Mean

Mean, also called the arithmetic mean or average, is obtained by dividing the sum of all the numbers by the quantity of the numbers.

Example

Find the mean of 3, 4 and 5


`(3 + 4 + 5)/6 = 12/6 = 2`


The basic formula of Mean is


`bar(X) = (sum_(i=1)^N X_i)/N `


Where,


`bar(X)` (read as "x" bar) - is the symbol for mean


∑ (the Greek letter sigma) - is the symbol for sum


`Χ_i` - is the sum of the numbers


N- is the quantity of numbers


Example
  `28.33`

Explanation:

We find the mean by applying the formula,


`bar(X) = (sumx)/N `


`bar(X)` = `(5 + 10 + 15 + 20 + 30 + 90) /6`


`bar(X)` = ` 170 /6`


`bar(X)` = `28.33`


So, the mean is `28.33`


Data can be represented as ungrouped or grouped.


Ungrouped data


It is data obtained in original / raw form.


Example

Marks obtained by 22 students in Science exam are as follows:


80, 72, 80, 75, 60, 40, 60, 44, 45, 47, 62, 46, 42, 42, 41,50, 51, 72, 73, 84, 87, 85


Grouped data


When we organize the raw data and put it in groups (known as classes), it is called grouped data.


Example

When we put the same data in a more condensed form, and mention the frequency of each group, then such a table is called grouped frequency distribution table.


Math scores Frequency (f)
80-89 5
70-79 4
60-69 3
50-59 2
40-49 8

Mean of ungrouped data


The formula for the calculation of mean of ungrouped data is given below:


` bar(X)`= `(sum_(i=1)^n x_i)/n `=` (x_1+x_2+...+x_n)/n`


Ungrouped Data Example

Example 1:
Math scores Frequency (f)
1 60
2 47
3 73
4 69
5 56
6 82
7 51
8 63
9 77
10 90
  `66.8`

Explanation:

This states that 10 students got 66.8 marks on an average.

The mean is calculated as:


`bar(X)= (sumx)/n `


`sumx= 60+47+73+69+56+82+51+63+77+90`


`bar(X)= 668/10`


`bar(X) = 66.8`

Example 2
  ` 200`

Explanation:

No. of observations, n = 7


z`sum_(i=1)^7 x_i= 50 + 100 + 150 + 200 + 250 + 300 + 350 `


`sum_(i=1)^7 x_i= 1400`


`barX= (sum_(i=1)^7 x_i)/n `


`barX= 1400/7`


`barX= 200`


Mean of Grouped Data


Data represented in the form of frequency distribution is called grouped data. The following table illustrates the result of an experiment in which a dice was rolled 20 times.Below values were recorded every time.


1,2,3,4,5,6


Math scores Frequency (f)
1 4
2 6
3 2
4 2
5 5
6 1
Total = 20

We find the mean using the below formula:

`bar(X)= (sum_(i=1)^n fx_i)/(sum_(i=1)^n f_i)`


Grouped Data Mean Example

Example 1
Math scores Frequency (f)
1 4
2 6
3 2
4 2
5 5
6 1
Total = 20
 `3`

Explanation:

The formula to determine the arithmetic mean of grouped data is :


`bar(X)= (sum_(i=1)^n fx_i)/(sum_(i=1)^n f_i)`


We find the values of f.x (by multiplying f with x) as shown below:


Number on the Dice (X) Frequency (f) f.X
1 4 4
2 6 12
3 2 6
4 2 8
5 5 25
6 1 6
`sum_(i=1)^6 f_i = 20` `sum_(i=1)^6 fx_i = 61`

Therefore


`bar(X)= (sum_(i=1)^6 fx_i)/(sum_(i=1)^6 f_i)`


`bar(X) = 61/( 20 )`


`bar(X) = 3.05 ~~ 3`


We round off the number to 3.

Example 2
Classes (X) Frequency (f)
0-9 7
10-19 4
20-29 8
30-39 5
40-49 9
50-59 7
60-69 10
Total= 50
 `37.7`

Explanation:

We need to find the midpoint of each of the class using the below formula.


Midpoint = `text(Upper limit + Lower Limit)/2`,


Using this formula, we find the midpoint of each group as shown in the table below. These values are denoted by x in the table.


Number on the dice (X) Frequency (f) f.X
1 4 4
2 6 12
3 2 6
4 2 8
5 5 25
6 1 6
`sum_(i=1)^6 f_i = 20` `sum_(i=1)^6 fx_i = 61`

Next, we multiple each midpoint value with its corresponding value of frequency. The "f.x" values are recorded in a new column as shown in the following table.


Number on the classes Frequency(f) f.X
0-9 7 4.5
10-19 4 14.5
20-29 8 24.5
30-39 5 34.5
40-49 9 44.5
50-59 7 54.5
60-69 10 64.5
Total= 50

Therefore,


`bar(X)= (sum_(i=1)^7 fx_i)/(sum_(i=1)^7 f_i) `


`= 1885/(50) = 37.7`






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