Mode

Mode is the value which occurs most frequently in a set of observations. For example, {6, 3, 9, 6, 6, 5, 9, 3} the Mode is 6, as it occurs most often.

Fact about Mode : 

1. Sometimes there can be more than one mode.Having two modes is called bimodal.Having more than two modes is called multimodal.
2. There is an empirical relationship between Mean, Median, and Mode. 
   Mean – Mode = 3 [ Mean – Median ]

1. Mode can be useful for qualitative data.
2. Mode can be located graphically.
3. Mode can be computed in an open-end frequency table.
4. Mode is not affected by extremely large or small values.

Formula for Mode of grouped data :  

👁 Image

How to find Mode?
Naive solution: 
Given an n sized unsorted array, find median and mode using counting sort technique. This can be useful when array elements are in limited range.

Examples:  

Input : array a[] = {1, 1, 1, 2, 7, 1}
Output : Mode = 1
 
Input : array a[] = {9, 9, 9, 9, 9}
Output : Mode = 9 

• Auxiliary(count) array before summing its previous counts, c[]: 
  Index: 0 1 2 3 4 5 6 7 8 9 10 
  count: 0 4 1 0 0 0 0 1 0 0 0
• Mode = index with maximum value of count. 
  Mode = 1(for above example)

Approach : 
Assuming size of input array is n: 
Step #1: Take the count array before summing its previous counts into next index. 
Step #2: The index with maximum value stored in it is the mode of given data. 
Step #3: In case there are more than one indexes with maximum value in it, all are results for mode so we can take any. 
Step #4: Store the value at that index in a separate variable called mode.

Below is the implementation:  

Output: 

mode = 1

Time Complexity = O(N + P), where N is the size of input array and P is size of the count array or maximum value in input array. 
Auxiliary Space = O(P), where the value of P is the size of auxiliary array. 
The above solutions works good when array element values are small. Please refer below post for efficient solutions. 
Most frequent element in an array

Basic Program related to Mode :  

• Finding Mean, Median, Mode in Python without libraries
• Median and Mode using Counting Sort