Math Formula for Median

Math Formula for Median

The median is a statistical measure that represents the middle value in a data set when the values are arranged in ascending or descending order. Unlike the mean, which can be heavily influenced by outliers, the median provides a better measure of central tendency for skewed distributions.

Calculating the Median

The method to calculate the median depends on whether the data set has an odd or even number of observations.

For an Odd Number of Observations

If the data set contains an odd number of observations, the median is the middle value.

Formula:

Median = x((n+1)/2)

where n is the number of observations, and x((n+1)/2) is the middle value.

For an Even Number of Observations

If the data set contains an even number of observations, the median is the average of the two middle values.

Formula:

Median = (x(n/2) + x(n/2 + 1)) / 2

where n is the number of observations, x(n/2) and x(n/2 + 1) are the two middle values.

Example Calculations

Example 1: Odd Number of Observations

Consider the data set: 3, 1, 4, 2, 5

1. Arrange the data in ascending order: 1, 2, 3, 4, 5
2. Number of observations (n) = 5 (odd)
3. The median position is: (5+1)/2 = 3
4. The median value is the 3rd value in the ordered list: 3

Example 2: Even Number of Observations

Consider the data set: 7, 3, 9, 1, 4, 6

1. Arrange the data in ascending order: 1, 3, 4, 6, 7, 9
2. Number of observations (n) = 6 (even)
3. The median positions are: 6/2 = 3 and 6/2 + 1 = 4
4. The median value is the average of the 3rd and 4th values: (4 + 6) / 2 = 5

Example 3: Median of a Frequency Distribution

For a grouped frequency distribution, the median can be found using the following formula:

Median = L + ((n/2 - CF) / f) * h

where:

• L is the lower boundary of the median class,
• n is the total number of observations,
• CF is the cumulative frequency of the class preceding the median class,
• f is the frequency of the median class,
• h is the class width.

Consider the following frequency distribution:

Class Interval	Frequency
0 - 10	5
10 - 20	8
20 - 30	12
30 - 40	6
40 - 50	3

1. Calculate the cumulative frequency:

Class Interval	Frequency	Cumulative Frequency
0 - 10	5	5
10 - 20	8	13
20 - 30	12	25
30 - 40	6	31
40 - 50	3	34

2. Determine the median class. Since n = 34, n/2 = 17. The median class is 20 - 30 because its cumulative frequency just exceeds 17.
3. Using the formula:

L = 20

CF = 13

f = 12

h = 10

Median = 20 + ((17 - 13) / 12) * 10 = 20 + (4 / 12) * 10 = 20 + (1/3) * 10 = 20 + 3.33 = 23.33

Therefore, the median is 23.33.

Conclusion

The median is a useful statistical measure that represents the center of a data set. It is less affected by outliers and skewed data compared to the mean. Whether dealing with raw data, an odd or even number of observations, or a frequency distribution, the median provides a valuable insight into the central tendency of the data.

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