We know that data can be grouped into two ways :
`color(red)((a))` `color(blue)(" Discrete frequency distribution,")`
`color(red)((b))` `color(blue)(" Continuous frequency distribution.")`
Let us discuss the method of finding mean deviation for both types of the data.
`color{red}("(a) Discrete frequency distribution")`
Let the given data consist of n distinct values `color{orange}(x_1, x_2, ..., x_n)` occurring with frequencies `color{orange}(f_1, f_2 , ..., f_n)` respectively.
This data can be represented in the tabular form as given below, and is `color(blue)("called discrete frequency distribution :")`
`color{orange}(x : " " x_1" "x_2" "x_3 ........ x_n)`
`color{orange}(f : " " f_1" "f_2" " f_3 ........ f_n)`
`color{red}("(i) Mean deviation about mean")`
First of all we find the mean `color(blue)(barx)` of the given data by using the formula
`color{orange}(barx = (sum_(i=1)^(n) x_i f_i)/(sum_(i=1)^(n)f_i) = 1/N sum_(i=1)^(n) x_i f_i,)`
where `color{navy}(sum_(i=1)^(n) x_i f_i,)` denotes the sum of the products of observations `x_i` with their respective frequencies `color{orange}(f_i)`
and `color{navy}(N = sum_(i=1)^(n) x_i f_i,)` is the sum of the frequencies.
Then, we find the deviations of observations `x_i` from the mean `barx` and take their absolute values, i.e., `color{orange}(|x_i - bar x|)` for all `i =1, 2,..., n.`
After this, find the mean of the absolute values of the deviations, which is the required mean deviation about the mean. Thus
`color{blue}(M.D .(barx) = (sum_(i=1)^(n) f_i|x_i-barx|)/(sum_(i=1)^(n) f_i) = 1/N * sum_(i=1)^(n) f_i|x_i - barx|)`
`color{red}("(ii) Mean deviation about median")`
To find mean deviation about median, we find the median of the given discrete frequency distribution.
For this the observations are arranged in ascending order. After this the cumulative frequencies are obtained. Then, we identify the observation whose cumulative frequency is equal to or just greater than `color{orange}(N/2)` N is the sum of frequencies. This value of the observation lies in the middle of the data, therefore, it is the required median. After finding median, we obtain the mean of the absolute values of the deviations from median.Thus,
`color{blue}(M.D .(M) = 1/N sum_(i=1)^(n) f_i |x_i - M|)`
`color{red}((b) "Continuous frequency distribution")`
A continuous frequency distribution is a series in which the data are classified into different class-intervals without gaps along with their respective frequencies.
For example, marks obtained by `100` students are presented in a continuous frequency distribution as follows :
`color{red}((i) "Mean deviation about mean")`
While calculating the mean of a continuous frequency distribution, we had made the assumption that the frequency in each class is centred at its mid-point.
Here also, we write the mid-point of each given class and proceed further as for a discrete frequency distribution to find the mean deviation.
`color{red}((ii) "Mean deviation about median")`
The process of finding the mean deviation about median for a continuous frequency distribution is similar as we did for mean deviation about the mean.
The only difference lies in the replacement of the mean by median while taking deviations.
Let us recall the process of finding median for a continuous frequency distribution. The data is first arranged in ascending order. Then, the median of continuous frequency distribution is obtained by first identifying the class in which median lies (median class) and then applying the formula
`color{blue}("Median" = l + (N/2-C)/fxxh)`
where median class is the class interval whose cumulative frequency is just greater than or equal to `color{orange}((N/2),N)` is the sum of frequencie `color{orange}( l , f , h)` and `color{orange}(C)` are, respectively the lower limit , the frequency, the width of the median class and C the cumulative frequency of the class just preceding the median class.
After finding the median, the absolute values of the deviations of mid-point `color{orange}(x_i)` of each class from the median i.e., `color{orange}(|x_i - M|)` are obtained.
Then `color{blue}(M.D. (M) = 1/N sum_(i=1)^(n) f_i |x_i-M|)`
We know that data can be grouped into two ways :
`color(red)((a))` `color(blue)(" Discrete frequency distribution,")`
`color(red)((b))` `color(blue)(" Continuous frequency distribution.")`
Let us discuss the method of finding mean deviation for both types of the data.
`color{red}("(a) Discrete frequency distribution")`
Let the given data consist of n distinct values `color{orange}(x_1, x_2, ..., x_n)` occurring with frequencies `color{orange}(f_1, f_2 , ..., f_n)` respectively.
This data can be represented in the tabular form as given below, and is `color(blue)("called discrete frequency distribution :")`
`color{orange}(x : " " x_1" "x_2" "x_3 ........ x_n)`
`color{orange}(f : " " f_1" "f_2" " f_3 ........ f_n)`
`color{red}("(i) Mean deviation about mean")`
First of all we find the mean `color(blue)(barx)` of the given data by using the formula
`color{orange}(barx = (sum_(i=1)^(n) x_i f_i)/(sum_(i=1)^(n)f_i) = 1/N sum_(i=1)^(n) x_i f_i,)`
where `color{navy}(sum_(i=1)^(n) x_i f_i,)` denotes the sum of the products of observations `x_i` with their respective frequencies `color{orange}(f_i)`
and `color{navy}(N = sum_(i=1)^(n) x_i f_i,)` is the sum of the frequencies.
Then, we find the deviations of observations `x_i` from the mean `barx` and take their absolute values, i.e., `color{orange}(|x_i - bar x|)` for all `i =1, 2,..., n.`
After this, find the mean of the absolute values of the deviations, which is the required mean deviation about the mean. Thus
`color{blue}(M.D .(barx) = (sum_(i=1)^(n) f_i|x_i-barx|)/(sum_(i=1)^(n) f_i) = 1/N * sum_(i=1)^(n) f_i|x_i - barx|)`
`color{red}("(ii) Mean deviation about median")`
To find mean deviation about median, we find the median of the given discrete frequency distribution.
For this the observations are arranged in ascending order. After this the cumulative frequencies are obtained. Then, we identify the observation whose cumulative frequency is equal to or just greater than `color{orange}(N/2)` N is the sum of frequencies. This value of the observation lies in the middle of the data, therefore, it is the required median. After finding median, we obtain the mean of the absolute values of the deviations from median.Thus,
`color{blue}(M.D .(M) = 1/N sum_(i=1)^(n) f_i |x_i - M|)`
`color{red}((b) "Continuous frequency distribution")`
A continuous frequency distribution is a series in which the data are classified into different class-intervals without gaps along with their respective frequencies.
For example, marks obtained by `100` students are presented in a continuous frequency distribution as follows :
`color{red}((i) "Mean deviation about mean")`
While calculating the mean of a continuous frequency distribution, we had made the assumption that the frequency in each class is centred at its mid-point.
Here also, we write the mid-point of each given class and proceed further as for a discrete frequency distribution to find the mean deviation.
`color{red}((ii) "Mean deviation about median")`
The process of finding the mean deviation about median for a continuous frequency distribution is similar as we did for mean deviation about the mean.
The only difference lies in the replacement of the mean by median while taking deviations.
Let us recall the process of finding median for a continuous frequency distribution. The data is first arranged in ascending order. Then, the median of continuous frequency distribution is obtained by first identifying the class in which median lies (median class) and then applying the formula
`color{blue}("Median" = l + (N/2-C)/fxxh)`
where median class is the class interval whose cumulative frequency is just greater than or equal to `color{orange}((N/2),N)` is the sum of frequencie `color{orange}( l , f , h)` and `color{orange}(C)` are, respectively the lower limit , the frequency, the width of the median class and C the cumulative frequency of the class just preceding the median class.
After finding the median, the absolute values of the deviations of mid-point `color{orange}(x_i)` of each class from the median i.e., `color{orange}(|x_i - M|)` are obtained.
Then `color{blue}(M.D. (M) = 1/N sum_(i=1)^(n) f_i |x_i-M|)`