Arithmetic Mean

The arithmetic mean is the measure of the average calculated by adding up a set of numbers and then dividing it by the count of the numbers. For example, if the given numbers are 10, 16, 20, or 35, the average is calculated by dividing the sum of the numbers by their count: \((10 + 16 + 20 + 35) ÷ 4 = 20.25.\) The arithmetic mean is 20.25. The important takeaways of the arithmetic mean are essential to understand. The arithmetic mean is calculated by dividing the total of the given numbers by their count.

Also, using the arithmetic mean in finance is not ideal because a single outlier can significantly affect the final result. For example: If you want to calculate the weekly expenses of your five friends. Four of them spend between $15 and $20 weekly, while the fifth friend spends $40 weekly. In finance, the geometric and harmonic means are among the most commonly used averages.

The arithmetic mean formula can be used to calculate the mean or average of a given data set. We use the symbol x̄ to denote the arithmetic mean. The arithmetic mean of a data set can be calculated by summing all observations and dividing by the number of observations. 

The arithmetic mean formula is represented as; 

\(\text{Arithmetic mean (x̄)} = \frac{\text{Sum of all observations}}{\text{Number of observations}}\)

Let us assume that there are n observations in a data set, namely \(n_1,\) \(n_2,\) \(n_3,\) \(n_4,\)…, \(n_n.\) Its arithmetic mean can be calculated by, 

\(A.M.=\frac{(n_1+n_2+n_3+....n_n)}{n}\)

If the frequency of various numbers in a data set is \(f_1\), \(f_2\), \(f_3\), \(f_4\), ..., \(f_n\) for the numbers \(n_1,\) \(n_2,\) \(n_3,\)..., \(n_n,\) 

\(AM = x̄ = \frac{(x₁f₁+x₂f₂+......+xₙfₙ)}{∑fi}\)

For an ungrouped data set, the arithmetic mean can be calculated using the following formula.

\(\text{Arithmetic mean (x̄)} = \frac{\text{Sum of all observations}}{\text{Number of observations}}\)

Let us try to understand the formula with the help of an example, as given below:

Find the mean of the first five even numbers. 

The first five even numbers are 0, 2, 4, 6, 8. 

Let us use the formula mentioned above. 

\(\text{Arithmetic mean (x̄)} = \frac{\text{Sum of all observations}}{\text{Number of observations}}\)

\(Mean (x̄) = \frac{(0 + 2 +  4 +  6 +  8)}{5}\)

\(Mean (x̄) = \frac{20}{5}\)

\(Mean (x̄) = 4\)

Therefore, the mean of the first five even numbers is 4.

Grouped data is presented as continuous intervals with accompanying frequency values for each class. There are three primary methods for finding the arithmetic mean for such data: the direct method, the assumed mean method, and the step-deviation method.

Depending on the values of frequencies and class midpoints, any of the three methods, namely, the direct method, assumed mean method, or step-deviation method, can be used to calculate the arithmetic mean for grouped data; now, let's explore each of these methods in detail.

The direct method

Let us assume that we have to find the mean of n observations x₁, x₂, x₃ ……xₙ, with frequency f₁, f₂, f₃ ……fₙ respectively. Then the formula to find the arithmetic mean is,

\(AM = x̄ = \frac{(x₁f₁+x₂f₂+......+xₙfₙ)}{∑fi}\)

Where x̄ represents the arithmetic mean, and \(f₁+f₂+....fₙ=∑fi\) represents the sum of all frequencies.

Assumed mean method

To find the arithmetic mean of a data set using the assumed mean method, follow these steps. 

Step 1: Calculate the midpoint of each class interval, denoted as \(x_i\).

Step 2: Select a value as the assumed mean, represented by A.

Step 3: Compute the deviation for each class midpoint as \(d_i=x_i−A.\)

Step 4: Apply the formula for the arithmetic mean as:

\(\bar{x} = A + \left( \frac{\sum f_i d_i}{\sum f_i} \right) \)

Step-deviation method

The step deviation method is also known as the scale method. We can easily find the arithmetic mean of a given set of data by the step-deviation method by following these steps. 

Step 1: Calculate the midpoint \(x_i\) for each class interval.

Step 2: Choose an assumed mean A.

Step 3: Compute \(u_i = \frac{(x_i-A)}{h},\) where h is the class width.

Step 4: Apply the formula to find the arithmetic mean.

\(\bar{x} = A + h\left( \frac{\sum f_i u_i}{\sum f_i} \right) \)

The geometric mean is widely used in finance as it accounts for probability, making it more precise for calculating investment returns. It works well for serially correlated data such as bond rates and stock returns. 

The arithmetic mean sums up the values and divides them by their count, whereas, the geometric mean multiplies them and calculates the nth root. The geometric mean will always be less than or equal to the arithmetic mean. 

For example, given a stock has a yearly return of 20%, – 8%, 10%, and 25% over four years:

Arithmetic Mean: \(\frac{(20 + (– 8) + 10 + 25)}{4} = 11.75\%\) per year.

Geometric Mean: \(\left[(1.20 \times 0.92 \times 1.10 \times 1.25)^{\tfrac{1}{4}}\right] - 1 = 10.84\% \) per year.

The arithmetic mean may not always be the appropriate measure, especially when a single outlier severely distorts it. For instance, if nine children receive between $10 and $12 per week as an allowance, but one child receives $60, the average expenditure is $16, which is misleading about what is typical for this group.

In this particular case, the median allowance might be a better measure. The arithmetic average is not always the appropriate measure of an investment portfolio, especially when dividends are compounded or reinvested. It is generally not used to calculate present or future cash flow, as reliance on it can produce misleading results.

Some of the important properties of arithmetic mean are listed below:
 

• The sum of deviations is always a zero. In simple words, when the mean value is subtracted from each data point, and the differences are added together, the result is zero. It can be mathematically expressed as:
  
  \(\sum { (x - \bar{x})} = 0\)
   
• Outliers in a dataset can influence the arithmetic mean, leading to inaccuracies. This sensitivity to outliers can be taken as an advantage to detect the presence of outliers in datasets.

• Adding or subtracting a constant: When the same number is added to or subtracted from each value in a dataset, the mean shifts by that number.

• When you multiply or divide all the values by the same number, the mean is multiplied or divided by that number.
   

The arithmetic mean helps summarize a dataset by representing its average value. Mastering it allows for quick and accurate analysis of numerical data.

• Understand that the arithmetic mean represents the average of all values in a dataset.
   
• Organize your numbers clearly to simplify addition and counting.
   
• Apply the formula \( \text{Mean} = \frac{\text{Sum of all values}}{\text{Number of values}} \) correctly.
   
• Watch out for outliers, as extreme values can affect the mean.
   
• Practice regularly with different problems to improve speed and accuracy.
   
• Teachers can start teaching the arithmetic mean using real objects rather than complex numbers at first. Use everyday items like candy, coins, pencils, and toys.
   
• Parents can help their children understand the concept by explaining it through a story. Ask them questions like, “Four friends bought snacks. How can they split the total cost equally?”
   
• Teachers can make use of number lines to explain the concept simply. The students arrange the given numbers on a number line in ascending order and show that the mean always lies between the smallest and the largest numbers in the data set. 
   
• Parents can tell their kids that the concept of the arithmetic mean talks about “fair sharing.” Also, start by giving them relatable, hands-on problems and then move to numbers and formulas.

Arithmetic mean is significantly used in different fields to determine the average of quantities. Let’s look at a few applications of arithmetic mean:

1. Finance: The arithmetic mean is often used in calculating the average income or expenses of employees.
    
2. Feature Scaling: The arithmetic mean is used in mean centering for machine learning. It is performed by subtracting the average so that the mean becomes zero.
    
3. Model Quality Check: We use the arithmetic mean in checking the model performance by applying absolute differences.
    
4. Hypothesis Testing: It is applied in t-tests, where the means of two means are compared to determine whether they are significantly different.
    
5. Demographics: The demographers apply the average mean to estimate the income, age, or population growth.