Provide Information Regarding Statistics & Econometrics : Median Lecture 09

Median

Lecture 09

Median is the most appropriate measure of central tendency if the data is highly skewed. The median is the middle value of the given list of data sorted in ascending or descending order of magnitude.

If the amount of the number is odd, median is the most middle value.

if n is odd

If the number of the number is even, the median is the average of the middle pair.

if n is even

Median for grouped data can be obtained by the following formula.

Where:

L1 and L2 are the lower and upper class boundaries, respectively. c is the cumulative frequency of the preceding class.

Example 3.29: A family has 7 children, aged 9, 17, 15, 12, 19, 21, and 25. What is the median?

Solution: ordering the children’s ages from least to greatest, we get:

Rank	1	2	3	4	5	6	7
Ages	9	12	15	17	19	21	25

The number of observations (n = 7) is odd. We use the following formula:

Example 3.30: Find the median for the following data:

Solution: Arrange the data in ascending order of magnitude.

Order	1	2	3	4	5	6	7	8	9	10
Value	7	15	18	25	26	37	45	46	50	65

The number of observations (n = 10) is even. We use the following formula:

Example 3.31: The following is a frequency table of the score obtained in a statistics test. Find the median score.

Score	0	5	6	7	8	9
No. of students	3	4	7	6	3	2

Solution:

Score	Frequency	Cumulative frequency
0	3	3
5	4	7
6	7	14
7	6	20
8	3	23
9	2	25
	25

As the sum of frequency 25 is odd, we are using the following formula.

Example 3.32: Find the median for the following mark distribution.

Mark	10 – 19	20 – 29	30 – 39	40 – 49	50 – 59	60-69	70 – 79
No. of students	10	16	31	22	41	12	18

Solution:

Mark	Class boundaries	f	Cumulative frequency
10 – 19	9.5 – 19.5	10	10
20 – 29	19.5 – 29.5	16	26
30 – 39	29.5 – 39.5	31	57
40 – 49	39.5 – 49.5	22	79
50 – 59	49.5 – 59.5	41	120
60-69	59.5 – 69.5	12	132
70 - 79	69.5 – 79.5	18	150
		150

To obtain median class, we divide total observation by 2.

Which approximately lies against 39.5–49.5.

Advantages and limitations of Median.

Some advantages of median are given below:

i. It is simple to understand, and in some cases it is obtained by inspection.

ii. It is the middle part of the data, and hence it is not affected by extreme values.

iii. It is a special average used in qualitative data.

iv. In grouped frequency distribution, it can be graphically located by ogive.

Some limitations of the median are given below:

I. It is not based on all the observations.

ii. It is not well defined.

iii. Its capability for further mathematical treatment is limited.

Determination of Median Graphically

The median can be determined graphically from ogive. This can be done in two ways.

i. Convert the frequency distribution into a “less than” or “more than” cumulative frequency distribution.

ii. Present the data graphically in the form of less than or more than ogive.

iii. The total number of observations is n. Find the (n/2) value and mark it on the y-axis. Draw a perpendicular line on the y-axis to the right (or left) to cut the object at point A.

iv. From point A, where the ogive is cut, draw a perpendicular on the x-axis. The point at which it touches the x-axis will be the median.

Example 3.33: Find the median graphically of the following data.

Height (cm)	150 - 155	155 - 160	160 – 165	165 - 170	170 - 175	175 - 180
No. of students	4	7	18	11	6	4

Solution:

Height (cm)	Frequency	Height	Cumulative frequency
150 - 155	4	Less than 150	0
155 - 160	7	Less than155	4
160 – 165	18	Less than 160	11
165 – 170	11	Less than165	29
170 – 175	6	Less than 170	40
175 - 180	4	Less than 175	46
		Less than 180	50

Median = 164

Empirical Relation between Mean, Median and Mode

In moderately skewed distribution, a very important relationship exists among these three measures of central tendencies. This relationship is given by: