**π‘ Problem Formulation:** When working with datasets in data analysis, we often need to combine different data series and compute aggregate statistics such as the mean and variance. This article addresses the problem of taking two data series and finding their combined mean and variance in Python, using the input of two numerical lists and producing the combined statistical measures as output.

## Method 1: Manual Calculation Using Basic Python

An easy yet manual approach would be to calculate the mean and variance for each series using Python’s sum and len functions, and then combining them using the pooled variance formula. This involves extending both series into a single list and then calculating the statistics as we would for any individual series.

Here’s an example:

series1 = [1, 2, 3] series2 = [4, 5, 6] # Calculate the means mean1 = sum(series1) / len(series1) mean2 = sum(series2) / len(series2) # Calculate the variances var1 = sum((x - mean1) ** 2 for x in series1) / (len(series1) - 1) var2 = sum((x - mean2) ** 2 for x in series2) / (len(series2) - 1) # Calculate the combined mean combined_mean = (sum(series1) + sum(series2)) / (len(series1) + len(series2)) # Calculate the combined variance combined_variance = ((len(series1) - 1) * var1 + (len(series2) - 1) * var2) / (len(series1) + len(series2) - 2) print(f"Combined Mean: {combined_mean}, Combined Variance: {combined_variance}")

Output:

Combined Mean: 3.5, Combined Variance: 3.5

This code snippet first calculates the mean and variance for each individual series. Then it computes the combined mean by adding the sums of both series and dividing by the total number of items. The combined variance uses the pooled variance formula, which is applicable for samples with the same population variance.

## Method 2: Using NumPy Library

NumPy, a popular Python library for numerical computations, provides functions to calculate mean and variance easily. This method is more efficient and concise compared to manual calculations.

Here’s an example:

import numpy as np series1 = np.array([1, 2, 3]) series2 = np.array([4, 5, 6]) combined_series = np.concatenate((series1, series2)) combined_mean = np.mean(combined_series) combined_variance = np.var(combined_series, ddof=1) print(f"Combined Mean: {combined_mean}, Combined Variance: {combined_variance}")

Output:

Combined Mean: 3.5, Combined Variance: 3.5

By using NumPy’s `mean`

and `var`

functions, which inherently handle series of numbers quite well, we skip the manual calculation steps. The `concatenate`

function merges the two series into one, while the `ddof`

parameter in `var`

is set to 1 to compute the sample variance.

## Method 3: Using Pandas Library

Pandas, a library built on top of NumPy, can simplify data aggregation tasks. It offers data structures like Series and DataFrame, which come with built-in methods for calculating statistics.

Here’s an example:

import pandas as pd series1 = pd.Series([1, 2, 3]) series2 = pd.Series([4, 5, 6]) combined_series = series1.append(series2) combined_mean = combined_series.mean() combined_variance = combined_series.var() print(f"Combined Mean: {combined_mean}, Combined Variance: {combined_variance}")

Output:

Combined Mean: 3.5, Combined Variance: 3.5

This snippet uses Pandas’ Series object and append method to merge two series. Calculating the mean and variance is straightforward with the `mean`

and `var`

methods. This method is efficient for larger datasets, and it simplifies the process as the dataset’s structural complexity increases.

## Method 4: Using Statistics Module

The statistics module in Python’s standard library provides functions for calculating mathematical statistics of numeric data. This module can be used for quick and direct calculations without additional dependencies.

Here’s an example:

import statistics series1 = [1, 2, 3] series2 = [4, 5, 6] combined_series = series1 + series2 combined_mean = statistics.mean(combined_series) combined_variance = statistics.variance(combined_series) print(f"Combined Mean: {combined_mean}, Combined Variance: {combined_variance}")

Output:

Combined Mean: 3.5, Combined Variance: 3.5

The code concatenates the two series lists using the `+`

operator to form a combined series. It then utilizes the `mean`

and `variance`

functions from the statistics module to obtain the desired statistical measures, making it a suitable choice for simple use-cases without requiring external libraries.

## Bonus One-Liner Method 5: Using SciPy Library

SciPy is another Python library used for scientific and technical computing. It provides many user-friendly and efficient numerical routines such as optimization, integration, interpolation, eigenvalue problems, and others, including statistics.

Here’s an example:

from scipy import stats series1 = [1, 2, 3] series2 = [4, 5, 6] combined_mean, combined_variance = stats.describe(series1 + series2)[2:4] print(f"Combined Mean: {combined_mean}, Combined Variance: {combined_variance}")

Output:

Combined Mean: 3.5, Combined Variance: 3.5

Here, the `describe`

function from the `stats`

module is used to extract descriptive statistics of the combined series. The function returns a tuple containing several statistics, from which we slice out the mean and variance. This method is concise and suitable for performing a set of statistical operations in just one line.

## Summary/Discussion

**Method 1:**Manual Calculation Using Basic Python. Strengths: No external dependencies, educative. Weaknesses: Verbose, prone to errors, not suitable for complex data structures.**Method 2:**Using NumPy Library. Strengths: Efficient, concise, suitable for large datasets. Weaknesses: Requires NumPy installation.**Method 3:**Using Pandas Library. Strengths: Simplifies handling complex data structures, good for data manipulation tasks. Weaknesses: Overhead for simple tasks, requires Pandas installation.**Method 4:**Using Statistics Module. Strengths: Built into standard library, easy to use for basic statistics. Weaknesses: Less functionality compared to specialized libraries.**Method 5:**Bonus One-Liner Using SciPy Library. Strengths: Quick and comprehensive statistical analysis in one line. Weaknesses: Requires SciPy installation, less readable.