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How to calculate the mode of a Data Set?

  • Last Updated : 21 Sep, 2021

The word probability or chance is extremely frequently utilized in day-to-day life. For example, we generally say, ‘He may come today or ‘probably it may rain tomorrow’ or ‘most probably he will get through the examination’. All these phrases involve an element of uncertainly and probability is a concept which measures the uncertainties. The probability when defined in the simplest way is the chance of occurring a certain event when expressed quantitatively, i.e., the probability is a quantitative measure of the certainty. Probability also means possibility. It is a branch of mathematics that deals with the occurrence of a random event. The value is expressed from 0 to at least 1.

The probability has its origin in the problems dealing with games of chance such as gambling, coin tossing, die throwing and playing cards. In all these cases the outcome of a trial is uncertain. These days probability is widely used in business and economies in the field of predictions for future.

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For example, when we toss a coin, either we get Head or Tail, only two possible outcomes are possible (H, T). But if we toss two coins within the air, there might be three possibilities of events to occur, like both the coins show heads or both show tails or one shows heads and one tail, i.e.(H, H), (H, T),(T, T).



Formula for Probability

The probability formula is defined because the possibility of an occasion to happen is adequate to the ratio of the amount of favorable outcomes and therefore the total number of outcomes.

Probability of event to happen P(E) = Number of favorable outcomes/Total Number of outcomes

Some important terms and concepts

1. Random experiment or Trial: The performance of an experiment is called a trial. An experiment is characterized by the property that its observations under a given set of circumstances do not always lead to the same observed outcome but rather to the different outcomes. If in an experiment all the possible outcomes are known in advance and none of the outcomes can be predicted with certainty, then such an experiment is called a random experiment.

2. Equally Likely Events: Events are said to be equally likely if there is no reason to accept anyone in preference to others. Thus, equally likely events mean the outcome is as likely to occur as the other outcome.

3. Simple and Compound Events: In the case of simple events we consider the probability of happening or non-happening of single events and in the case of compound events we consider the joint occurrence of two or more events.

4. Exhaustive Events: It is the total number of all possible outcomes of any trial.



5. Algebra of Events: If A and B are two events associated with sample space S, then

  • A ∪ B is that the event that either A or B or both occur.
  • A ∩ B is the event that A and B both occur simultaneously.

6. Mutually Exclusive Events:- In an experiment, if the occurrence of an event precludes or rules out the happening of all the other events in same experiment.

7. Probability of an Event:- Assume an event E can occur in r ways out of a sum of n probable or possible equally likely ways. Then the probability of happening of the event or its success is expressed as;

P(E) = r/n

The probability that the event won’t occur or referred to as its failure is expressed as:

P(E’) = (n-r)/n = 1-(r/n)

E’ represents that the event won’t occur.

Therefore, now we can say;

P(E) + P(E’) = 1

This means that the total of all the probabilities in any random test or experiment is equal to 1.



How to calculate the mode?

The mode is that the value that is repeatedly occurring during a given set or we will say that the amount during a data set, which features a high frequency or appears more frequently is named mode or mode.

For example, In the given set of data: {2,4,2,3,5,2,6,2} the mode of the data set is 2 since it has appeared in the set thrice.

Bimodal, Trimodal & Multimodal

  • When there are two modes in a data set, then the set is called bimodal.

For example, The mode of Set A = {1,1,1,3,4,4,6,6,6} is 1 and 6, because both 1 and 6 is repeated three times in the given set.

  • When there are three modes in a data set, then the set is called trimodal.

For example, the mode of Set A = {2,2,2,3,4,4,6,6,6,7,9,9,9} is 2, 6 and 9

  • When there are four or more modes during a data set, then the set is named multimodal

For example, the mode of Set A = {1,1,1,3,4,4,6,6,6,7,9,9,9,11,11,11} is 1, 6, 9 and 11.

Sample Problems

Question 1. Find the mode in the given set of data: 3, 6, 7, 15, 21, 23, 40, 23, 41, 23, 14, 12, 60, 23, 28

Answer:

First arrange the given set of data in ascending order: 



3, 6, 7, 12, 14, 15,  21, 23, 23, 23, 23, 28, 40, 41, 60

Therefore, the mode of the data set is 23 since it has appeared in the set four times.

Question 2. Find the mode in the given set of data: 1, 3, 3, 3, 6, 6, 6, 4, 4, 10

Answer:

First arrange the given set of data in ascending order: 

1, 3, 3, 3, 4, 4, 6, 6, 6, 10

Therefore, the mode of the data set is 3 and 6, because both 3 and 6 is repeated three times in the given set.

Question 3. Find the mode in the given set of data: 15, 8, 26, 35, 15, 33, 20

Answer:

First arrange the given set of data in ascending order: 

8, 15, 15, 20, 26, 33, 35

Therefore, the mode of the data set is 15 since it has appeared in the set twice.

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