Median
Median is the middle number in a sorted list of numbers. To determine the median value in a sequence of numbers, the numbers must first be arranged in value order from lowest to highest. If there is an odd amount of numbers, the median value is the number that is in the middle, with the same amount of numbers below and above. If there is an even amount of numbers in the list, the middle pair must be determined, added together and divided by two to find the median value. The median can be used to determine an approximate average, or mean. The median is sometimes used as opposed to the mean when there are outliers in the sequence that might skew the average of the values. The median of a sequence can be less affected by outliers than the mean.
To find the median value in a list with an odd amount of numbers, one would find the number that is in the middle with an equal amount of numbers on either side of the median.
Example type 1
To find the median, first arrange the numbers in order from lowest to highest: List: 3, 13, 2, 34, 11, 26, 47
Arranged in order, the list becomes: 2, 3, 11, 13, 26, 34, 47
The median is the number in the middle: 2, 3, 11, 13, 26, 34
13 is the median in the list of numbers since there are 3 numbers on either side.
Example type 2
with an even amount of numbers things are slightly different.
In that case we find the middle pair of numbers, and then find the value that is half way between them. This is easily done by adding them together and dividing by two.
Suppose 3, 13, 7, 5, 21, 23, 23, 40, 23, 14, 12, 56, 23, 29 is the number series
When we put those numbers in order we have: 3, 5, 7, 12, 13, 14, 21, 23, 23, 23, 23, 29, 40, 56
There are now fourteen numbers and so we don’t have just one middle number, we have a pair of middle numbers:
3, 5, 7, 12, 13, 14, 21, 23, 23, 23, 23, 29, 40, 56
middle numbers are 21 and 23.
To find the value halfway between them, add them together and divide by 2:
21 + 23 = 44 then 44 ÷ 2 = 22
So the Median in this example is 22.
Mode
Mode is the value which occurs most frequently in a set of observations. Simply put, it is the number which is repeated most, i.e. the number with the highest frequency. In the field of statistics, it is an important tool to interpret data in a relevant manner. Now it is possible for the data set to be multimodal (have more than one mode) which means more than one observation has the same number of frequencies.
Example: Let us find the Mode of the following data
4, 89, 65, 11, 54, 11, 90, 56
Here in these varied observations the most occurring number is 11, hence the Mode = 11
Mode of Grouped Data
As we know that Mode is the most frequently occurring number of a data set. This is easily recognizable in an ungrouped dataset, but if the data set is presented in class intervals, this can get a bit tricky. So how can we calculate Mode of grouped data?
Solution method
- Create a table with two columns
- In column 1 write your class intervals
- In column 2 write the corresponding frequencies
- Locate the maximum frequency denoted by fm
- Determine the class corresponding to fm , this will be your Modal class
- Calculate the Mode using given formula
Mode = L +fm−f1 / (2fm−f1−f2) × h
Where,
L = lower limit of Modal Class
fm = frequency of modal class
h = width of modal class
f1 = frequency of pre modal class
f2 = frequency of post modal class