##### Mean

Mean, Median and Mode
Introduction: In this world, we need a large amount of data to be collected in the forms of groups or individual.

For this purpose statistical tests are applied which helps us to generalize the data in groups and to compare with others group.

Statistical measuring tests include Mean, Median and Mode.

These three are commonly used to analyze data.

Mean: It refers to average, the sum of all the data items are divided by the number of data items.

Example: Mean is the average of numbers
To calculate we just add up all the numbers and divided by the how many numbers there are.

Also known as average.

The formula for calculating a mean is
Mean= (X1 +X2 +X3 +…+XN)/N
Example 1: Where X1, X2, X3 are the values of the given observation average and N = the number of observation.

Let’s assume that you like to find the mean price of company XYZ for the last four year. Here is the stock price for each of the four years:
Year 1 Rupees 10
Year 2 Rupees 15
Year 3 Rupees 20
Year 4 Rupees 25
Using this information and the formula above, we can calculate that the mean price of company XYZ is (Rupees 10 + Rupees 15 + Rupees 20 + Rupees 25) 4= Rupees 17.50
The mean is always between the smallest and the largest of the number in the set.

Example 2: What is mean of 3, 6 and 9?
Divide by how many number (i.e we added 3 numbers) 18/3=6
So the Mean is 6.

MEDIAN
Median: is the middle number in any set of data which is in the form of lower to higher or least to greatest.

If there is even number in a set of data we have to take average of two middle numbers to find median
First arrange numbers from lower to highest.

Find the middle number
If data is even take mean (average) of two middle numbers
Important Note for Formula in case of Odd and Even.

When the number of values in the data set in odd, the median is the middle value when the data is ordered. Thus the median is (N+1÷2) th
When the number values in the data set is even there are two middle values in the ordered. Thus the median is the average of (N÷ 2)th and (N÷2+1) th .

Example 1: The ages of 15 people living in old age home are 65,62,78,82,89,90,73,69,70,70,71,72,78,68 and 72. Find the median of the data set given.

Solution:
Step 1: Arrange the data in ascending order.

62, 65, 68, 69, 70, 70, 71, 72, 72, 73, 78, 78, 82, 89, 90
Step 2: The number of data values N =15. Hence median is the 8 value in the ordered array.

With formula:
(N+1÷ 2) th N is 15 which is total figures of number is given above
SO (N+1÷ 2) th if we put the value in N then,
15+1÷ 2
16÷2= 8
To find Median age of the inmates.

62, 65, 68, 69, 70, 70, 71, 72, 72, 73, 78, 78, 82, 89, 90
(72 )is THE Median age.

Example 2: Find the median from below following frequency distribution.

Prize of a dress items in Rupees. 60 75 90 100 110 120
Number of items sold. 20 18 22 18 16 18
Solution:
We rewrite the table including a column for cumulative frequency.

Prize…X Number of items sold …F Cumulative frequency
60 20 20
75 18 38
90 22 60 ?Median class containing the 56 th and 57 th items
100 18 78
110 16 94
120 18 112
N=112 The total numbers of items sold N=112. Hence the median is the average of 112÷2th item and 112÷2+1 th item in the order. That is the average of the 56 th and 57 th items. The row containing the cumulative frequency 60 will contain these two items whose prizes are both 90 Rupees.

Hence the median price of the items sold = 90+90÷2=90 Rupees.

By formula:
(N÷ 2)th and (N÷2+1)
N is 112
Cumulative frequency 60, whose prizes are 90 Rupees.

So (N÷ 2)th and (N÷2+1)
(112÷ 2)=56 and (112÷ 2+1)=57
56 and 57 is the Median.

And the median is 90.

Example 3: Find the median
{13, 11, 9, 23, 16, 26, 10}
Arranging from lower to highest
{9, 10, 11, 13, 16, 23, 26}
13 is Median
If there are two middle numbers
Example: {10, 8, 6, 12, 14, 16}
Arranging {6, 8, 10, 12, 14, 16}
10+12/2=11
11 is the Median
MODE
Mode: mode is the most repeated number in any set or group of data
Example 1: The following is the number problem’s that Mr. Kamal Aftab Choudhary assigned for home work on 10 different days. What is the mode?
8, 11, 9, 14, 9, 15, 18, 6, 9, 10
Solution: Ordering data from least to greatest, we get:
6, 8, 9, 9, 9, 10, 11, 14, 15, 18
Ans: The mode for above is 9.

Example 2:{6,3,9,3,6,5,9,6,11}
6 is the most repeated number so it is mode
Example 3: A marathon race completed by 6 participants. What is the mode of these times given in hours below?
2.7HR, 8.3HR, 3.5HR, 5.1HR, 4.9HR, 3.5HR.

Solution: Ordering the data from least to greatest we get,
2.7HR, 3.5HR, 3.5HR, 4.9HR, 5.1HR, 8.3HR,
Ans: The mode for these above solution is 3.5HR.
Example 3: Mr. Kamal Aftab Choudhary asked students in his class how many siblings they each had.

Find the mode of the data:
0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 5
Look for the value that occurs the most:
0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 5
The Mode is (1) sibling.