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Median

The median is the middle value when the data is arranged in order of size.
In other words, the median divides the whole set of values in two parts such that half of the observations are less than or equal to it and half are more than or equal to it.

Find the median of the followin set of data: 2, 3, 5, 3, 4, 3, 6
Step 1. Rewrite the numbers in ascending order: 2, 3, 3, 3, 4, 5, 6.
Step 2. There are 7 values in the data set. The median is the fourth value.

The median is 3.
• If the total number of given values n, is an odd number, then there exists only one middlemost value, namely the th value in the arrangement and it represents the median of the values.

• Find the median for the following set of values:

2 4 3 5 1 4 5 3 3

Step 1. Rank the data in ascending order as follows:

1 2 3 3 3 4 4 5 5

Step 2. Because the number of values in this set is odd (nine), there are four values less than and four values greater than the median. Therefore, the median is teh fifth value, 3.

• If the total number of given values n, is an even number, median may not be ubiquely determined. In fact, any possible value between the two middle values, namely, the th and the th values in the ordered arrangement, may be takes as median. But in order to obtain a definite value, the arithmetic mean of the th and the th values is regarded as the median of te set of values, by convention.

Find the median for the following set of values:

0 2 3 5 1 4 5 3

Step 1. Rank the data in ascending order as follows:

0 2 3 3 4 4 5 5

Step 2. Because the number of values in this set is even (eight), the median is the midpoint between the fourth and the fifth values, 3 and 4.

The median for grouped data is slightly more difficult to compute. We know that the median occurs in the particular class interval for which the cumulative frequency is . On observing the less-than type, say, cumulative frequencies, we can obtain the class interval that contains te median. In fact, the cumulative frequency for this interval is just more than or equal to .

The value of the median for grouped data can be approximately obtained by the following formula: (Proof):
where l and u respectively denote the lower and upper class-boundaries of the class in which the median lies and Fl and Fu represent the corresponding cumulative frequencies.
or
where and c respectively represent the frequency and the width of the class-interval containing the median.

Find the median of following grouped data:

 Marks Number of students 0-10 2 10-20 12 20-30 22 30-40 8 40-50 6

Let us construct a cumulative frequency table of less than type for the above data to find the particular class interval where the median occurs.
 Marks Number of students Less than 10 2 Less than 20 14 Less than 30 36 Less than 40 44 Less than 50 50

After than we calculate =25, with its help we determine the class whose cumulative frequency is nearly equal to =25. This class is known as median class. Then, the median is calculated by
the following formula:

The median is 25.

Properties:
• If two variables x and y be linearly related in the form y=a+bx, then Me(y)=a+b·Me(x)