**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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