Standard Deviation – Introduction

1.1 Description:

In statistics, the lowercase σ (sigma) generally denotes the population standard deviation. Standard deviation quantifies how far, on average, each value in a data set deviates from the mean (μ). The smaller σ is, the more tightly the data cluster around μ; a larger σ indicates greater spread.

\[ \sigma \;=\; \sqrt{\frac{1}{N}\,\sum_{i=1}^{N}\bigl(x_i - \mu\bigr)^2} \]

Example. For the integers 1–4:

\[ \mu \;=\; \frac{1+2+3+4}{4}\;=\;2.5 \quad\Longrightarrow\quad \sigma \;=\;\sqrt{\frac{(1-2.5)^2+(2-2.5)^2+(3-2.5)^2+(4-2.5)^2}{4}} \approx 1.118 \]

The Java program delivered in this project automates this calculation and displays every intermediate step for full transparency.

1. Introduction

1.1 Description

The Standard-Deviation Calculator is a compact Java Swing application that computes the population standard deviation (σ) of every integer in the range [a, b]. It reveals each intermediate step so students can document their methodology in lab reports.

1.2 Persona

Name: Julia Chen – 20-year-old Mechanical-Engineering student.

Devices: Windows 11 laptop, OpenJDK 21.
Goals: Obtain σ rapidly and view the mean & variance steps.
Pain-points: Black-box calculators, accidental input errors.

1.2 Persona – Joe Ramirez

Age	23
Major	Electrical Engineering (B.Eng.)
Location	Montreal, Canada
Tech Stack	macOS 14, OpenJDK 21, IntelliJ IDEA Edu

Background. Joe is in his final undergraduate year and currently works as a teaching assistant for a first-year “Engineering Computation” course. He frequently prepares tutorial slides and lab demos where students must calculate statistical measures, but many of his tutees struggle to interpret the black-box outputs of conventional calculators.

Goals & Motivations

Produce clear, step-by-step demonstrations of population standard deviation.
Save prep time by re-using exported logs instead of typing formulas by hand.
Reduce students’ arithmetic errors by providing an intuitive GUI they can replicate at home.

Pain Points

Scientific calculators hide intermediate results, making it hard to teach “why it works.”
Spreadsheet software feels overkill and distracts beginners with unrelated UI clutter.
Manual LaTeX derivations are time-consuming and error-prone.

Typical Use Scenario

Joe launches the Standard-Deviation Calculator before a lab session, inputs the dataset bounds a = 5, b = 30, and presses Evaluate. The application streams the mean, variance, and each squared deviation into a scrollable log. Joe clicks Save Steps, drags the exported .txt file into Overleaf, and instantly embeds the derivation in his slide deck. During class, when a student suggests a different range, Joe simply edits the numbers, re-runs, and projects the updated steps live—eliminating chalkboard arithmetic.

Personality Snapshot

Trait	Level / Note
Tech Enthusiasm	High – builds hobby IoT projects
Teaching Passion	Very High – enjoys mentoring peers
Patience for Errors	Moderate – wants instant, clear feedback
Design Sensitivity	Prefers minimalist UIs over cluttered dashboards

“If students can see every calculation unfold, they stop guessing and start understanding.”

2. Implementation

2.1 GUI in Swing

The single-window interface (JFrame) uses BorderLayout:

North panel: text fields for a and b plus an Evaluate button.
Center: scrollable JTextArea that streams the step-by-step log.
South: status bar displaying the final σ value.

2.2 Input

Lower bound a (integer)
Upper bound b (integer)

Live document listeners flag non-numeric keys; if a > b, a dialog offers to swap the bounds.

2.3 Output

Running totals, μ, variance σ²
Final standard deviation σ
Optional Save Steps button to export the log

3. Requirements

3.1 Software Requirements Specification

ID	Requirement	Rationale / Note
FR-01	The system shall compute the population standard deviation σ for the integer range `[a, b]`.	Core functionality.
FR-02	The system shall display each intermediate step (partial sums, mean μ, variance σ², final σ) in chronological order.	Transparency for learning.
UR-01	The GUI shall validate numeric input in real time and highlight errors.	Reduces user frustration.
UR-02	Error dialogs shall propose corrective actions (e.g., “Swap bounds?”).	Helpful feedback mandate.
IR-01	The implementation shall rely solely on core Java 21 and Java Swing—no external libraries.	“From scratch” rule.
QR-01	The codebase shall compile with `javac` on any JDK 21 without IDE-specific tooling.	Tool-agnostic build.
QR-02	The system shall throw a custom `InvalidBoundException` when `a > b`.	Explicit exception handling.
PR-01	For \|b − a\| ≤ 10⁶, total computation time shall be < 500 ms on a 2 GHz CPU.	Performance ceiling.
PR-02	Peak RAM usage shall remain below 100 MB.	Ensures laptop compatibility.
UR-03	The GUI shall offer a `Save Steps` export option.	Stakeholder request.
QR-03	The test suite shall achieve ≥ 95 % statement coverage.	Quality assurance.

3.2 Example Inputs and Results

Input (a, b)	σ Result
1, 5	1.414 213 562
10, 20	3.162 277 660
100, 150	14.577 379 736