Median
- Another frequently used measure of center is the median.
- Essentially, the median of a data set is the number that divides the bottom 50% of the data from the top 50%.
- The median of a data set is the middle value in its ordered list.
Steps to obtain median
- Arrange the data in increasing order. Let n be the total number of observations in the dataset.
- If the number of observations is odd, then the median is the observation exactly in the middle of the ordered list, i.e. n+1 observation
- If the number of obsevations is even, then the median is the mean of the two middle observations in the ordered list, i.e. mean of n / 2 and n / 2 + 1 observation
Example
- 2, 12, 5, 7, 6, 7, 3
- Arrange the data in increasing order 2, 3, 5, 6, 7, 7, 12
- n = 7 odd, median is the n+1 / 2 = 8 / 2 = 4th observation, “6”.
- 2,105, 5, 7, 6, 7, 3
- Arrange the data in increasing order 2, 3, 5, 6, 7, 7, 105
- n = 7 odd, median is the n+1 / 2 = 8 / 2 = 4th observation,“6”.
- 2, 105, 5, 7, 6, 3
- Arrange the data in increasing order 2, 3, 5, 6, 7, 105
- n = 6 even, median is the average of n / 2 and n / 2 + 1 observation = 5+6 / 2 = 5.5.
The sample mean is sensitive to outliers, whereas the sample median is not sensitive to outliers. the median is not very sensitive to the outliers the way the mean was.
Adding a constant
- Let yi = xi + c where c is a constant then new median = old median + c
- Example: Recall the marks of students 68,79,38,68,35,70,61,47,58,66.
- Arranging in ascending order 35,38,47,58,61,66,68,68,70,79
- The median for this data is the average of n / 2 and n / 2 + 1 observation which is 61+66 / 2 = 127 / 2 = 63.5
- Suppose the teacher has decided to add 5 marks to each student.
- Then the data in ascending order is 40,43,52,63,66,71,73,73,75,84
- The median of the new dataset is 66+71 / 2 = 137 / 2 = 68.5
- Note 68.5=63.5+5
Multiplying a constant
- Let yi = xi c where c is a constant then new median = old median × c
- Example: Recall the marks of students 68,79,38,68,35,70,61,47,58,66.
- We already know median for this data is 63.5
- Suppose the teacher has decided to scale down each mark by 40%, in other words each mark is multiplied by 0.4.
- Then the data becomes 27.2, 31.6, 15.2, 27.2, 14, 28, 24.4, 18.8, 23.2, 26.4
- The ascending order is 14, 15.2, 18.8, 23.2, 24.4, 26.4, 27.2, 28, 31.6
- The median of new dataset is 24.4+26.4 / 2 = 50.8 /2 = 25.4
- Note 25.4 = 0.4 × 63.5
Mode
- Another measure of central tendency is the sample mode.
- The mode of a data set is its most frequently occurring value.
- If no value occurs more than once, then the data set has no mode.
- Else, the value that occurs with the greatest frequency is a mode of the data set.
Example
- 2, 12, 5, 7, 6, 7, 3; 7 occurs twice, hence 7 is mode
- 2. 2, 105, 5, 7, 6, 7, 3 7 is mode
- 3. 2, 105, 5, 7, 6, 3 no mode
Adding a constant
- Let yi = xi + c where c is a constant then new mode = old mode + c
- Example: Recall the marks of students 68,79,38,68,35,70,61,47,58,66.
- The mode for this data is 68
- Suppose the teacher has decided to add 5 marks to each student.
- Then the data in ascending order is 40,43,52,63,66,71,73,73,75,84
- The mode of the new dataset is 73
- Note 73 = 68 + 5
Multiplying a constant
- Let yi = xi c where c is a constant then new mode = old mode × c
- Example: Recall the marks of students 68,79,38,68,35,70,61,47,58,66.
- We already know mode for this data is 68
- Suppose the teacher has decided to scale down each mark by 40%, in other words each mark is multiplied by 0.4.
- Then the data becomes 27.2, 31.6, 15.2, 27.2, 14, 28, 24.4, 18.8, 23.2, 26.4
- The mode of new dataset is 27.2
- Note 27.2 = 0.4 × 68
Mode is not affected by the outliers in the data.
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