The Scatter Diagram (Scatter Plot) is the simplest visual and graphical technique for inspecting bivariate data. By plotting paired numerical observations $(x_1, y_1), (x_2, y_2), \dots, (x_n, y_n)$ as discrete points on a two-dimensional Cartesian plane, the scatter plot visually reveals the presence, direction, linearity, and approximate strength of correlation before undertaking any intensive algebraic calculations.
Perfect Positive Correlation ($r = +1$): All data points lie exactly on a single straight line rising upward from the lower-left to the upper-right quadrant.
High Degree Positive Correlation ($0.75 \le r < 1$): Points cluster tightly along an upward-sloping straight band from bottom-left to top-right.
Low / Moderate Positive Correlation ($0 < r < 0.75$): Points show a discernible upward trend but are widely dispersed across the plane.
Perfect Negative Correlation ($r = -1$): All data points lie exactly on a single straight line sloping downward from upper-left to lower-right.
High Degree Negative Correlation ($-1 < r \le -0.75$): Points cluster tightly along a downward-sloping linear band from top-left to bottom-right.
Zero Correlation ($r = 0$): Points are scattered haphazardly over the entire plot without any directional drift (forming a circular or diffuse cloud), or form a strictly horizontal or vertical line.
Curvilinear (Non-Linear) Correlation ($r \approx 0$ linearly): Points follow a well-defined curved path (e.g., U-shape, inverted U, or parabola). While linear correlation $r$ may compute to zero, a strong non-linear functional dependency exists!
Merits & Limitations of Scatter Diagrams
Merits
Extremely simple and intuitive visual inspection requiring zero mathematical calculations.
Instantly identifies whether the underlying relationship is linear or non-linear.
Easily highlights extreme data outliers and anomalous entry points.
Unaffected by mathematical scale distortions.
Limitations
Purely qualitative: cannot provide an exact, quantified numerical value for correlation (e.g., cannot distinguish between $r = 0.78$ and $r = 0.84$).
Subjective interpretation: different observers may judge the dispersion of the point band differently.
Unsuitable when multiple observations share identical $(x_i, y_i)$ coordinates (overplotting).
Worked Benchmark Examples
Example 1 (Interpreting Direction and Degree):
A scatter plot of employee years of experience ($X$) versus annual salary ($Y$) reveals that all points form a narrow, tightly clustered band rising upward from the bottom-left to the top-right. What can you infer about:
(a) The direction of correlation?
(b) The degree of correlation?
(c) The approximate numerical range of Karl Pearson's coefficient $r$?
Show Step-by-Step Solution
• (a) Direction: Because the band rises from bottom-left to top-right, the correlation is Positive (Direct).
• (b) Degree: Because the points form a narrow, tightly clustered band rather than a wide diffuse cloud, the degree of correlation is High.
• (c) Numerical Range: For a high degree positive linear correlation, Karl Pearson's coefficient lies approximately in the range $\mathbf{+0.75 \le r < +1.0}$.
Example 2 (The Curvilinear Trap):
Consider the bivariate dataset:
$$(-3, 9), \quad (-2, 4), \quad (-1, 1), \quad (0, 0), \quad (1, 1), \quad (2, 4), \quad (3, 9)$$
(i) Plotting these points on a scatter diagram reveals what geometric curve?
(ii) If Karl Pearson's linear correlation coefficient computes to $r = 0$, does this mean there is no relationship between $X$ and $Y$?
Show Step-by-Step Solution
• (i) Geometric Shape: The points follow a symmetric U-shaped parabolic curve defined by the exact functional rule $Y = X^2$.
• (ii) Interpretation: No! A linear correlation coefficient of $r = 0$ only means there is no linear (straight-line) relationship. Here, there is a perfect non-linear deterministic relationship ($Y = X^2$). A scatter diagram is vital because it reveals non-linear patterns that linear correlation formulas completely miss!
CHECK YOUR PROGRESS 11.2
Graded textbook exercises on Scatter Diagram Interpretations from CBSE Class 11 Applied Mathematics.
Problem 1Visual Archetype Identification
State the type and degree of correlation indicated by each of the following scatter diagram patterns:
(a) All plotted points lie exactly on a straight line falling from upper-left to lower-right.
(b) Points are widely scattered in a circular cloud with no discernible trend.
(c) Points cluster closely along a straight line rising from lower-left to upper-right.
Show Step-by-Step Solution
• (a) Perfect Negative Correlation ($r = -1$).
• (b) Zero / No Correlation ($r = 0$).
• (c) High Degree Positive Correlation ($0.75 \le r < 1$).
Problem 2Plot Construction & Inspection
Given the paired observations of temperature ($X$ in °C) and sales of hot beverages ($Y$ in cups):
$X$: $10, \quad 15, \quad 20, \quad 25, \quad 30, \quad 35$
$Y$: $90, \quad 80, \quad 65, \quad 50, \quad 35, \quad 20$
Describe the appearance of its scatter plot and specify the type of correlation.
Show Step-by-Step Solution
• As $X$ increases by uniform steps of $+5$ ($10 \to 35$), $Y$ decreases consistently:
$(10, 90), (15, 80), (20, 65), (25, 50), (30, 35), (35, 20)$
• The plotted points form a downward-sloping linear band falling from upper-left to lower-right.
• Conclusion: The scatter plot indicates a High Degree Negative Correlation.
Problem 3Qualitative vs Quantitative
Why is the scatter diagram method termed a 'qualitative' rather than a 'quantitative' method of correlation analysis?
Show Step-by-Step Solution
• A scatter diagram reveals the visual nature and general direction (positive/negative/zero) of the relationship, but does not yield a precise mathematical number.
• It cannot quantify whether correlation is $0.82$ or $0.88$, and hence cannot be used in formal statistical hypothesis testing without algebraic coefficients.
Problem 4Outlier Detection on Scatter Plots
In a scatter plot of study hours vs marks, 24 students lie in a tight positive linear band, while 1 student who studied 15 hours scored 10 marks. How does this point appear on the scatter plot and what is its statistical term?
Show Step-by-Step Solution
• The point $(15, 10)$ appears isolated far away in the bottom-right corner, completely detached from the general upward-trending cluster.
• This point is termed an Outlier (or Influential Point). Scatter diagrams are especially valuable because they make such anomalies instantly visible.
Problem 5Slope & Degree Distinction
Does a steeper upward slope on a scatter plot imply a higher degree of correlation $r$? Explain clearly.
Show Step-by-Step Solution
• NO. The slope of the line reflects the rate of change ($b = Delta Y / Delta X$) and depends on the scale/units of measurement.
• The degree of correlation ($r$) depends strictly on how closely the points cluster around the line, not how steep the line is. Points clustered tightly around a gentle slope have $r approx 0.95$, while points widely dispersed around a steep slope may have $r approx 0.40$.
Comprehensive Practice Worksheet: Scatter Diagram Method
Graded problems structured according to CBSE Class 11 Applied Mathematics examination blueprint.
Q1: What is the value of Karl Pearson's correlation coefficient $r$ if all points on a scatter diagram lie strictly on a downward-sloping straight line?
Show Answer & Scheme
$$\mathbf{r = -1} \quad [1 \text{ Mark}]$$
Q2: State one primary advantage of using a scatter diagram before computing mathematical correlation coefficients.
Show Answer & Scheme
Answer [1 Mark]: It immediately reveals whether the underlying relationship is linear or non-linear, preventing the misuse of linear correlation formulas on non-linear data.
SECTION B: 2-Mark Short Answer Problems
Q3: Describe the visual appearance of a scatter plot when there is: (i) High positive correlation, and (ii) Zero correlation.
Show Answer & Scheme
• (i) High Positive Correlation [1 Mark]: The plotted points form a narrow, tightly concentrated band running from lower-left to upper-right.
• (ii) Zero Correlation [1 Mark]: The points are dispersed diffusely in a circular or chaotic cloud across the coordinate plane with no discernible direction.
Q4: "A scatter diagram can indicate whether a relationship exists, but it cannot measure its numerical strength." Discuss this statement with reasons.
Show Answer & Scheme
Discussion [2 Marks]:
The statement is accurate. A scatter plot provides a qualitative visual snapshot of the direction and linearity of association. However, because it relies on visual judgment without a numerical index, it cannot quantify the exact strength of association or differentiate subtle variations (e.g., $r = 0.75$ vs $r = 0.85$).
SECTION C: 4-Mark Comprehensive Scenario Drill
Q5: An automobile manufacturer tests the braking performance of 6 test vehicles under varying speeds:
• Speed ($X$ in km/h): $20, \quad 40, \quad 60, \quad 80, \quad 100, \quad 120$
• Stopping Distance ($Y$ in metres): $6, \quad 16, \quad 36, \quad 64, \quad 100, \quad 144$
(i) Plotting $(X, Y)$ reveals what shape of scatter diagram?
(ii) Is the relationship linear or non-linear? Explain by examining ratios $\Delta Y / \Delta X$.
(iii) What is the physical/mathematical law connecting $X$ and $Y$?
Show Step-by-Step Solution & Marking Scheme
Part (i) Shape of Scatter Diagram [1 Mark]:
• The points curve upward steeply, forming a convex parabolic curve rising from bottom-left to top-right.
Part (ii) Linearity Inspection [1.5 Marks]:
• $\Delta X = +20$ km/h consistently.
• $\Delta Y$: $16-6=10, 36-16=20, 64-36=28, 100-64=36, 144-100=44$.
• $\Delta Y / \Delta X$ increases steadily ($0.5 \to 1.0 \to 1.4 \to 1.8 \to 2.2$).
• Because the slope accelerates, the relationship is strictly Non-Linear.
Part (iii) Mathematical Law [1.5 Marks]:
• In physics, kinetic energy $E_k = \frac{1}{2}mv^2$. Work done by braking force $F \cdot d = \frac{1}{2}mv^2 \implies d \propto v^2$.
• Here $Y = 0.01 X^2$ (e.g. $0.01 \times 100^2 = 100$, $0.01 \times 120^2 = 144$).
• The scatter plot perfectly exposes this quadratic physical law.
High-Yield Quick Revision: Scatter Diagram Method
Summary of visual archetypes, correlation boundaries, and diagnostic checks for Chapter 11.2.
Visual Archetype Summary Table
Pattern Description
Visual Trend
Correlation Category
Approx. $r$ Value
Straight line rising right
Bottom-left to top-right
Perfect Positive
$r = +1$
Narrow band rising right
Bottom-left to top-right
High Positive
$+0.75 \le r < +1$
Straight line falling right
Top-left to bottom-right
Perfect Negative
$r = -1$
Narrow band falling right
Top-left to bottom-right
High Negative
$-1 < r \le -0.75$
Circular cloud of points
No directional drift
Zero Correlation
$r = 0$
Crucial Exam Distinction
Do not confuse slope with correlation tightness. A steep line with widely scattered dots has a lower correlation than a gentle slope with dots resting almost directly on the line. Correlation measures the tightness of fit around the line, not the inclination of the line.
Standard timed test: 4 Objective MCQs / Assertion-Reason with instant automatic grading, plus 6 multi-mark subjective problems with CBSE marking rubrics.
PART 1: Objective Section (4 Marks)
Q1: If all the points on a scatter diagram lie on a straight line falling from upper left to lower right, the correlation coefficient is:
Solution: (B) Perfect negative linear correlation corresponds to $r = -1$.
Q2: A scatter diagram in which the points are scattered haphazardly in a circular cloud indicates:
Solution: (C) Diffuse circular cloud indicates zero linear correlation ($r = 0$).
Q3: Which of the following is an inherent limitation of the Scatter Diagram Method?
Solution: (B) A scatter plot is qualitative and cannot provide an exact numerical value for $r$.
Q4 (Assertion-Reason): Assertion (A): A linear correlation coefficient of $r = 0$ does not guarantee that there is no relationship between two variables. Reason (R): A scatter diagram can show points forming a perfect parabolic curve where linear correlation computes to zero despite an exact deterministic functional dependency.
Solution: (A) Both Assertion and Reason are true and Reason highlights the exact non-linear pitfall ($Y = X^2$).
PART 2: Subjective Section (Q5 - Q10)
Q52 Marks
Define a 'Scatter Diagram' and state two practical situations where constructing a scatter plot is recommended prior to formal modeling.
Show Step-by-Step Solution & Marking Scheme
• Definition [1 Mark]: A scatter diagram is a Cartesian coordinate plot where paired bivariate observations $(x_i, y_i)$ are plotted as individual points to visualize their relationship.
• Situations [1 Mark]: (1) Checking whether the data satisfies the linearity assumption required for linear regression; (2) Detecting extreme outliers and data entry errors.
Q62 Marks
Sketch or describe how points are distributed on a scatter diagram when:
(a) $r = +1$
(b) $r = -1$.
Show Step-by-Step Solution & Marking Scheme
• (a) When $r = +1$, every single point lies precisely on an upward-sloping straight line ($45^circ$ or positive slope) [1 Mark].
• (b) When $r = -1$, every single point lies precisely on a downward-sloping straight line (negative slope) [1 Mark].
Q73 Marks
State three distinct merits and two limitations of the Scatter Diagram Method in statistical quality control.
Show Step-by-Step Solution & Marking Scheme
• Merits [1.5 Marks]: (1) Visual and instantly understandable by non-mathematicians; (2) Unaffected by extreme values in terms of distorting formula calculations; (3) Reveals non-linear curvature.
• Limitations [1.5 Marks]: (1) Fails to yield an exact numerical index of correlation; (2) Subject to human visual interpretation bias.
Q83 Marks
Explain why a scatter diagram where points lie on a horizontal line ($Y = c$) indicates zero correlation, even though all points lie perfectly on a straight line.
Show Step-by-Step Solution & Marking Scheme
• Mathematical Analysis [1.5 Marks]: If $Y = c$ (constant), then for every observation $y_i = c$ and mean $ar{Y} = c$. The deviations $(y_i - ar{Y}) = 0$ for all points.
• Correlation Implication [1.5 Marks]: The covariance $sum (x_i - ar{X})(y_i - ar{Y}) = 0$, and the variance $sigma_y = 0$. Since $Y$ does not vary at all with changes in $X$, knowing $X$ provides zero discriminatory information about changes in $Y$. Hence, $r = 0$.
Q9 & Q10Case Study (4 Marks Total)
Case Context: Chemical Synthesis Yield vs Temperature
A pharmaceutical researcher tests the yield of an active drug ingredient ($Y$ in grams) at different reaction temperatures ($X$ in °C):
$(20^circ, 30g), \quad (40^circ, 70g), \quad (60^circ, 100g), \quad (80^circ, 75g), \quad (100^circ, 35g)$.
Q9 (2 Marks): Describe the geometric shape formed by these points on a scatter diagram. Q10 (2 Marks): If linear correlation $r$ is computed for this data, explain why $r$ is approximately zero, and explain the chemical phenomenon behind the curve.
Show Step-by-Step Solution & Marking Scheme
Solution to Q9 [2 Marks]:
• The points rise to an optimal peak at $(60^circ, 100g)$ and then symmetrically decline at higher temperatures.
• The scatter plot forms an inverted U-shape (parabolic arc).
Solution to Q10 [2 Marks]:
• Linear $r approx 0$: Because the strong positive correlation in the first half ($20^circ o 60^circ$) is cancelled out by the equally strong negative correlation in the second half ($60^circ o 100^circ$), Karl Pearson's linear $r$ averages out to near zero.
• Chemical Phenomenon: Reaction rates accelerate with temperature up to an optimum ($60^circ$), beyond which thermal denaturation/decomposition degrades the active compound.