Different types of mean

The statistical mean, commonly known as the mean or average, is a measure of central tendency. It represents the total sum of all values in a dataset divided by the number of values. This calculation yields a single value that summarizes the central or typical value of the dataset.

There are several types of mean, each serving different purposes in statistical analysis. Each type of mean is appropriate for different types of data and analytical contexts, enabling statisticians to select the most suitable measure of central tendency based on the specific requirements of their analysis.The most common types include:

1. Arithmetic Mean:

The arithmetic mean is the sum of all values divided by the number of values, making it the most commonly used type of mean.The formula for calculating the arithmetic mean (or average) of a set of numbers is:

In words, you add up all the values in the dataset and then divide the sum by the total number of values.

An example of calculating the arithmetic mean (or average) can be demonstrated with a simple dataset. Let’s consider a dataset representing the daily temperatures (in degrees Celsius) recorded over a week:

Temperatures: 20,22,25,24,23,21,20

To find the arithmetic mean:

Add up all the values: 20+22+25+24+23+21+20=15520+22+25+24+23+21+20=155

Divide the sum by the total number of values (which is 7 in this case): Mean= 155 / 7 ≈ 22.14

So, the arithmetic mean (average) temperature over the week is approximately 22.14 degrees Celsius.

2. Geometric Mean:

The geometric mean is defined as the nth root of the product of all values in a dataset. It is frequently used for data that is multiplicative or exhibits exponential growth. Mathematically, Geometric mean is given by:

In words, you multiply all the numbers together and then take the nth root of the product, where n is the total number of values in the dataset.

Let’s consider an example where the geometric mean is used to calculate the average growth rate of an investment over multiple periods.

Suppose you invest in a stock that has the following annual growth rates over a 3-year period:

• Year 1: 5%
• Year 2: 8%
• Year 3: 10%

To find the geometric mean of these growth rates:

Convert the growth rates to decimal form (i.e., divide by 100):

• Year 1: 5% = 0.05
• Year 2: 8% = 0.08
• Year 3: 10% = 0.10

Let’s put the values into the formula

n = 3 , representing 3 years.

= ∛(0.05×0.08×0.10) = ∛(0.0004) ≈0.063

Convert back to percentage form (multiply by 100) to interpret the result: Geometric mean ≈ 6.3%

So, the geometric mean growth rate of the investment over the 3-year period is approximately 6.3% per year. This indicates the constant annual growth rate that would yield the same cumulative return over the period as the variable growth rates observed.

3. Harmonic Mean:

The harmonic mean is the reciprocal of the arithmetic mean of the reciprocals of the values. It is particularly useful for calculating averages of rates and ratios. The formula for calculating the harmonic mean of a set of numbers is:

In words, you take the reciprocal of each value in the dataset, calculate the sum of these reciprocals, and then divide the total number of values by this sum of reciprocals.

Let’s consider an example where the harmonic mean is used to calculate the average speed for a journey with multiple segments. Suppose a car travels three segments of a journey at different speeds:

• Segment 1: Speed = 60 km/h
• Segment 2: Speed = 40 km/h
• Segment 3: Speed = 80 km/h

To find the harmonic mean of these speeds:

1. Calculate the reciprocal of each speed: 1/60,1/40,1/80
2. Find the arithmetic mean of these reciprocals:

Arithmetic mean of reciprocals = (1/60+1/40+1/80) / 3 = (1/15)/3 = 1/45

Take the reciprocal of the result from step 2 to find the harmonic mean: Harmonic mean = 1/Arithmetic mean of reciprocals = 1/(1/45) = 45

So, the harmonic mean speed for the journey is 45 km/h. This means that if the car had maintained a constant speed of 45 km/h throughout the journey, it would have covered the same distance in the same amount of time as it did with the varying speeds.

4. Weighted Mean:

The weighted mean is calculated by multiplying each value by a corresponding weight that reflects its importance, summing these products, and then dividing by the total sum of the weights.The formula for calculating the weighted mean of a set of numbers is:

In words, you multiply each value by its corresponding weight, sum these weighted values, and then divide the total sum of the weighted values by the sum of the weights.

Let’s consider an example where we want to calculate the weighted mean of exam scores, where each exam has a different weight based on its importance.

Suppose we have the following data:

• Exam 1 score: 85, Weight: 0.2
• Exam 2 score: 90, Weight: 0.3
• Exam 3 score: 80, Weight: 0.5

To find the weighted mean:

Multiply each exam score by its corresponding weight:

Exam 1 weighted score=85×0.2=17

Exam 2 weighted score=90×0.3=27

Exam 3 weighted score=80×0.5=40

Sum the weighted scores:

Sum of weighted scores=17+27+40=84

Sum the weights:

Sum of weights = 0.2+0.3+0.5 = 1

Divide the sum of the weighted scores by the sum of the weights:

Weighted Mean = 84/ 1 = 84

So, the weighted mean of the exam scores is 84. This means that when considering the importance of each exam, the average score across all exams is 84.

5. Quadratic Mean (Root Mean Square):

The quadratic mean, or root mean square, is the square root of the average of the squares of the values. It is commonly used in engineering and physics applications.The formula for calculating the quadratic mean, also known as the root mean square (RMS), of a set of numbers is:

In words, you square each value in the dataset, sum the squares, divide by the total number of values, and then take the square root of the result.

For example, suppose we have a dataset of speeds measured in km/h:

• Speed 1: 40 km/h
• Speed 2: 60 km/h
• Speed 3: 80 km/h

To calculate the quadratic mean:

1. Square each value: 40 ^ 2 = 1600, 60 ^ 2 = 3600, 80 ^ 2 = 6400.
2. Sum the squares: 1600 + 3600 + 6400 = 11600.
3. Divide by the total number of values (3): 11600 / 3 = 3866.67.
4. Take the square root of the result: √(3866.67) ≈ 62.19.

How to select the appropriate type of mean ?

The selection of an appropriate type of mean hinges upon the unique attributes of the dataset and the objectives of the analysis. Here’s a guideline delineating the optimal application scenarios for each mean:

Arithmetic Mean:

• Opt for the arithmetic mean when the dataset demonstrates a symmetrical distribution without pronounced outliers.
• Best suited for interval or ratio scale data where the distances between values hold significance.
• Widely employed across diverse domains to compute averages due to its simplicity and versatility.

Geometric Mean:

• Employ the geometric mean when analyzing datasets featuring multiplicative relationships or exponential growth patterns.
• Particularly suitable for data exhibiting exponential growth or decay, such as in finance, biology, and economics.

Harmonic Mean:

• Utilize the harmonic mean when dealing with rates or ratios, such as speed or efficiency calculations.
• Ideal for datasets characterized by an inverse relationship between values and weights, or when reciprocal relationships among data points are prominent.

Quadratic Mean (Root Mean Square):

• Employ the quadratic mean, also known as the root mean square (RMS), when assessing the magnitudes of values.
• Appropriate for datasets where deviations from a reference point are crucial, as seen in physics, engineering, and signal processing applications.

Weighted Mean:

• Opt for the weighted mean when the dataset comprises data points with varying levels of significance or importance.
• Ideal for situations where certain observations possess higher reliability or precision than others, or when amalgamating data from diverse sources or indices.

In summary, the selection of the most suitable mean type should be predicated on a thorough consideration of the dataset’s attributes and the analytical objectives. Each mean type presents distinct strengths and weaknesses, necessitating a nuanced assessment of the contextual requirements before arriving at a decision.