{n}},\]where ‘n’ is the total number of value, then arithmetic mean

\[(\overline{x})=\frac{{{x}_{1}}+{{x}_{2}}+.....+{{x}_{n}}}{n}=\frac{1}{n}\sum\limits_{i=1}^{n}{{{x}_{i}}}\]

- Direct method (when frequencies are given) o finding arithmetic mean: If \[{{x}_{1}},{{x}_{2}},.....,{{x}_{n}}\]are the ‘n’ variables with corresponding frequencies \[{{f}_{1}},{{f}_{2}},.....,{{f}_{n}}\]respectively, then the arithmetic mean is given by

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**\[\bar{x}=\frac{{{f}_{1}}{{x}_{1}}+{{f}_{2}}{{x}_{2}}+.....+{{f}_{n}}{{x}_{n}}}{{{f}_{1}}+{{f}_{2}}+....{{f}_{n}}}=\sum\limits_{i=1}^{n}{\frac{{{f}_{i}}{{x}_{i}}}{{{f}_{i}}}}\]**

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**Properties of arithmetic mean:**

(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 \[\frac{{\bar{x}}}{a}\].

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**Note: Class mark of mid-value of a class \[\mathbf{=}\frac{\mathbf{Upper}\,\mathbf{limit+Lower}\,\mathbf{limit}}{\mathbf{2}}\]**

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**Methods of finding the mean of a grouped data:**Mean of a grouped data is calculated using (a) Direct method (b) Assumed mean method and (c) Step deviation method.

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**Direct method:**the observations\[{{x}_{1}},{{x}_{2}}....,{{x}_{n}}\]appear with frequencies \[{{f}_{1}},{{f}_{2}}....,{{f}_{n}}\], then mean is given by \[x=\frac{1}{N}\sum\limits_{i=1}^{n}{{{f}_{i}}{{x}_{i}}}\]where \[N={{f}_{1}}+{{f}_{2}}+....+{{f}_{n}}\]

**Note:** It is time-consuming and tedious to find the product \[{{f}_{i}}{{x}_{i}}\]

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**Assumed mean method:**In this method, we choose an assumed mean 'a' and subtract it from each of the values\[{{x}_{i}}\]. The reduced value \[{{x}_{i}}-a\] is called the deviation of \[{{x}_{i}}\]from ‘a’. the deviations are then multiplied by corresponding frequencies to get \[{{f}_{i}}{{d}_{i}}\]. Then the sum of \[{{f}_{i}}{{d}_{i}}\]is computed.

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.

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**(c) Step deviation method:**The deviation method is further simplified on dividing the deviation by width of the class interval h. Then arithmetic mean\[\bar{x}=a+\frac{\Sigma {{f}_{i}}{{u}_{i}}}{\Sigma {{f}_{i}}}\times h\] where\[{{u}_{i}}=\frac{{{x}_{i}}-a}{h}\].

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**Mode:**The value among the observations that occurs most frequently is called mode.

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**Multimodal data:**Data having more than one value with maximum frequency is said to be multimodal data.

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**Modal class:**In a grouped frequency distribution, the class with the maximum frequency is called the modal class.

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**Note:** The mode of the data lies in the modal class.

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**Mode of a grouped frequency distribution:**The mode of a grouped frequency is given by

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- Mode\[=l+\left[ \frac{{{f}_{1}}-{{f}_{0}}}{2{{f}_{1}}-{{f}_{0}}-{{f}_{2}}} \right]\times h\], where

\[l=\]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.

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**Types of modal series:**

**(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.

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**Note:** Ill-defined mode: If a series has more than one mode, then the mode is said to be ill-defined.

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**Median (Ungrouped data):**The measure of the central tendency when the observations are arranged in ascending or descending order of magnitude is called the median.

If 'n' is odd, median = value of\[{{\left( \frac{n+1}{2} \right)}^{th}}\]term.

If 'n' is even, median = value of \[\frac{1}{2}\left[ {{\left( \frac{n}{2} \right)}^{th}}term+{{\left( \frac{n+1}{2} \right)}^{th}}term \right]\]

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- Median of grouped frequency distribution: The median of grouped frequency distribution is given by

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- Median \[=l+\frac{\frac{N}{2}-C}{f}\times h\]where,

\[l=\]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

\[\text{N}={{\text{f}}_{1}}+{{\text{f}}_{2}}+{{\text{f}}_{3}}+......+{{\text{f}}_{n}}\]

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**Cumulative frequency distribution is of two types –**

(i) Lesser than cumulative frequency

(ii) Greater than cumulative frequency

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- Lesser than cumulative frequency is computed from the top of the table and greater than cumulative frequency is computed from the bottom of the table [when frequencies are added to get cumulative frequencies].

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- The empirical relationship among the three measures of central tendency is 3 median = mode + 2 mean (or) Mode = 3 median - 2 mean. Mode = 3 median - 2 mean

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- Graphical representation of cumulative frequency distribution:

(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'.

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- (b) The free hand smooth curve on a graph paper obtained by plotting the lower limits of the class intervals on the X-axis and the corresponding cumulative frequencies on the Y-axis choosing a convenient scale is called a cumulative frequency curve (of greater than type).

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**Locating median from an 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.

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**Method II:**

(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.

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**Note:** To draw ogives, the class-intervals must be continuous.

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