There are different formulas used to calculate mean. Following are some.
- Arithmetic mean
- Weighted mean
- Geometric mean
- Harmonic mean
- Quadratic Mean (Root Mean Square)
- Trimmed mean
Arithmetic mean
It is also called as average of the numbers. Formula to calculate the mean.
Here,
- μ represents the mean.
- xi represents each individual value in the dataset.
- n is the total number of values in the dataset.
For example, I have the values [1, 2, 3, 4, 5] and mean is calculated like below.
Mean = (Sum of all values)/(total number of values)
= (1 + 2 + 3 + 4 + 5)/5
= 15 / 5
= 3
You can go for Arithmetic mean when,
- When the data is evenly distributed around the mean like a symmetric, bell-shaped normal distribution.
- When the data is not skewed, in a skewed data, the data points are not evenly distributed, and the curve is distorted in one direction.
- When the data representing values like height, weight, or temperature you can go with Arithmetic mean.
Examples of Arithmetic mean
- Calculate the student’s average grades
- Average salaries of employee by their experience
- Average income per each person in India
- Average satisfaction score of the customers who use X product.
- Player’s average scores
- Average blood pressure of people by their age
- Average travel time for an employee to reach office in peak traffic time
- Average temperatures in Bangalore in May month
- Average height of group of people
- Average running time of animals by category
- Average student scores in a Mathematical test
- Average fuel efficiency of cars
- Average price of a house in given locality
Weighted mean
Here we add certain weights to each data point and then calculate the weighted sum.
Formula
|
value |
1 |
2 |
3 |
4 |
5 |
|
Weight |
2 |
4 |
2 |
4 |
2 |
Weighted mean = ((1*2) + (2*4) + (3*2) + (4*4) + (5*2))/(2+4+2+4+2)
= (2 + 8 + 6 + 16 + 10)/14
= 42/14
= 3
Weighted mean is useful, when some data points are more significant than others. You can go with weighted mean when not all the data points are equally important.
For example,
- While taking the survey on a given product, you might give more weightage to the loyal customers to your company than the occasional ones.
- While calculating the average grade of a student, we can give more weightage to some complex or practical courses than others.
Geometric mean
It is the nth root of the product of all values.
Geometric mean is suitable for the data that follows exponential growth.
Geometric mean is more appropriate for the multiplicative data. In multiplicate data, values are multiplied together to predict the current data.
For example, the growth rate of a company’s stock price is multiplicative.
Following are some other examples, where you can go for Geometric mean.
- While calculating population growth rate during certain time periods.
- To calculate average compound interest earned on an investment.
- To calculate average inflation rate over a period of time.
Harmonic mean
It is used for data with reciprocals. It is used in the situations where the average of rates or speeds is needed.
Harmonic mean is used when the data is inversely proportional. In inversely proportional data, when one variable increases other variable decrease and vice versa.
For example, take the formula ‘xy=k’. In this formula, the variable, when x increases, y will be decreased to make sure that the product of xy is equal to k.
xy = 20
When
- x is 2, y is 10
- x is 4, y = is 5
- x is 5, y is 4
Other real time examples
- Time it takes to travel a fixed distance is inversely proportional to the speed of travel. When the speed of travel is more, then we reach the destination in less time.
- The brightness of a light bulb and the distance from the light bulb. When we are near to the light bulb the brightness is more, when we are far away from the light bulb, the brightness is less.
Quadratic Mean (Root Mean Square)
It is also known as root mean square (RMS) or the quadratic average, and calculated with below formula.
Trimmed mean
Trimmed mean is calculated after removing a certain percentage of extreme values from the dataset.
Trimmed mean is your buddy, when you want to remove specific percentage of extreme values (outliers) from the data points. For example,
- While calculating the average income of a population, we might need to ignore a few very wealthy people, who would skew the distribution of the data.
- While calculating the average test score of a class, we might need to ignore a few very low or very high scores who would skew the distribution of the data.
- While analysing customer feedback ratings or reviews, you may trim the highest and lowest ratings to focus on opinions of majority of customers.
Find the below working application.
means_comparison.py
import matplotlib.pyplot as plt import numpy as np from scipy.stats import gmean from scipy import stats # Sample dataset data = np.array([20, 400, 32, 456, 98, 500, 23, 45, 235]) # Sample weights weights = np.array([2, 4, 2, 4, 2, 4, 3, 4, 3]) # Calculate the mean arithmetic_mean = np.mean(data) # Calculate the weighted mean weighted_mean = np.average(data, weights=weights) # Calculate the geometric mean geometric_mean = gmean(data) # Calculate the harmonic mean manually harmonic_mean = len(data) / np.sum(1.0 / data) # Calculate the quadratic mean manually quadratic_mean = np.sqrt(np.mean(data ** 2)) # Calculate the trimmed mean by trimming 10% from both ends trimmed_mean = stats.trim_mean(data, proportiontocut=0.20) # Create a scatter plot of data points plt.scatter(data, range(len(data)), label='Data Points', color='blue', marker='o') # Add a vertical line at the mean plt.axvline(x=arithmetic_mean, color='red', label='Arithmetic Mean') plt.axvline(x=weighted_mean, color='magenta', label='Weighted Mean') plt.axvline(x=geometric_mean, color='yellow', label='Geometric Mean') plt.axvline(x=harmonic_mean, color='black', label='Harmonic Mean') plt.axvline(x=quadratic_mean, color='cyan', label='Quadratic Mean') plt.axvline(x=trimmed_mean, color='blue', label='Trimmed Mean') # Add labels and a title plt.xlabel('Actual value') plt.ylabel('Index') plt.title('Scatter Plot with Mean') # Add a legend plt.legend() # Show the plot plt.show() print(f'arithmetic_mean : {arithmetic_mean}') print(f'weighted_mean : {weighted_mean}') print(f'geometric_mean : {geometric_mean}') print(f'harmonic_mean : {harmonic_mean}') print(f'quadratic_mean : {quadratic_mean}') print(f'trimmed_mean : {trimmed_mean}')
Output
arithmetic_mean : 201.0 weighted_mean : 238.5 geometric_mean : 103.7371576736313 harmonic_mean : 53.53864688040538 quadratic_mean : 276.23319456172857 trimmed_mean : 184.14285714285714
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