Question

What is the mean, median, mode and range of 5, 30, 35, 20, 5, 25, 20 ?

Answer 1

Hint: To calculate the mean, median and mode of the given data set, we will use their respective formulas. The mean of a data set is the sum of all the data upon total number of data entries. The median is the middle most term of the data set when represented in ascending order. Mode is the number that occurs most frequently and range is equal to the difference between the maximum and minimum value of the data set.

Complete step by step solution:
Let us first rearrange our given set of data in ascending order. It can be done as follows:
5, 5, 20, 20, 25, 30, 35.
Let the mean of this data set be ‘X’.
Let the median of this data set be ‘Y’. And,
Let the mode of this data set be ‘Z’.
Now, the mean of this data set can be calculated as follows:
$\begin{align}
  & \Rightarrow X=\dfrac{5+5+20+20+25+30+35}{7} \\
 & \Rightarrow X=\dfrac{140}{7} \\
 & \therefore X=20 \\
\end{align}$
Now, the median of this data set is the middle-most term. So, out of the 7 terms in our data set, the middle most term is the fourth term. Therefore, the median of our data set is equal to:
$\Rightarrow Y=20$
Now, we can calculate the Mode of the given data set with the help of the following formula:
$\Rightarrow \text{Mode = 3Median - 2Mean}$
Putting the values of mean and median calculated above, we get the Mode as:
$\begin{align}
  & \Rightarrow Z=3Y-2X \\
 & \Rightarrow Z=3\times 20-2\times 20 \\
 & \therefore Z=20 \\
\end{align}$
Now, lastly the range of our data set is equal to:
$\begin{align}
  & \Rightarrow R=35-5 \\
 & \therefore R=30 \\
\end{align}$
Hence, the mean, median, mode and range of 5, 30, 35, 20, 5, 25, 20 comes out to be 20, 20, 20 and 30 respectively.

Note: If we looked carefully, then by definition the mode should have been both 5 as well as 20 as both the items were repeated the maximum number of times. To solve this problem, Karl Pearson gave the formula relating mean, median and mode of a data set which was applied while calculating the mode of our data set.