Mean

Statistics deal with data analysis. The condensation of data is the first step in rendering comprehensible a long series of individual observations. This was made possible by classifying and tabulating the data and then presenting it in the form of frequency distributions. Once the frequency distributions are formed, the next step is to study their characteristics, so that it can be compared with each other.

Frequency distribution may be studied in two ways:

1) By diagrams and graphs
2) By measuring quantitatively.

Study by the first method is less accurate and therefore any conclusion drawn from it may not be relied upon to a great extent. So the second method is considered more accurate than the first one as it involves numbers which can be compared easily i.e. which is the greater of any 2 or more given numbers. Therefore it is necessary that such a representative number should truly represent the series. For this purpose different authorities have selected those numbers that measure the central tendency of the individual items of the series. The measure of this tendency is called the measure of the central tendency or averages. There are five types of averages which are commonly known:

1. Arithmetic average or mean
2. Median
3. Mode
4. Geometric mean
5. Harmonic mean

An average or mean, in whatever way it may be defined is merely a particular value of the variable. Therefore it is expressed necessarily in the same unit in which the series is. For example if the variable is a weight in kilogram, the average will also be a weight in kilogram.This type of average is very commonly used in various types of study. It is very easy to calculate. If the marks of a tutorial group of five students are 2, 4, 6, 8 and 10 then to find out average marks of these students, we shall merely add these five values and divide the total. The total number of items in this case is thirty and dividing it by 5 we get 6 which is the statistical mean. This is termed as the arithmetic average or mean of the series. Thus mean in math is calculated by adding values of all the items and dividing their total by the number of items. In the case of discrete and continues series, the values of the frequencies are also taken into account.

Arithmetic mean may either be:
1. Simple arithmetic mean
2. Weighted arithmetic mean

Mean definition

Mean or Arithmetic mean is not only a fundamental concept but a very important concept covered in statistics. The definition of mean is as simple as “the mathematical average for the set of numbers” provided the data is in the numerical form. The arithmetic mean, mean and average all these terms are used to define the same term.

Let’s now derive the formula for mean. As the definition says, mean is the mathematical average for the set of numbers, so the sample mean formula is as derived below:

The mean of a series is the quotient obtained by dividing the sum of the values by the number of items.

If x₁, x₂, ......x_n are the N values of a variate X, then the Arithmetic Mean($\bar{X}$) is defined as the quotient of the sum of the values divided by their number . Symbolically,

$\bar{X}$ = $\frac{1}{N}$$[ {x_{1} + x_{2}+ ......+ x_{n}}]$ = $\frac{\sum X }{N}$
Or
$\frac{1}{N}$ $\left ( \sum X \right)$= $\frac{ \sum X}{N}$

Arithmetic Average or $\bar{X}$ ( read as X bar) = $\frac{ \sum X}{N}$

Where
X = values of the variables:
$\sum$ = total of the values and read as summations;
N = Number of items.
Steps: The formula for mean given above suggests certain steps in calculating the mean.
They are:
1. Ascertain the value of $\sum X$ by adding together the various values of X
2. Ascertain the number of items or observations, i.e. N
3. Divide $\sum X$ by N.
Let’s solve some simple examples to understand mean:

Example 1:
The following table gives the monthly income of 10 families in a town.

Family	A	B	C	D	E	F	G	H	I	J
Daily Income in dollars	30	70	10	75	500	8	42	250	40	36

Calculate the average income per month.

Solution: Computation of Arithmetic Average

Families	Income(X) in dollars
A	30
B	70
C	10
D	75
E	500
F	8
G	42
H	250
I	40
J	36
N = 10	$\sum X$ = 1061

$\bar{X}$ = $\frac{\sum X}{N}$ = $\frac{1061}{10}$ = 106.10 dollars

Example 2:
Eight coins were tossed together and the number of heads resulting was observed. The operation was performed 256 times and the frequencies that were obtained for the different values of X, the number of heads are shown in the following table:

X	0	1	2	3	4	5	6	7	8
Frequency	1	9	26	59	72	52	29	7	1

Find out the arithmetic mean.

Solution: Computation of arithmetic average.

X	Frequency(f)	fX
0	1	0
1	9	9
2	26	52
3	59	177
4	72	288
5	52	260
6	29	174
7	7	49
8	1	8

$\bar{X}$ = $\frac{\sum \left ( fx \right )}{N}$ = $\frac{1017}{256}$ = 3.97

Example 3:
Calculate the arithmetic mean of the following.

Marks obtained	Number of students
0 - 9	9
10 - 19	42
20 ­- 29	61
30 - 39	140
40 - 49	250
50 - 59	102
60 - 69	71
70 - 79	23
80 - 89	2

Solution: Computation of the average marks of students.

Marks	f	Mid value of marks(m)	fm
0 - 9	9	4.5	40.50
10 - 19	42	14.5	607.90
20 ­- 29	61	24.5	1494.50
30 - 39	140	34.5	4830.00
40 - 49	250	44.5	11125.00
50 - 59	102	54.5	4559.00
60 - 69	71	64.5	4579.50
70 - 79	23	74.5	1713.50
80 - 89	2	84.5	169.00
	N = $\sum f$ = 700		$\sum fm$ = 30100

$\bar{X}$ = $\frac{\sum fm}{N}$ = $\frac{30100}{700}$ = 43 marks

1) The sum of the deviations of the values in the variable from the mean is zero. This property of arithmetic mean is important as this helps in making the calculation simpler.

2) If $\bar{X}$ and $\bar{Y}$ are the arithmetic mean of 2 series of N observations each, then the corresponding terms of the 2 series are added. To find the arithmetic average which is given by
$\bar{Z}$ = $\bar{X} + $\bar{Y}$
It is to be noted that the total number of observations in each series is constant.

3) Another mathematical characteristic of mean is that the sum of the squared deviations of the items from the mean is minimum, which is less than the sum of the squared deviations of the items from any other values.

4) If each of the items in the series is replaced by the mean, the sum of these substitutions will be equal to the sum of individual items.

The meaning of the arithmetic average is now clear to us. It is easily calculated and determined in every case. It is capable of algebraic manipulation as it is a computed average. The average of two or more series can be obtained from the averages of two or more series. Another merit, which is not shared by other average, lies in the fact that it gives weight to all items in direct proportion to their case.

This is very useful when the information desired is per capita consumption or production or wealth regardless of its distribution. The arithmetic mean takes into consideration all given values. It is rigidly defined by a mathematical equation. It is least effected by fluctuation of sampling. Thus if random samples of the same size are drawn repeatedly from the same population and if for each sample value the mean median and mode are computed, means of different samples will show less variation among them, than medians and modes of various samples. It doesn’t require the arraying of data. It is a value, so selected out of a group that if all the items in the group were uniform in size they would each be equal to the arithmetic mean, thus the total deviation of the items from the arithmetic mean is zero. The only disadvantage or demerit is that the arithmetic mean may not be represented in the actual data. It can hardly be located by inspection, whereas mode and median can be. It is affected to a greater degree by unduly large or small items.

Example 1

Estimate the mean value for the following set of data.

x	11	12	14	16	19
p(x)	0.14	0.15	0.16	0.32	0.23

Solution:

x	10	20	50	70	80
p(x)	0.13	0.22	0.14	0.33	0.18

Solution:

x	1	5	8	12	17
p(x)	0.18	0.23	0.21	0.25	0.13

Solution: