Question

Calculate mean deviation from median for the following:

Answer 1

Hint:
Here, we have to use the concept of the median and mean deviation in this question. So firstly we have to convert the given cumulative frequency in the data into the normal frequency and then calculate the median of the data by using the formula of median. Then we have to calculate the mean deviation from the median by using the basic formula of the mean deviation.

Complete Complete Step by Step Solution:
Firstly we have to write the given data in the class interval and then we will convert the cumulative frequency given in the question into the normal frequency and also find out the middle point of the class intervals i.e. \[{{\text{x}}_{\text{i}}}\]. Therefore, we get

Now we can clearly see that \[\Sigma {{\text{f}}_{\text{i}}} = 100\]which means total number of students is equal to 100 i.e. \[{\text{N = 100}}\].
We can clearly see that from the above table that the median class is 40-50.
Now we have to calculate the median of the data by using the formula of the median.
Median\[{\text{ = L + }}\left( {\dfrac{{\dfrac{{\text{N}}}{2} - {\text{cf}}}}{{\text{f}}}} \right){\text{h}}\]where
L is the lower limit of the median class i.e. 40.
\[{\text{cf}}\]is the cumulative frequency i.e. 32.
f is the frequency of the median class i.e. 28.
h is the gap of class interval i.e. 10.
Now we have to put all the values in the formula of the median, we get
Median \[{\text{ = 40 + }}\left( {\dfrac{{\dfrac{{100}}{2} - {\text{32}}}}{{{\text{28}}}}} \right){\text{10 = 40 + }}\left( {\dfrac{{50 - {\text{32}}}}{{{\text{28}}}}} \right) = 40 + \left( {\dfrac{{18}}{{{\text{28}}}}} \right) = 40 + 6.43 = 46.43\]
Now, we have to simply calculate the mean deviation from median from the formula. Therefore,
Mean deviation from median \[ = \dfrac{{\Sigma {{\text{f}}_{\text{i}}}\left| {{{\text{x}}_{\text{i}}} - {\text{M}}} \right|}}{{\Sigma {{\text{f}}_{\text{i}}}}}\]where
M is the median of the data.
\[{{\text{x}}_{\text{i}}}\]is the median of the class interval.
\[{{\text{f}}_{\text{i}}}\]is the frequency of the class interval.
So, now we have to calculate the value of the\[\Sigma {{\text{f}}_{\text{i}}}\left| {{{\text{x}}_{\text{i}}} - {\text{M}}} \right|\]for all the classes. Then we get

From the data, we get\[\Sigma {{\text{f}}_{\text{i}}}\left| {{{\text{x}}_{\text{i}}} - {\text{M}}} \right| = 1428.6\]
So now we have to put the values in the formula of the mean deviation from median, we get
Mean deviation from median\[ = \dfrac{{1428.6}}{{100}} = 14.286\]
So, 14.286 is the mean deviation of the data from median.


Note:
1) Statistics is the science of collecting some data in the form of the number and studying it to forecast or predict its future possibility. Some definitions we should know.
2) Mean is equal to the ratio of sum of the total numbers and total count of the numbers. Mean is also known as the average of the numbers.
3) Mode is the most common or most repeating number.
4) Median is the middle value of the given list of numbers or it is the value which is separating the data into two halves i.e. upper half and lower half.