Defining the Central Tendency of Data
Upon observing the image, notice that mean, mode and median are used to measure the central tendency or to find a value in a dataset that acts as a representation of the entire dataset. Let us look into mode and median specifically, and find the median using two formulas.
List of Formulas for Mode and Median
The list below gives us the formulas that are applied to calculate the mode and median of a dataset.
Let us consider a dataset with n terms.
- Median
- When n is an odd number of terms, then, median = Middle Term
- When n is an even number of terms, then median = \(\frac{\text{sum of two middle terms}}{2}\)
Note: To find the median the data should always be arranged in either ascending or descending order.
- Mode:
In a given data set the Mode/Modes = Most repeated term/terms.
Further Describing the Median
In a given dataset, if there is an odd number of terms, then the median will be the middle term. And if a given set of data has an even number of terms, then the median will be the mean of the two middle terms.
Further Describing the Mode
The mode of a given dataset is defined as the value or values that occur the most number of times or in statistical terms, the mode is described to be a value that has the highest frequency in a dataset.However, f a data set has multiple values that occur the same number of times, and there is no other value that occurs more number of times compared to these values, then, we will have multiple modal values. There is a possibility that all values occur only once in a dataset. In this case there will be no mode.
So, we can conclude that a given dataset can have one mode, more than one mode, or no mode at all.
Solved Examples - Median formula and Mode of Data
Example 1: Find the median of the following scores.
80 | 40 | 65 | 80 |
|---|---|---|---|
50 | 95 | 75 | 60 |
Solution:
Arranging the data in descending order,
40, 50, 60, 65, 75, 80, 80, 95
As we know, for a set containing an even number of terms, the median will be the mean of the two middle values.
So, the median \(=~\frac{65~+~75}{2}\)
\(=~\frac{140}{2}\)
\(=70\)
Hence, the median for the given data is 70.
Example 2: Find the mode of the data.
30, 3, 16, 7, 8, 4, 16, 12
Solution:
Arranging the data in descending order,
3, 4, 7, 8, 16, 16, 19, 30
As we know, the value which occurs most often in a data set is defined as the mode of that data.
So here, the value 16 is repeated most often.
Hence, the mode for the given data is 16.
Example 3: The heights of of Liam, Noah, Lucas, Oliver, and Noah are:
155 cm, 155cm, 158 cm, 164 cm and 168 cm respectively. Find the median and the mode of their heights.
Solution:
Let us arrange the heights of Liam, Noah, Lucas, Oliver, and Noah in ascending order.
155 cm, 155cm, 158 cm, 164 cm,168 cm
As we know, in a dataset containing an odd number of terms, the median will be the middle value.
So, the median is 158 cm
To find the mode we know that the value which occurs most often in a data set is defined as the mode of that data.
So here, the value 155 is repeated most often.
Therefore, the mode for the data is 155 cm.
Hence, the median is 158 cm , and the mode for the given data is 155cm.
Example 4: This list shows the favorite type of sport that students like to participate in, from a particular grade.
Organize this data in a table or tabular form. Find the mode of the data.
Game | Number of students |
Baseball | 9 |
Athletics | 6 |
Tennis | 5 |
Soccer | 8 |
Solution:
As we know, the value which occurs most often in a data set is defined as the mode of that data.
So here, the number of students who like baseball is the maximum.
Hence, the mode of the data is 9.
Example 5: The total hours Jerry spent studying in a week is as follows:
8, 8, 9, 10, 13, 14, 15
Find the median and mode of the data.
Solution:
The data is already arranged in ascending order.
As we know, for a data set containing an odd number of terms, the median will be the middle value.
So, the median is = 10 (As the middle term is the 4th term)
As we know, the value which occurs most often in a data set is the mode of that data.
So here, the value 8 is repeated most often.
Therefore, the mode for the given data is 8.
Hence, the median and the mode for the given data are 10 and 8 respectively.
The measures of center are mean, median, and mode.
The value which occurs most often in a data set is defined as the mode of that data.
The median of the data is defined in two ways:
- For a set containing an odd number of terms, the median will be the middle value.
- For a set containing an even number of terms, the median will be the mean of the two middle values.
A dataset can have one mode, more than one mode, or it may even have no mode.
the same as that of the improper fraction. Thus, the required fraction is obtained.
In such a dataset there will be no mode as there are no repeating values.