# 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

## 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

## 10 students scored the following marks in their biology exam. Find the mean of their marks.

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

## 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

## Explanation:

No. of observations, n = 7

zsum_(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)

## 1. Find the arithmetic mean of this grouped data.

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

## 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.

## Find the arithmetic mean of the continuous grouped data as shown below :

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

## 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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