Average of Averages – Definition, Applications, and Examples

This article dives deep into the mathematical and statistical implications of averaging averages, elucidating potential pitfalls, best practices, and the underlying principles that guide this practice. 

Defining Average of Averages

The “average of averages” refers to a calculation where individual averages from separate subsets of data are combined to produce a single overall average. In essence, it means taking the average of several different averages.

Mathematically, if we have $n$ subsets, with the i–th subset having an average $\bar{x}_i$​, the average of averages$A$ is: $\bar{x}_i$. The average of averages $\bar{x}$A is given by:

A = (1/n) $\sum^n_{i=1} \bar{x}_i$​

However, it’s important to note that this “average of averages” may not always represent the overall average of all data points combined, especially if the subsets are of different sizes. The true average of all data points, $\bar{X}$, is given by:

$\bar{X}$ = (1/N) $\sum^n_{i=1} \sum_{j=1}^{m_i} x_{ij}$

Here, N is the total count of data points, ​ represents the number of data points in the i-th subset, and ${x}_i$​​ denotes the j-th data point within the i-th subset.

If the subsets (sample sizes) are uniform in size, the average of averages will match the overall average. However, if there’s variability in the sizes of these subsets, the two calculated values might differ.​

Properties

The “average of averages” is a concept that can often lead to misunderstandings if not approached with caution. Here are some of the fundamental properties and nuances associated with the average of averages:

Equal Subsets

When all subsets have an equal number of elements, the average of averages is equal to the average of the entire data set.

Mathematically, if each subset $S_i$ has $k$ elements and their averages are:

A = $\frac{1}{n} \sum_{i=1}^{n} \bar{x}_i= \frac{1}{N} \sum_{i=1}^{N} x_i$

where, N = n*k

Unequal Subsets

When the subsets do not have the same number of elements, the average of averages can differ from the average of the entire dataset. In such cases, it represents a simple mean of the subset averages rather than a weighted average, which would account for the size of each subset.

Weighted Average of Averages

A more representative average can be found by weighting each subset’s average by its size (number of elements) and then dividing by the total number of elements across all subsets.

$A_{\text{weighted}} = \sum_{i=1}^{n} (m_i \times \bar{x}_i)$

where $m_i$ is the number of elements in the subset i and is the total number of elements across all subsets.

Potential Misinterpretations

The average of averages can sometimes lead to misleading interpretations, especially when dealing with uneven subsets. This is because large subsets with particularly high or low averages can disproportionately influence the overall average.

Subset Variability

High variability in the sizes of the subsets can lead to greater discrepancies between the simple average of averages and the overall dataset average.

Application-Dependent

Whether using the average of averages is appropriate or not largely depends on the context. In some scenarios, especially where each subset’s importance is equal regardless of its size, the average of averages might be more meaningful.

Relation with Central Tendency

Just like the mean, the average of averages is also susceptible to extreme values in the data, which can skew the resultant value. When subsets have outliers, it’s essential to be cautious about interpreting the average of averages.

Subset Independence

If subsets have some form of dependence on each other, the average of averages might not encapsulate the full essence of the data dynamics.

In essence, while the average of averages can be a valuable tool, it’s essential to use it judiciously, understanding its properties and the implications it carries, especially in contexts where subset sizes vary significantly.

Exercise 

Example 1

Equal Subsets

Data:

• Subset 1: {2, 4, 6}
• Subset 2: {8, 10, 12}

Solution

Average of Subset 1 = 2+4+6/3  = 4

Average of Subset 2 = 8+10+12/3​ = 10

Average of Averages = 4+10/2​ = 7

Actual Overall Average = 2+4+6+8+10+12/6​ = 7

Example 2

Unequal Subsets

Data:

• Subset 1: {2, 4}
• Subset 2: {8, 10, 12, 14}

Solution

Average of Subset 1 = 2+4/2​ = 3

Average of Subset 2 = 8+10+12+14/4 = 11

Average of Averages = 3+11/2​ = 7

Actual Overall Average = 2+4+8+10+12+14/6​ = 8.33

Example 3

Equal Subsets with Outliers

Data:

• Subset 1: {1, 2, 99}
• Subset 2: {3, 4, 5}

Solution

Average of Subset 1 = 1+2+99/3​ = 34

Average of Subset 2 = 3+4+5/3​ = 4

Average of Averages = 34+4/2​ = 19

Actual Overall Average = 1+2+99+3+4+5/6= 19

Example 4

Weighted Average

Data:

• Subset 1: {1, 2}
• Subset 2: {8, 9, 10, 11}

Solution

Average of Subset 1 = 1+2/2​ = 1.5

Average of Subset 2 = 8+9+10+11/4​ = 9.5

Weighted Average = (2×1.5)+(4×9.5)/6 = 7.33

Example 5

Zero Elements

Data:

• Subset 1: {0, 0, 0}
• Subset 2: {10, 10, 10}

Solution

Average of Subset 1 = 0

Average of Subset 2 = 10

Average of Averages = 0+10/2​ = 5

Actual Overall Average = 0+0+0+10+10+10/6= 5

Example 6

Negative Numbers

Data:

• Subset 1: {-5, -3, -1}
• Subset 2: {1, 3, 5}

Solution

Average of Subset 1 = −5−3−1/3 = -3

Average of Subset 2 = 1+3+5​ = 3

Average of Averages = 0

Actual Overall Average = 0

Example 7

Single Data Point Subsets

Data:

• Subset 1: {4}
• Subset 2: {8}

Solution

Average of Subset 1 = 4

Average of Subset 2 = 8

Average of Averages = 4+8/2 = 6

Actual Overall Average = 4+8/2​ = 6

Example 8

Fractional Numbers

Data:

• Subset 1: {0.5, 1.5, 2.5}
• Subset 2: {3.5, 4.5}

Solution

Average of Subset 1 = 0.5+1.5+2.5/3​ = 1.5

Average of Subset 2 = 3.5+4.5/2 = 4

Average of Averages = 1.5+4/2= 2.75

Actual Overall Average = 0.5+1.5+2.5+3.5+4.5/5​ = 2.5

Applications