Combined Mean | Formula and Examples

The combined mean is the average of two or more data groups taken together.

• It is used when each group has a different number of observations and a different mean.
• Instead of finding the average of all values individually, the combined mean is calculated using the mean and the number of items in each group.

Example

• Class A has 20 students with an average score of 70.
• Class B has 30 students with an average score of 80.

\bar{x}=\frac{20\times70+30\times80}{20+30} =\frac{1400+2400}{50} =\frac{3800}{50} =76

The combined mean of the two classes is 76.

Formula

If there are two groups:

• Group 1: Mean = \bar{x}_1​, Number of observations = n₁
• Group 2: Mean = \bar{x}_2​, Number of observations = n₂​

Then the combined mean is:

\bar X_{12}=\frac{\bar X_1.N_1+\bar X_2.N_2}{N_1+N_2}

Solved Examples

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

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})=6.54

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.