# Grand Mean (Statistics) – Calculation & Examples

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### Definition of Grand Mean/Combined Mean:

The grand mean(also known as combined mean) refers to the average of the entire data when the data has been subdivided into many groups.

Suppose you want to know the average height of students in a school. The students in the school are naturally subdivided according to their grade such as 1st grade, 2nd grade, etc. Using the mean height of students in each grade we can calculate the grand mean, that is, the average height of all the students in the school.

### Formula to calculate the Grand Mean/Combined Mean:

The grand mean can be calculated using the formula,

### Steps to calculate the Grand Mean:

1. Find out the sample sizes n1, …, nk of the subdivisions.
2. Calculate the mean for each of the above subdivisions.
3. Find out the grand mean using the above formula.

#### Example 1:

Calculate the Grand Mean for the following data subdivided into 2 parts:

[5, 4, 6] [8, 3, 6, 7]

Solution:

Step 1: We note that the sample sizes are n1=3 and n2=4.

Step 2: We calculate the mean for each of the subdivisions as follows,

\bar{x}_1 = \frac{5+4+6}{3} = \frac{15}{3} = 5 \bar{x}_2 = \frac{8+3+6+7}{4} = \frac{24}{4} = 6

Step 3: Using the formula for grand mean we get,

\text{Grand mean }= \frac{n_1 \bar{x}_1 + n_2 \bar{x}_2 }{n_1+ n_2} = \frac{3\times 5 + 4\times 6}{7} = \frac{15+24}{7} = \frac{39}{7} = 5.57

#### Example 2:

Calculate the Grand Mean for the following data subdivided into 3 parts:

[1, 3, 2] [7, 4, 5, 8] [7, 5]

Solution:

The sample sizes are n1=3 and n2=4 and n3=2. The mean for each of the subdivisions is,

\bar{x}_1 = \frac{1+2+3}{3} = \frac{6}{3} = 2 \bar{x}_2 = \frac{7+4+5+8}{4} = \frac{24}{4} = 6 \bar{x}_3 = \frac{7+5}{2} = \frac{12}{2} = 6 \text{Grand mean }= \frac{n_1 \bar{x}_1 + n_2 \bar{x}_2 + n_3 \bar{x}_3}{n_1+ n_2 +n_3} = \frac{3\times 2 + 4\times 6 + 2\times 6}{9} = \frac{6+24+12}{9} = \frac{45}{9} = 5

### Proof of the Formula for Grand Mean/Combined Mean:

Suppose that the data is divided into k parts each of size n_i. Let x_{i1},x_{i2},\ldotsx_{in_i} be the sample values for the i^{th} group. By the defintion of arithmetic means we have that,

\bar{x_i}=\frac{\sum_{j=1}^{n_i} x_{ij}}{n_i} \implies \sum_{j=1}^{n_i} x_{ij} = n_i\bar{x_i} \text{ for all }i=1,2,\ldots,k

Now for calculating the grand mean we sum over all values of x_{ij} and divide by the total number of data values. Therefore we have that,

\begin{align*}\text{Grand Mean }&=\frac{\sum_{i=1}^{k}\sum_{j=1}^{n_i} x_{ij}}{n_1+n_2+\ldots+n_k} \\ &= \frac{\sum_{i=1}^{k}n_i\bar{x_i}}{n_1+n_2+\ldots+n_k} \text{ (from above equation)} \\ &= \frac{n_1\bar{x_1}+ n_2\bar{x_2}+\ldots+n_k\bar{x_k}}{n_1+n_2+\ldots+n_k} \end{align*}

This proves the formula for the combined mean.

### Grand Mean in ANOVA:

The Grand Mean in ANOVA(Analysis of Variance) is used to calculate the total sum of squares TSS. The TSS is calculated by summing the squares of deviations of all data points from the grand mean and then dividing the result by the total number of data points.

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