Combined Mean | Formula and Examples

What is Combined Mean?

When two or more series having different arithmetic means and number of items are combined together, the combined mean of all the series can be calculated using,

Combined Mean i.e. \bar X_{12}=\frac{\bar X_1.N_1+\bar X_2.N_2}{N_1+N_2}

Combined Mean i.e. \bar X_{123}=\frac{\bar X_1.N_1+\bar X_2.N_2+\bar X_3.N_3}{N_1+N_2+N_3}

Where,

\bar X_{12}= Combined Mean

\bar X_1= Mean of the first series

N₁ = Number of items in the first series

\bar X_2= Mean of the second series

N₂ = Number of items in the second series

\bar X_3= Mean of the third series

N₃ = Number of items on the third series

Examples of Combined Mean

Example 1:

Find out the combined mean when \bar X_1=12 , N₁ = 6, \bar X_2=18 , N₂ = 9.

Solution:

Combined Mean (\bar X_{12})=\frac{\bar X_1.N_1+\bar X_2.N_2}{N_1+N_2}

Combined Mean (\bar X_{12})=\frac{12\times6+18\times9}{6+9}

Combined Mean (\bar X_{12})=15.6

Example 2:

Find out the combined mean when

	Series 1	Series 2
Mean	6	7
No. of Items	12	14

Solution:

Combined Mean (\bar X_{12})=\frac{\bar X_1.N_1+\bar X_2.N_2}{N_1+N_2}

Combined Mean (\bar X_{12})=\frac{6\times12+7\times14}{12+14}

Combined Mean (\bar X_{12})=6.53

Example 3:

Class A has 15 students with mean marks of 60, and Class B has 12 students with mean marks of 48. Calculate the combined mean.

Solution:

For Class A, \bar X_1=60 , N₁ = 15

For Class B, \bar X_2=48 , N₂ = 12

The required combined mean \bar X_{12}=\frac{\bar X_1.N_1+\bar X_2.N_2}{N_1+N_2}

\bar X_{12}=\frac{60\times 15+48\times 12}{15+12}

Combined Mean \bar X_{12}=54.67

Example 4:

Assume that group 1 has 25 employees with an average salary of ₹82, group 2 has 32 employees with an average salary of ₹45, and group 3 has 77 employees. If the combined salary of the three groups is 70.86, find out the average salary of group 3.

Solution:

For Group 1, \bar X_1=82 , N₁ = 25

For Group 2, \bar X_2=45 , N₂ = 32

For Group 3, Assume average salary be \bar X_3=m and it is given that N₃ = 77

So, Combined Mean \bar X_{123}=\frac{\bar X_1.N_1+\bar X_2.N_2+\bar X_3.N_3}{N_1+N_2+N_3}

70.86=\frac{(82\times25)+(45\times32)+(m\times77)}{25+32+77}

70.86\times134=2,050+1,440+77m

9,495.24 = 3,490 + 77m

77m = 6005.24

m = ₹77.99 or ₹78

Hence, the average salary of group 3 is ₹78.