Question

Find the median and mode of 14, 14, 14, 14, 17, 18, 18, 18, 22, 23, 25, 28.

Answer 1

Hint:
We will first arrange the given data in either ascending or descending order. Then, we will count the number of values in the data. We will then use the appropriate formula of median and find the value of the median. Also, we will check the most occurring value in the given data to get the mode of the data.

Formula used:
We will use the following formulas:
1) Median \[ = \dfrac{{{{\left( {\dfrac{n}{2}} \right)}^{th}} + {{\left( {\dfrac{n}{2} + 1} \right)}^{th}}}}{2}\]observation, where \[n\] is even.
2) Mode \[ = \] the most occurring data

Complete step by step solution:
The data given to us is 14,25,14,28,18,17,18,14,23,22,14,18.
Let us first find the median.
We know that the median of the data is the middle value.
Now, we will arrange the given data in the ascending order. We have
14,14,14,14,17,18,18,18,22,23,25,28
Here, the total number of values is \[12\], which is even.
For, even number of values, there are two medians. But, we need only one value as the median. So, we will take the average of the two middle values.
Substituting \[n = 12\] in the formula Median \[ = \dfrac{{{{\left( {\dfrac{n}{2}} \right)}^{th}} + {{\left( {\dfrac{n}{2} + 1} \right)}^{th}}}}{2}\] observation, we get

Median \[ = \dfrac{{{{\left( {\dfrac{{12}}{2}} \right)}^{th}} + {{\left( {\dfrac{{12}}{2} + 1} \right)}^{th}}}}{2}\]
Dividing the terms, we get
\[ \Rightarrow \] Median \[ = \dfrac{{{6^{th}} + {{\left( {6 + 1} \right)}^{th}}}}{2}\]
Adding the terms, we get
\[ \Rightarrow \] Median \[ = \dfrac{{{6^{th}} + {7^{th}}}}{2}\]
Hence, the median is the average of the \[{6^{th}}\] and the \[{7^{th}}\] observations.
We observe from the given data that the \[{6^{th}}\] observation is 18 and the \[{7^{th}}\] observation is also 18. Therefore, we get the median as the average of 18 and 18 i.e.,
\[ \Rightarrow \] Median \[ = \dfrac{{18 + 18}}{2} = \dfrac{{36}}{2}\]
Dividing 18 by 2, we get
\[ \Rightarrow \] Median \[ = 18\]
Now, we have to find the mode of the data. We know that the mode is the most frequently occurring data. Here, we see that 14 occurs 4 times, which is the maximum. Hence,
Mode \[ = 14\]

So, the median of the given data is 18 and the mode is 14.

Note:
We know that the median is defined as the middle value of the given list of numbers or data. Here, we need to keep in mind that the formula of median for even number of terms and odd number of terms is different. That’s why we have applied the formula of a median number of terms. Before finding the median we need to arrange the given set in either descending order or ascending order. If we will arrange the given set we will get the wrong answer.
We know that there are different types of modes. If there are three modes in a given set of data, then it is called the trimodal mode. If there are four or more than four modes then it is called a multimodal mode. If in the data there are two modes, then the set is called bimodal.