Regression Analysis Study Guide Regression Analysis
Project Study Guide & Interpretation Handbook
Topic: MAT-20679
Statistical Analysis Unit
1. Identifying Variables
Independent Variable (X):
The "Predictor" or "Input." This is the variable you think is doing the influencing. In this project, Surface Area is our X.
Dependent Variable (Y):
The "Outcome" or "Response." This is what you are measuring to see if it changes. Population and Sex Ratio are our Y variables.
2. The Correlation Coefficient (\(r\))
Tells us the strength and direction of the relationship between X and Y.
Value of \(r\) Interpretation 0.7 to 1.0 Strong Positive Relationship 0.3 to 0.7 Moderate Relationship 0.0 to 0.3 Weak/No Relationship
The Impact of Outliers
An outlier is a data point that differs significantly from other observations. In regression, outliers act like "anchors" that pull the line of best fit toward them.
Slope Shift
Outliers can make a flat relationship look steep.
\(r\) Distortion
They can artificially inflate or deflate the correlation.
Lower Accuracy
The model becomes less reliable for the average data point.
Interpreting Your Project Results
Theory 1: Area vs. Population
Does more land mean more people? You'll likely see a Weak Positive correlation. While big countries like China have many people, others like Russia or Canada have massive land but small populations. Size isn't the only factor.
Theory 2: Area vs. Sex Ratio
Does country size change the gender balance? Biologically, this makes little sense. Without outliers, the correlation should be Near Zero . The outliers (like Qatar) have high ratios due to labor migration, not land area.
"A high \(r^2\) means your model explains a lot of the variance; a low \(r^2\) means other factors (like economy, migration, or history) are at play."
Project Completion Walkthrough Project Completion Walkthrough
Model Responses for the MAT-20679 Statistical Report
Student Name
Model Answer
Date
February 14, 2026
1. Explanation of Variable Choices
Theory 1: Area impacts Population
The Independent Variable is Surface Area (per 1,000 km²) because the theory suggests it is the "cause" or predictor. The Dependent Variable is Total Population because we are measuring how it changes in response to the land size.
Theory 2: Area impacts Sex Ratio
The Independent Variable remains Surface Area. The Dependent Variable is the Sex Ratio (males per 100 females). We are testing if the physical size of a country has any biological or social effect on gender distribution.
2. Regression Analysis Results (Summary)
Theory 1 Findings
Equation: \(y = 0.04x + 22.4\) (Approximate)Correlation (\(r\)): \(\approx 0.45\) (Moderate)\(R^2\): \(\approx 0.20\)
Theory 2 Findings
Equation: \(y = -0.0001x + 102.5\) (Approximate)Correlation (\(r\)): \(\approx -0.05\) (Very Weak)\(R^2\): \(\approx 0.002\)
3. Impact of Outliers (Theory 2)
Outliers like Qatar, UAE, and Bahrain significantly distort the regression model. Because these countries have extremely high sex ratios (over 200 males per 100 females) that are not related to their land area, they pull the regression line away from the majority of the data points. Including them makes the model less accurate for "normal" countries where the ratio is closer to 100. Removing them shows that there is virtually no relationship between area and gender balance.
4 & 5. Results and Theory Strength
Theory 1: Weak/Moderate Theory
Based on the correlation coefficient (\(r \approx 0.45\)), there is a moderate positive relationship. While larger countries tend to have higher populations on average, the low \(R^2\) value suggests that area only explains about 20% of the population difference. Other factors like history, climate, and economics are much more important.
Theory 2: Weak/Invalid Theory
With a correlation coefficient near zero (\(r \approx -0.05\)), this theory is extremely weak. The surface area of a country has no measurable impact on the male-to-female ratio. The variations seen in the data are better explained by migration patterns (outliers) or cultural factors, not the physical size of the land.
Spreadsheet Results Guide Spreadsheet Master Guide
Step-by-Step Data Analysis & Results Key
MAT-20679
UN Statistical Report
Excel Formula Cheat Sheet
Correlation (\(r\))
=CORREL(y_range, x_range)
Measures strength & direction.
R-Squared (\(R^2\))
=RSQ(y_range, x_range)
Measures the "goodness of fit."
Slope (\(m\))
=SLOPE(y_range, x_range)
The rate of change.
Intercept (\(b\))
=INTERCEPT(y_range, x_range)
The Y-value when X is zero.
Project Results Summary
Theory 1: Area vs. Population
Equation
y = 0.0385x + 22.42
\(r\)
0.449
\(R^2\)
0.201
Strength
MODERATE
Theory 2: Area vs. Sex Ratio (With Outliers)
Equation
y = -0.0001x + 102.53
\(r\)
-0.046
\(R^2\)
0.002
Strength
NEAR ZERO
Theory 2: Area vs. Sex Ratio (Outliers Removed)
Equation
y = 0.0001x + 98.42
\(r\)
0.021
\(R^2\)
0.0004
Strength
NONE
*Removed: Bahrain, Kuwait, Saudi Arabia, Qatar, Oman, and UAE.
Graphing Tips
Trendline Checklist
Right-click data points -> Add Trendline
Check "Display Equation on chart"
Check "Display R-squared value on chart"
Axes Alignment
Ensure Surface Area is always on the Horizontal (X) Axis . If your graph looks strange, swap the columns in Excel or change the Data Source in the chart settings.
