We'll walk through the process of analyzing real-world sales data using probability distributions such as the normal distribution. Analyzing sales trends is the most common business use case, but it can be applied almost equally to sales data as well as other predictive or effectiveness data. 

Sales trend analysis is a critical component of effective business planning and decision-making. In this guide, we'll walk you through a detailed, step-by-step process to analyze real-world sales data using normal distribution and other statistical techniques.

Step 1: Understand the business context and objectives

Before diving into the data analysis, it's crucial to understand the business context and objectives. Sales trend analysis can address various goals, such as identifying seasonality patterns, forecasting future sales, evaluating the effectiveness of marketing campaigns, and optimizing inventory levels. Your analysis approach will depend on the specific objectives you aim to achieve.

Step 2: Collect and preprocess the sales data

To analyze sales trends, you'll need historical sales data. This data may come from various sources, such as transactional databases, enterprise resource planning (ERP) systems, or customer relationship management (CRM) systems. Once you've gathered the data, preprocess it by cleaning and organizing it into a structured format, such as a time-series dataset with daily sales figures.

Step 3: Perform exploratory data analysis (EDA)

Begin by conducting an EDA to gain insights into the data's characteristics. Plot the daily sales data, look for outliers, and examine the data distribution. At this stage, you can check whether the sales data follows a normal distribution or another type of distribution. If the data roughly follows a normal distribution, you can calculate the mean and standard deviation to describe its central tendency and variability.

Step 4: Identify trends, seasonality, and cyclical patterns

To detect underlying patterns in the sales data, calculate moving averages using different window sizes (e.g., 7-day, 30-day, or 90-day moving averages). These moving averages help smooth out daily fluctuations and reveal trends, seasonality, and cyclical patterns. Seasonality refers to recurring patterns in the data due to factors such as holidays, weather, or time of year. Cyclical patterns are longer-term fluctuations that are not tied to a fixed calendar-based timeline.

Step 5: Forecast future sales (if applicable)

If one of your objectives is to forecast future sales, you can use various techniques, such as time-series forecasting models like ARIMA (Autoregressive Integrated Moving Average) or exponential smoothing. These methods may consider the normal distribution and other statistical properties of the sales data to make predictions.

Step 6: Evaluate the impact of promotions, campaigns, or external factors

You can use the insights from the sales trend analysis to evaluate the effectiveness of marketing campaigns, promotions, or other external factors. For instance, you can compare sales data before, during, and after a promotional campaign to determine its impact on sales. You can also analyze the sales data across different product categories, regions, or customer segments to understand how these factors affect sales performance.

Step 7: Communicate the insights and recommendations

After completing the sales trend analysis, it's essential to communicate your findings and recommendations effectively. Create visualizations, such as line charts or bar charts, to illustrate the trends, seasonality, and other patterns in the data. Summarize the key insights and outline the implications for the business, such as inventory management, marketing, and staffing decisions.

Step 8: Monitor and adjust

Sales trend analysis is an ongoing process. Continuously monitor the sales data, update your analysis, and adjust your business strategies accordingly. As the business environment and customer behavior evolve, your sales analysis should reflect these changes to help you make data-driven decisions.

In summary, sales trend analysis involves understanding the business context, collecting and preprocessing sales data, performing EDA, identifying trends and patterns, forecasting future sales (if applicable), evaluating the impact of external factors, communicating insights, and monitoring the data continuously. Normal distribution plays a role in describing the central tendency and variability of the sales data, as well as in some forecasting models. By following this step-by-step process, data analysts can gain valuable insights into sales performance and make informed decisions to optimize business operations.

Let's take the above scenario and break it down into Python code.  

You can see your daily sales data along with the 7-day and 30-day moving averages, as shown in the graph above. The 7-day moving average can be used to smooth out daily fluctuations to predict weekly seasonality patterns, while the 30-day moving average can be used to take a longer-term view by predicting annual upward trends.

By reviewing your moving averages, you can gain insights into your sales patterns, such as seasonality and trends. This information can be useful for making strategic decisions, such as planning promotions, managing inventory, and allocating resources.

When using real data, as in this example, you might need to adjust parameters such as the moving average window size to better capture the underlying patterns in your data. You can then use the insights derived from your real data to make informed decisions and improve business performance.