Elementary Statistics and Computer Application

Arithmetic mean or mean

• Arithmetic mean or simply the mean of a variable is defined as the sum of the observations divided by the number of observations. It is denoted by the symbol If the variable x assumes n values x1, x2 … xn then the mean is given by
 


• This formula is for the ungrouped or raw data.
Example 1

• Calculate the mean for pH levels of soil 6.8, 6.6, 5.2, 5.6, 5.8
Solution

Grouped Data

• The mean for grouped data is obtained from the following formula

• where x = the mid-point of individual class
• f = the frequency of individual class
• n = the sum of the frequencies or total frequencies in a sample.
Short-cut method



Where
A = any value in x
n = total frequency
c = width of the class interval
Example 2

• Given the following frequency distribution, calculate the arithmetic mean
Marks	64	63	62	61	60	59
Number of Students	8	18	12	9	7	6

Solution

X	F	Fx	D=x-A	Fd
64	8	512	2	16
63	18	1134	1	18
62	12	744	0	0
61	9	549	-1	-9
60	7	420	-2	-14
59	6	354	-3	-18
	60	3713		-7

Direct method


Short-cut method

Here A = 62


Example 3

• For the frequency distribution of seed yield of seasamum given in table, calculate the mean yield per plot.
Yield per plot n(in g)	64.5-84.5	84.5-104.5	104.5-124.5	124.5-144.5
No of plots	3	5	7	20


Solution
Yield ( in g)	No of Plots (f)	Mid X		Fd
64.5-84.5	3	74.5	-1	-3
84.5-104.5	5	94.5	0	0
104.5-124.5	7	114.5	1	7
124.5-144.5	20	134.5	2	40
Total	35			44


A=94.5
The mean yield per plot is
Direct method:


= =119.64 gms

Shortcut method



Merits and demerits of Arithmetic mean
Merits

• It is rigidly defined.
• It is easy to understand and easy to calculate.
• If the number of items is sufficiently large, it is more accurate and more reliable.
• It is a calculated value and is not based on its position in the series.
• It is possible to calculate even if some of the details of the data are lacking.
• Of all averages, it is affected least by fluctuations of sampling.
• It provides a good basis for comparison.
Demerits

• It cannot be obtained by inspection nor located through a frequency graph.
• It cannot be in the study of qualitative phenomena not capable of numerical measurement i.e.
• Intelligence, beauty, honesty etc.,
• It can ignore any single item only at the risk of losing its accuracy.
• It is affected very much by extreme values.
• It cannot be calculated for open-end classes.
• It may lead to fallacious conclusions, if the details of the data from which it is computed are
• not given