The Median

The median (Med) is the measure of central tendency that indicates the middle of the data series. It is therefore the value that separates an ordered distribution into |2| groups that contain the same number of data.

|\text{Median rank}=\dfrac{n+1}{2}|
where
|n:| number of data values in the distribution

When using the formula, there are 2 possible cases:

• If |n| is an odd number, then |\dfrac{n+1}{2}| is a whole number. In this case, the median can be determined directly from the data.

• If |n| is an even number, then |\dfrac{n+1}{2}| is a decimal number. In this case, the median is determined by calculating the mean (average) of the |2| middle data of the distribution.

This formula does not directly indicate the median. It provides the rank of the median, which is the median's position in the distribution. To locate the median, the data in the distribution must be placed in ascending order.

1. Place the data in the distribution in ascending order.

2. Calculate the rank of the median.

3. Find the median.

Here is the number of kilometres Victor travels each day.

|192,| |196,| |134,| |185,| |201,| |188,|​ |197|

What is the median of this distribution?

1. Place the data in the distribution in ascending order.
   We get the following:

   |134,| |185,| |188,| |192,| |196,| |197,| |201|

2. Calculate the rank of the median.
   The median rank formula is used with |n=7,| since there are |7| data values in the distribution.
   ||\begin{align}\text{Median rank}&= \dfrac{7+1}{2}\\&= 4\end{align}||
   The median is therefore the 4th data value in the ordered distribution.

3. Find the median.

   |134,| |185,| |188,| |\boldsymbol{192},| |196,| |197,| |201|

   The 4th data value in the distribution is |192,| so the median is |192.|
   In other words, there are the same number of days on which Victor travelled less than |192\ \text{km}| as there are days on which he travelled more than |192\ \text{km}.|

Here is the number of kilometres Victor travels each day.

|192,| |196,| |134,| |185,| |201,| |188,|​ |197,| |199|

What is the median of this distribution?

1. Place the data in the distribution in ascending order.
   We get the following:

   |134,| |185,| |188,| |192,| |196,| |197,| |199,| |201|

2. Calculate the rank of the median.
   The median rank formula is used with |n=8,| since there are |8| data values in the distribution.
   ||\begin{align}\text{Median rank}&= \dfrac{8+1}{2}\\&= 4.5\end{align}||
   Since the rank obtained is not a whole number, this means that the mean (average) of the 4th and 5th data values must be calculated to find the median.

3. Find the median.

   |134,| |185,| |188,| |\boldsymbol{192},| |\boldsymbol{196},| |197,| |199,| |201|

   The mean of |192| and |196,| is calculated,which is the 4th and 5th data values.
   ||\begin{align}\text{Median}&=\dfrac{192+196}{2}\\&=194\end{align}||
   The median of the distribution is |194.| In other words, Victor travelled less than |194\ \text{km}| half the time and more than |194\ \text{km}| during the other half.

1. Calculate the rank of the median.

2. Add the frequencies to determine the location of the median.

3. Find the median.

What is the median of this condensed data distribution?

1. Calculate the rank of the median.
   The median rank formula is used with |n=25,| since there are |25| data values in the distribution.
   ||\begin{align}\text{Median Rank}&= \dfrac{25+1}{2}\\&= 13\end{align}||

2. Add the frequencies to determine the location of the median.

   Value	Frequency	Cumulative frequency
   |1|	|6|	|6|
   |2|	|12|	|\boldsymbol{18}|
   |3|	|5|	|23|
   |4|	|2|	|25|
   Total	|\boldsymbol{25}|

   With the cumulative frequency, we can conclude that the first |6| data in the distribution are |1s,| and the data from the 7th to the 18th position inclusively are |2|s and so on.

3. Find the median.
   We are interested in the 13th data value, which corresponds to the number |2,| since the 13th data value is located between the 7th and the 18th position. Therefore, the median of the distribution is |2.|

What is the median of this condensed data distribution?

1. Calculate the rank of the median.
   The median rank formula is used with |n=50,| since there are |50| data values in the distribution.
   ||\begin{align}\text{Median rank}&= \dfrac{50+1}{2}\\&= 25.5\end{align}||
   Since the rank is a decimal number, the median is the average of the 25th and 26th data values.

2. Add the frequencies to determine the location of the median.

   Value	Frequency	Cumulative frequency
   |1|	|9|	|9|
   |2|	|16|	|\boldsymbol{25}|
   |3|	|19|	|\boldsymbol{44}|
   |4|	|6|	|50|
   Total	|\boldsymbol{50}|

3. Find the median.
   From the cumulative frequency column, we see that the value |2| is associated with positions 10 to 25 inclusively. Therefore, the 25th data item is |2.| The value |3| is associated with positions 26 to 44 inclusively. So the 26th data item is |3.| Finally, we calculate the average of these 2 data values.
   ||\begin{align}\text{Median}&=\dfrac{2+3}{2}\\&=2.5\end{align}||
   Therefore, the median of this distribution is |2.5.|

1. Calculate the rank of the median.

2. Add the frequencies to determine the location of the median.

3. Find the medial class.

What is the medial class of the following classed data distribution?

1. Calculate the rank of the median.
   The median rank formula is used with |n=41,| since there are |41| data values in the distribution.
   ||\begin{align}\text{Median Rank}&= \dfrac{41+1}{2}\\&= 21\end{align}||
   The median data value is ranked 21st in the distribution.

2. Add the frequencies to determine the location of the median.

   Value	Frequency	Cumulative frequency
   |[0,10[|	|7|	|7|
   |[10,20[|	|12|	|19|
   |[20,30[|	|8|	|\boldsymbol{27}|
   |[30,40[|	|14|	|41|
   Total	|\boldsymbol{41}|

   According to the cumulative frequency column, the data in the 21st position is found to be between the 20th and 27th positions.

3. Find the medial class.
   The medial class is therefore the interval |[20, 30[.|

   To estimate the median, find the middle value of this class.
   ||\begin{align}\text{Median}&\approx\dfrac{20 + 30}{2}\\&= 25\end{align}||

What is the medial class of the following classed data distribution?

1. Calculate the rank of the median.
   The median rank formula is used with |n=124,| since there are |124| data values in the distribution.
   ||\begin{align}\text{Median rank}&= \dfrac{124+1}{2}\\&= 62.5\end{align}||
   So, the median is the mean of the 62nd and 63rd data values.

2. Add the frequencies to determine the location of the median.

   Value	Frequency	Cumulative frequency
   |[0,5[|	|32|	|32|
   |[5,10[|	|28|	|60|
   |[10,15[|	|41|	|\boldsymbol{101}|
   |[15,20[|	|23|	|124|
   Total	|\boldsymbol{124}|

   According to the cumulative frequency column, the 62nd and 63rd data values are located in the same interval.

3. Find the medial class.
   The medial class is |[10,15[.|

It is hard to draw conclusions from the analysis of a distribution based only on its median. However, it does provide some nuance to the analysis of the mean, since the median is not influenced by outliers, unlike the mean.