In-depth Interpretation of the Seven QC Tools · Scatter Diagram

Introduction

"I think there is a relationship" and "Data proves there is a relationship" are two different things.

The scatter diagram is one of the seven QC tools used to "verify relationships."

In previous issues, we learned:

• Fishbone diagram: Hypothesize causes (qualitative)
• Pareto chart: Focus on key issues (ranking)
• Stratification: Discover differences (classification)

The scatter diagram addresses: Do two variables have a relationship? How strong is the relationship? What is the direction of the relationship?

Chapter 1: The Essence of Scatter Diagrams

1.1 What is a Scatter Diagram?

Scatter Diagram (Scatter Diagram) is a chart that plots the data of two variables as points on a Cartesian coordinate system to observe and analyze whether there is a correlation and the form of the relationship.

Core Logic:
  X-axis = One variable (e.g., temperature)
  Y-axis = Another variable (e.g., defect rate)
  Each point = A data pair (temperature, defect rate)
  
  Determine by the distribution of points: Do the two variables have a relationship?

1.2 Six Forms of Correlation

① Positive Correlation
    ↑ Y          •••
    |         •••••
    |       •••••
    |      •••
    +————→ X
  As X increases, Y also increases

② Negative Correlation
    ↑ Y       •••
    |      •••••
    |    •••••
    |   •••
    +————→ X
  As X increases, Y decreases

③ Strong Positive Correlation
    ↑ Y       ••••
    |       ••••••
    |      ••••••
    |      ••••
    +————→ X
  Points are tightly distributed around a straight line

④ Weak Positive Correlation
    ↑ Y     •  •••
    |     ••• • ••
    |    •• •• ••
    |   • •••
    +————→ X
  Points are loosely distributed, trend is not obvious

⑤ No Correlation
    ↑ Y   • • • • • •
    |    • • • • • •
    |   • • • • • •
    |  • • • • • •
    +————→ X
  Points are randomly distributed, no pattern

⑥ Non-linear Correlation (Curvilinear Correlation)
    ↑ Y       •••
    |      ••  ••
    |    ••    ••
    |   •      ••
    +————→ X
  Points are distributed in a curvilinear pattern

1.3 Three Major Functions of Scatter Diagrams

1.4 Scatter Diagram vs. Correlation Analysis

Scatter Diagram (Qualitative)                      Correlation Coefficient (Quantitative)
────────────────────────────────────────────────────────────────────────────────────────────
  Observe the distribution pattern of points      Calculate the r value using mathematical methods
  Can determine: whether there is a relationship,  Can only determine: the strength of linear correlation
  direction, and form                              Cannot identify non-linear correlation
  Can identify outliers                             Outliers can significantly affect the accuracy of the r value

Best Practice: Draw a scatter diagram first, then calculate the correlation coefficient!

Chapter 2: Drawing and Analyzing Scatter Diagrams

2.1 Standard Drawing Steps

Step 1: Determine the Analysis Object
  Identify the two variables to be analyzed
  X variable (independent variable/cause variable)
  Y variable (dependent variable/result variable)

Step 2: Collect Data Pairs
  Collect at least 30-50 data pairs
  Data must be paired (X and Y values under the same conditions)

Step 3: Set the Coordinate Axes
  X-axis: Horizontal axis (cause variable)
  Y-axis: Vertical axis (result variable)
  The scale range should cover all data points

Step 4: Plot the Points
  Each data pair corresponds to one point

Step 5: Analyze and Judge
  Observe the distribution pattern of the points
  Determine whether there is a correlation

2.2 Data Collection Requirements

Key Points for Data Collection:

  ① Pairing Principle
    Each data point must correspond to the same sample/condition
    ✓ Correct: Temperature + defect rate of the same batch of products
    ✗ Incorrect: Today's temperature + yesterday's defect rate

  ② Sample Size
    At least 30, preferably more than 50
    Too few samples lead to high randomness, making conclusions unreliable

  ③ Data Range
    Cover the entire possible range of the variables
    If only the middle range is measured, extreme value relationships may be missed

  ④ Data Accuracy
    The measurement system must be reliable (perform MSA first)
    Incorrect data will inevitably lead to incorrect conclusions

2.3 Quick Reference for Correlation Coefficients (r Values)

Correlation Coefficient r (Pearson Correlation Coefficient):
  r = 1.0        Perfect positive correlation
  r = 0.8～1.0    Strong positive correlation
  r = 0.5～0.8    Moderate positive correlation
  r = 0.3～0.5    Weak positive correlation
  r = -0.3～0.3   No correlation (or no linear correlation)
  r = -0.5~-0.3  Weak negative correlation
  r = -0.8~-0.5  Moderate negative correlation
  r = -1.0~-0.8  Strong negative correlation
  r = -1.0       Perfect negative correlation

⚠ Precautions:
  1. The r value can only measure linear correlation, not non-linear correlation
  2. The r value is highly affected by outliers
  3. Correlation ≠ Causation (a relationship between two variables does not mean one causes the other)

Chapter 3: Practical Cases of Scatter Diagrams

Case 1: Manufacturing — Relationship Between Temperature and Defect Rate

Background: A plastic injection molding workshop suspects that mold temperature affects product shrinkage defects
Collected 30 data pairs of mold temperature and defect rate

Data Analysis:
  Higher mold temperature → Lower defect rate
  Scatter diagram shows: obvious negative correlation distribution
  Correlation coefficient r = -0.82 (strong negative correlation)

Conclusion:
  There is a strong negative correlation between mold temperature and defect rate
  Appropriately increasing the mold temperature can reduce the defect rate

Action:
  Adjust the mold temperature setting from 180°C to 200°C
  Defect rate decreased from 4.5% to 2.1%

Case 2: Service Industry — Relationship Between Waiting Time and Customer Satisfaction

Background: A bank branch suspects that waiting time affects customer satisfaction

Data Analysis:
  Longer waiting time → Lower satisfaction
  Scatter diagram: moderate negative correlation (points are more loosely distributed)
  Correlation coefficient r = -0.65

Conclusion:
  Waiting time is indeed an important factor affecting satisfaction
  But it is not the only factor (other factors are also at play)

Action:
  Set up a waiting time alert (automatic reminder if waiting time exceeds 15 minutes)
  Increase the number of service windows during peak hours
  Satisfaction score improved from 3.2 to 4.1

Case 3: Software Industry — Relationship Between Lines of Code and Defects

Background: A development team suspects that module size (lines of code) is related to the number of defects

Data Analysis:
  More lines of code → More defects
  Scatter diagram: moderate positive correlation
  Correlation coefficient r = 0.72

Identified Outliers:
  Two modules with very few lines of code but many defects
  → Further investigation revealed that these were written by new developers

Action:
  ── Set a code review threshold for large modules
  ── Conduct additional reviews for code written by new developers
  ── Reduced module defect rate by 35%

Case 4: Non-linear Correlation — Chemical Reaction

Background: The relationship between reaction temperature and yield of a chemical product

Data Analysis:
  Too low temperature → Low yield
  Moderate temperature → Highest yield
  Too high temperature → Yield decreases

  Scatter diagram: shows a clear inverted U-shaped curve
  Correlation coefficient r = 0.12 (does not reflect the true relationship!)

⚠ Key Lesson:
  If only the r value is considered, it may be misjudged as "no correlation"
  But drawing a scatter diagram reveals: a clear curvilinear relationship
  → This highlights the importance of "drawing the diagram first, then calculating the value"

Chapter 4: Common Misconceptions About Scatter Diagrams

Misconception 1: Confusing Correlation with Causation

× Incorrect Inference:
  "Ice cream sales and drowning incidents are positively correlated (r=0.95)"
  → Conclusion: Eating ice cream leads to drowning

✓ Correct Understanding:
  Both are results of warmer weather (common cause)
  Correlation ≠ Causation
  A scatter diagram can only prove "there is a relationship," not "who causes who"

Misconception 2: Relying Only on the Correlation Coefficient

× Incorrect Practice:
  Directly calculate the r value using software
  r=0.02 → No correlation, end analysis

✓ Correct Practice:
  Draw a scatter diagram first
  Possible Discoveries:
  → Random distribution (no correlation)
  → Curvilinear relationship (non-linear correlation, r value will be close to 0)
  → An outlier is pulling down the r value

Misconception 3: Insufficient Sample Size

× Incorrect Practice:
  Only collect 5-10 data pairs
  Judge the relationship between two variables based on a few points

✓ Correct Practice:
  Collect at least 30 data pairs, preferably more than 50
  More samples lead to more reliable conclusions

Misconception 4: Narrow Data Range

× Incorrect Practice:
  Collect data only under normal production conditions
  Temperature range: 180-190°C (normal range)

✓ Correct Practice:
  Appropriately expand the data collection range
  Temperature range: 170-210°C (including critical values)
  To fully understand the true relationship between the two variables

Misconception 5: Ignoring Outliers

× Incorrect Practice:
  Directly delete outliers when seen

✓ Correct Practice:
  Mark outliers and investigate the cause
  Outliers may reveal important special causes
  → Such as: equipment failure, operational errors, measurement errors

Chapter 5: Combining Scatter Diagrams with Other Tools

5.1 Scatter Diagram + Fishbone Diagram

Best Practice:

Step 1: Fishbone Diagram Brainstorming
  → Hypothesize N possible causes

Step 2: Verify with Scatter Diagram
  → Draw a scatter diagram for each hypothesized cause variable and the result variable
  → Verify whether there is a true correlation

Step 3: Screen Key Variables
  → Significantly correlated → Include in improvement scope
  → Not significantly correlated → Exclude to save resources

5.2 Scatter Diagram + Stratification

Combined Use:

Step 1: Use stratification to categorize data by different dimensions
Step 2: Draw a scatter diagram for each category
Step 3: Compare whether the scatter diagrams of each category are consistent

Example:
  Relationship between temperature and defect rate
  Categorized by shift:
  → Day shift: strong negative correlation (r=-0.85)
  → Night shift: weak correlation (r=-0.30)
  → Indicates: there are other interfering factors during the night shift

5.3 Scatter Diagram + Control Chart

Combined Use:

Step 1: Control Chart → Determine if the process is in control
Step 2: Scatter Diagram → Analyze variable relationships
Step 3: Control Chart → Verify the effectiveness of improvements

Note:
  If the process is out of control (special causes present),
  The data in the scatter diagram is mixed with the effects of special causes,
  The correlation conclusions may be unreliable.

5.4 Scatter Diagram + Regression Analysis

Scatter Diagram → Qualitative judgment (whether there is a relationship)
Regression Analysis → Quantitative modeling (what kind of relationship)

Upgrade Path:
  Step 1: Draw a scatter diagram → Identify a linear relationship
  Step 2: Calculate the correlation coefficient r = 0.85
  Step 3: Perform regression analysis → Y = 0.5X + 10
  Step 4: Use the regression equation → For prediction and control

Chapter 6: Advanced Uses of Scatter Diagrams

6.1 Grouped Scatter Diagram

Use different colors/symbols to distinguish different groups on the same chart
Can simultaneously see: overall trend + differences between groups

Example: Temperature-defect rate relationship for different shifts
  ● Day shift: clear trend, well-controlled
  ▲ Night shift: points are scattered, other interfering factors present
  → Further analysis needed for the night shift

6.2 Time Series Scatter Diagram

Label the time sequence on the scatter diagram
Can observe whether the variable relationship changes over time

Method:
  Label the time sequence next to each point
  Or use color gradients to indicate the time sequence

Value:
  If the distribution of early points differs from later points
  → Indicates a structural change in the relationship
  → Requires re-analysis

6.3 Multi-variable Matrix Scatter Diagram

Simultaneously display the relationships between multiple variables
One matrix chart = N×(N-1)/2 scatter diagrams

Advantages:
  ── Quickly identify key variables
  ── Discover interactions between variables
  ── Provide input reference for DOE (Design of Experiments)

Chapter 7: Evaluation Standards for Scatter Diagrams

Summary: The "Way" and "Method" of Scatter Diagrams

Method (How to Draw):
  ── Collect 30-50 paired data points
  ── X-axis = cause variable, Y-axis = result variable
  ── Plot points and observe the distribution pattern
  ── Combine with the r value for comprehensive judgment

Way (Why to Draw):
  ── Not to draw a "trendy chart"
  ── To "replace intuition with data"
  ── To "confirm the direction before making improvements"

The greatest value of a scatter diagram is not to "prove a relationship," but to "avoid wasting resources in the wrong direction."

In quality management, the most expensive mistake is not doing something wrong, but spending a lot of resources to improve a cause that is not related at all.

The scatter diagram is a tool to help you avoid such mistakes.

Preview of the Next Issue

Issue 5: Scatter Diagram (Scatter Diagram / Scatter Plot)

We will discuss data distribution — not just the average, but also the dispersion, distribution pattern, and process capability.

Knowledge code: 5.2.4
Version: v20260503
Author: Quality Think Tank