Unit IV Β· Lecture 27 of 40 Β· B.Sc. Data Science, 1st Year, Semester II
One variable measured per observation.
Examples:
Heights of 50 students
Daily temperatures for a month
Exam scores of a class
Tools: Mean, variance, histograms, frequency tables
Two variables measured on each observation.
Examples:
Height & weight of students
Advertising spend & sales
Study hours & exam score
Tools: Scatter diagrams, correlation, regression
Three or more variables per observation.
Examples:
Height, weight & age
Temperature, humidity & rainfall
Multiple exam scores per student
Tools: Multiple regression, partial correlation
Bivariate data consists of pairs of observations (xα΅’, yα΅’) for i = 1, 2, β¦, n, where two different measurements are recorded on each subject or experimental unit. The goal is to study the relationship between the two variables β how changes in one variable are associated with changes in the other.
Univariate statistics is like studying a single clue β it tells you about one piece of evidence. Bivariate statistics is like comparing two clues together β it reveals connections between them. Does the suspect's alibi hold up when compared with the timeline? Do two measurements move together? This is where real insight begins.
1. Is there a relationship? Do x and y tend to move together?
2. How strong is it? Is the association weak, moderate, or strong?
3. What direction? When x increases, does y increase or decrease?
4. What is the form? Is the relationship linear, curved, or random?
5. Can we predict? Given x, can we estimate y?
Height & Weight: Taller people tend to weigh more (positive).
Price & Demand: Higher prices tend to reduce demand (negative).
Study Hours & Score: More study tends to improve scores (positive).
Temperature & Ice Cream Sales: Hotter days increase sales (positive).
Age & Reaction Time: Reaction time tends to increase with age (positive after maturity).
A bivariate frequency table (or two-way table) shows the frequency of occurrence of each combination of values for two variables. Each cell shows how many observations fall into that particular (x-class, y-class) combination.
| Height β \ Weight β | 40β50 | 50β60 | 60β70 | 70β80 | Total |
|---|---|---|---|---|---|
| (yβ) | (yβ) | (yβ) | (yβ) | ||
| 150β160 (xβ) | 4 | 3 | 1 | 0 | 8 |
| 160β170 (xβ) | 2 | 6 | 5 | 1 | 14 |
| 170β180 (xβ) | 0 | 3 | 7 | 4 | 14 |
| 180β190 (xβ) | 0 | 0 | 2 | 2 | 4 |
| Total | 6 | 12 | 15 | 7 | 40 |
The row totals give the marginal distribution of height (x): 8, 14, 14, 4.
The column totals give the marginal distribution of weight (y): 6, 12, 15, 7.
These are just the univariate distributions of each variable separately.
The high-frequency cells (6, 7, 5) cluster along the diagonal from top-left to bottom-right. This indicates a positive association β taller students tend to weigh more.
If the high frequencies were along the opposite diagonal, it would indicate a negative association.
A scatter diagram (or scatter plot) is a graphical representation of bivariate data where each observation (xα΅’, yα΅’) is plotted as a point on a two-dimensional coordinate system. The horizontal axis (x-axis) represents one variable and the vertical axis (y-axis) represents the other.
A scatter diagram is the first thing you should draw when analysing bivariate data. Before computing any correlation coefficient or fitting any regression line, look at the scatter plot. It reveals the shape, direction, strength, and outliers of the relationship β information that summary statistics alone cannot fully capture. Anscombe's Quartet (coming up) proves this dramatically.
Step 1: Identify the two variables. Choose which is x (horizontal) and which is y (vertical). Convention: x = independent/explanatory, y = dependent/response.
Step 2: Draw axes with appropriate scales. Include labels and units.
Step 3: Plot each (xα΅’, yα΅’) pair as a dot on the graph.
Step 4: Look for patterns β direction, form, strength, outliers.
Direction: Positive (upward trend) or negative (downward trend)?
Form: Linear (straight line), curved, or no pattern?
Strength: How tightly do points cluster around a pattern?
Outliers: Any points far from the main cluster?
Clusters: Are there distinct groups in the data?
Points cluster tightly around an upward line. r β +0.9 to +1.0
Points show upward trend with scatter. r β +0.5 to +0.8
Points scattered randomly. r β 0
Points cluster tightly around a downward line. r β β0.9 to β1.0
Non-linear pattern. Pearson r β 0 (misleading!)
One point far from cluster. Can distort r significantly.
Slight upward trend, much scatter. r β +0.2 to +0.4
All points on exact line. r = +1.0
Statistician Francis Anscombe created four datasets that have identical summary statistics (same mean of x, mean of y, variance of x, variance of y, correlation r = 0.816, and same regression line) β but look completely different when plotted. This proves that you must always visualise your data before computing statistics.
Linear relationship. Classic scatter. The regression line fits well.
Curvilinear pattern. A straight line is NOT the right model here.
One outlier pulls the line. Without it, r would be 1.0.
All x's are the same except one. One point controls everything.
Enter paired data and see the scatter diagram. Try different patterns to build intuition:
| Type | Both Variables | Example | Analysis Method |
|---|---|---|---|
| Both Quantitative | Continuous or discrete numerical | Height & Weight | Pearson's r, regression |
| Both Qualitative | Categorical (nominal/ordinal) | Gender & Eye colour | Chi-square test, CramΓ©r's V |
| One of Each | One numerical, one categorical | Gender & Height | Point-biserial correlation, t-test |
| Both Ranked | Ordinal or rank-ordered | Rank in class & Rank in sport | Spearman's rank correlation |
Karl Pearson's correlation (Lecture 28) is designed for both quantitative variables.
Spearman's rank correlation (Lecture 30) works with ranked or ordinal data, or when the relationship is monotonic but not necessarily linear.
Chi-square and CramΓ©r's V (covered elsewhere) handle both qualitative variables.
Choosing the right method depends on the type of data you have.
The variable that is controlled, manipulated, or used to predict. Also called the explanatory variable, predictor variable, or regressor.
Placed on the horizontal axis of a scatter plot.
Example: Study hours, advertising spend, temperature, height.
The variable whose value depends on or is predicted by the independent variable. Also called the response variable, outcome variable, or regressand.
Placed on the vertical axis of a scatter plot.
Example: Exam score, sales, ice cream sales, weight.
Finding a strong correlation between two variables does not mean one causes the other. Ice cream sales and drowning deaths are positively correlated β but ice cream doesn't cause drowning. Both are caused by a third variable (hot weather). Always consider confounding factors and the possibility of coincidence.
Q1: Bivariate data involves measuring:
Q2: In a scatter diagram, the independent variable is placed on:
Q3: If points in a scatter plot cluster tightly around a line sloping upward from left to right, this indicates:
Q4: Anscombe's Quartet demonstrates that:
Q5: "Correlation does not imply causation" means:
Q6: The row totals in a bivariate frequency table give:
1. The following data gives marks in Statistics (x) and marks in Mathematics (y) for 8 students. Draw a scatter diagram and describe the pattern.
x: 30, 35, 40, 45, 50, 55, 60, 65
y: 32, 38, 42, 48, 50, 54, 58, 62
2. Construct a bivariate frequency table for the following data on hours of TV watched (x) and GPA (y) for 20 students. Use classes for both variables.
3. For each scatter pattern below, state whether Pearson's r would be close to +1, β1, 0, or misleading:
(a) Perfect upward line (b) U-shaped curve (c) Random cloud (d) Downward line with one outlier
4. Explain why a scatter diagram should always be drawn before computing a correlation coefficient.
5. BONUS: Look up Anscombe's Quartet data online. Compute the mean and variance of x and y for all four datasets and verify they are identical.
Lecture 28: Karl Pearson's Coefficient of Correlation β the most widely used measure of linear association.