---
title: "A Mean-ingful Guide to Averages"
author: "Ada Lovelace"
date: 2026-01-15
date-format: "long"
format:
  typst: 
    number-sections: true
    link-citations: true
    citeproc: true
  html: default
  docx: default
abstract: |
  Calculating the average of a set of numbers is a foundational concept in statistics. This mini paper defines averages and provides examples and applications that illustrate their importance in data analysis.
bibliography: references.bib
csl: csl/chicago-author-date.csl
---

# Introduction

An average, or *mean*, represents the **central tendency** of a set of data, offering a single value that summarizes the data. The mathematical formula for the mean of a dataset $\{x_1, x_2, \dots, x_n\}$ is:

$$
\bar{x} = \frac{1}{n} \sum_{i=1}^n x_i
$$

Here, $\bar{x}$ represents the mean, $n$ is the number of values in the dataset, and $x_i$ are the individual data points.

# How to Calculate an Average

To compute the mean of a dataset:

1. **Calculate the Sum of All Values**: Compute $\sum_{i=1}^n x_i$, the total of all data points.
2. **Count the Number of Values**: Determine $n$, the total number of data points.
3. **Divide the Sum by the Count**: Use the formula $\bar{x} = \frac{1}{n} \sum_{i=1}^n x_i$.

## Example: Test Scores

Consider the following dataset representing a student’s test scores: $\{85, 90, 78, 92, 88\}$. To find the mean:

1. Compute the **sum** of the scores:

$$
\sum_{i=1}^5 x_i = 85 + 90 + 78 + 92 + 88 = 433
$$

2. Count the total **number** of scores:

$$
n = 5
$$

3. **Divide** the sum by the number of scores:

$$
\bar{x} = \frac{433}{5} = 86.6
$$

Thus, the mean test score is 86.6.

# Applications of Averages

Averages are crucial for summarizing and interpreting data in various fields:

- **Education**: Teachers use averages to evaluate class performance.
- **Finance**: Families calculate average expenses to manage budgets.
- **Science**: Researchers use averages to analyze experimental data.

Understanding averages allows for meaningful comparisons and informed decision-making.

# Conclusion

The mathematical concept of the average is a versatile tool in data analysis. By using the formula $\bar{x} = \frac{1}{n} \sum_{i=1}^n x_i$, you can calculate and interpret averages effectively.

# Appendix

Here's a picture of Ada Lovelace [@lovelace1842], the fictional writer of this goofy little guide (but she was real! [read more about her here](https://en.wikipedia.org/wiki/Ada_Lovelace))

![Ada Lovelace](img/ada-lovelace.png){width="30%"}

# References
