# 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 subsets, with the subset having an average $\bar{x}_i$, the average of averages 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, is the total count of data points, represents the number of data points in the subset, and ${x}_i$ denotes the data point within the i- 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 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

The concept of calculating the average of averages, also known as a weighted average, is applied in various fields to provide a more accurate representation of data when different groups or subsets have varying significance or sizes. Here are some applications of the average of averages in different fields:

In education, teachers might give different weightings to quizzes, tests, and assignments. By calculating a weighted average, the overall grade reflects the varying importance of each component.

### Economics and Financial Analysis

In finance, stock indices like the S&P 500 use weighted averages to represent the performance of a group of stocks. Companies with larger market capitalization have a greater influence on the index’s movement.

### Marketing and Consumer Behavior

When analyzing customer feedback ratings or reviews, a weighted average might be used to reflect the importance of different products or aspects based on their popularity or significance to customers.

### Quality Control and Manufacturing

In manufacturing, different defects might be assigned different levels of severity. By calculating a weighted average of defect rates, companies can prioritize areas for improvement.

### Healthcare and Medical Research

In medical studies, researchers might consider studies with larger sample sizes as more significant. By assigning weights to study sizes, they can calculate a weighted average to summarize findings.

### Polling and Survey Analysis

In polling, different demographic groups might be given different weights based on their representation in the population. This creates a more accurate estimate of public opinion.

### Sports Rankings and Ratings

In sports rankings, a weighted average might be used to give more importance to recent performance or higher-stake matches, resulting in a more dynamic and accurate ranking.

### Climate and Environmental Studies

Researchers might calculate weighted averages of temperature or pollution levels across different regions to determine overall trends while considering the size or significance of each region.

### Project Management

In project management, task completion times might be weighted differently based on their impact on the overall project timeline.

### Population and Demographics

When analyzing population statistics, a weighted average can be used to provide a more accurate representation of characteristics that might differ significantly between subgroups.

The concept of the average of averages or weighted averages is versatile and finds applications in various fields where different subsets have varying levels of importance or influence on the overall outcome.