Statistics 

Mean, Median and Mode
for Ungropued Data and for Grouped Data

Ungrouped Data (Raw Data): The information collected systematically regarding a population or a sample survey is called an ungrouped data. It is also called raw data.

Grouped Data (Classified Data): When a frequency distribution is obtained by dividing an ungrouped data in a number of strata according to the value of variate under study, such an information is called gruoped data or classified data.

Measures of central tendency:
There are several measures of central tendency. Out of these, the following 3 are used more often
1. Mean, 2. Median and 3. Mode


Ungroped Data

Mean of ungrouped data:

Let x1, x2, x3, ..., xn be n observations then mean is obtained by dividing the sum of n observations by n. It is denoted by
X' =
Xi/n
For Eg: Find the mean of 4,6,8,6,7,8
Sol:
Mean X' =
Xi/n
= (4 + 6 + 8 + 6 + 7 + 8)/6
= 39/6
= 6.5

Median of ungrouped data:
If the observations of an ungrouped data are arranged in increasing or decreasing order of their magnitude, a value which divides these ordered observations into two equal parts is called the median of the data. It is denoted by M.

If the number of observations (n) is an odd integer, then
M = Value of (n+1)/2 th observation in the arrangement of observations in increaseing order.

If the number of observations (n) is an even integer, then
M = (Value of n/2 th observation + Value of (n/2 + 1) th observation)/2

For Eg:
Find the median of the following observations
4,6,8,6,7,8,8


Sol: Observations in the ascending order are :
4, 6, 6, 7, 8, 8, 8
Here, n = 7 is odd.
Median :
M = Value of (n+1)/2 th observation
= Value of (7+1)/2 th observation
= Value of 4 th observation
= 7


Mode of ungrouped data:
An observation occurring most frequently in the data is called mode of the data. It is denoted by Z.

For Eg:
Find the median of the following observations
4,6,8,6,7,8,8
Sol:
In the given data, the observation 8 occurs maximum
number of times (3)
\ Mode(Z) = 8



Grouped Data

1. The information regarding the no of children per family is given in the following table. Find the Mean, Median, Mode of the data
No of children 0 1 2 3 4 5
No of families 3 20 15 8 3 1

HINT: For input
For this type of example select X & Freq Option, Input the values like

X Freq
0 3
1 20
2 15
3 8
4 3
5 1


2. The frequency distribution of the marks obtained by 100 students in a test of Mathematics carrying 50 marks is given below. Find the Mean, Median, Mode of the data.

Marks obtained 0 - 9 10 - 19 20 - 29 30 - 39 40 - 49
No of students 8 15 20 45 12

HINT: For input
For this type of example select X,Y & Freq Option, Input the values like

X Y Freq
0 9 8
10 19 15
20 29 20
30 39 45
40 49 12