Median Mode Median Mode The median of a set of data values is the middle value of the data set when it has been arranged in ascending order. That is, from the smallest value to the highest value. Example 1 :- The marks of nine students in a geography test that had a maximum possible mark of 50 are given below : 47 35 37 32 38 39 36 34 35 Find the median of this set of data values. Solution :- Arrange the data values in order from the lowest value to the highest value: 32 34 35 35 36 37 38 39 47 The fifth data value, 36, is the middle value in this arrangement. In general : If the number of values in the data set is even, then the median is the average of the two middle values. Example 3 :- Find the median of the following data set :12
18
16
21
10
13
17
19
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Solution : Arrange the data values in order from the lowest value to the highest value: 10 12 13 16 17 18 19 21 The number of values in the data set is 8, which is even. So, the median is the average of the two middle values. Mode The mode of a set of data values is the value(s) that occurs most often. The mode has applications in printing. For example, it is important to print more of the most popular books; because printing different books in equal numbers would cause a shortage of some books and an oversupply of others. Likewise, the mode has applications in manufacturing. For example, it is important to manufacture more of the most popular shoes; because manufacturing different shoes in equal numbers would cause a shortage of some shoes and an oversupply of others. Example 4 :- Find the mode of the following data set : 48
44
48
45
42
49
48
Solution : The mode is 48 since it occurs most often. Note : It is possible for a set of data values to have more than one mode. If there are two data values that occur most frequently, we say that the set of data values is bimodal. If there is no data value or data values that occur most frequently, we say that the set of data values has no mode. Analysing Data : The mean, median and mode of a data set are collectively known as measures of central tendency as these three measures focus on where the data is centred or clustered. To analyse data using the mean, median and mode, we need to use the most appropriate measure of central tendency. The following points should be remembered: Learn More What is Central Tendency
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Mutually Exclusive Events Mutually Exclusive Events Definition of Mutually Exclusive Events Two or more events are said to be mutually exclusive if they cannot occur at the same time. In other words, events that have no outcomes in common are said to be mutually exclusive events or disjoint events.If two events A and B are mutually exclusive, then the probability of the occurrence of A or B is the sum of the their individual probabilities. Events are mutually exclusive if they cannot happen at the same time. For example, if we toss a coin, either heads or tails might turn up, but not heads and tails at the same time. Similarly, in a single throw of a die, we can only have one number shown at the top face. The numbers on the face are mutually exclusive events. If A and B are mutually exclusive events then the probability of A happening OR the probability of B happening is P( A) + P( B). P( A or B) = P( A) + P( B) Example 1 : A die is rolled. Let us define event E1 as the set of possible outcomes where the number on the face of the die is even and event E2 as the set of possible outcomes where the number on the face of the die is odd. Are event1 E1 and E2 mutually exclusive?
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Solution to Example 1: We first list the elements of E1 and E2. E1 = {2,4,6} E2 = {1,3,5} E1 and E2 have no elements in common and therefore are mutually exclusive. Another way to answer the above question is to note is that if you roll a die, it shows a number that is either even or odd but no number will be even and odd at the same time. Hence E1 and E2 cannot occur at the same time and are therfore mutually exclusive. Example 2: A die is rolled. Event E1 is the set of possible outcomes where the number on the face of the die is even and event E2 as the set of possible outcomes where the number on the face of the die is greater than 3. Are event E1 and E2 mutually exclusive? Solution to Example 2: The subsets E1 and E2 are given by. E1 = {2,4,6} E2 = {4,5,6} Subsets E1 and E2 have 2 elements in common. If the die shows 4 or 6, both events E1 and E2 will have occured at the same time and therefore E1 and E2 are not mutually exclusive.
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