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

Introduction

"I think there's a relationship" and "data proves there's 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: Assume causes (qualitative)
• Pareto Chart: Focus on priorities (ranking)
• Stratification: Discover differences (classification)

The scatter diagram addresses: Is there a relationship between two variables? 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 Plot) is a chart that plots data points of two variables on a Cartesian coordinate system to observe and analyze whether there is a correlation and the pattern 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 pattern: Is there a relationship?

1.2 Six Patterns 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 show a curvilinear distribution 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 mathematically
  Can determine: presence of relationship,         Can only determine: strength of linear correlation
  direction, and pattern                           Cannot identify non-linear correlation
  Can identify outliers                            Outliers can significantly affect the accuracy of 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 Up the Axes
  X-axis: horizontal axis (cause variable)
  Y-axis: vertical axis (result variable)
  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 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
    Minimum 30 pairs, recommended 50 pairs or more
    Too few samples can lead to unreliable conclusions due to high randomness

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

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

2.3 Quick Reference for Correlation Coefficients (r Value)

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

⚠ Important Notes:
  1. The r value can only measure linear correlation, not non-linear correlation
  2. The r value is highly sensitive to outliers
  3. Correlation ≠ Causation (a relationship between two variables does not imply 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: clear negative correlation
  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 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 scattered)
  Correlation coefficient r = -0.65

Conclusion:
  Waiting time is indeed an important factor affecting satisfaction
  However, it is not the only factor (other factors are also influencing)

Action:
  Set up an over-time waiting alert (automatically remind 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

Outliers Identified:
  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 you only look at the r value, you might misjudge it as "no correlation"
  But by drawing the scatter diagram, you can see: there is a clear curvilinear relationship
  → This highlights the importance of "drawing the diagram first, then calculating the value"

Chapter 4: Common Misunderstandings of Scatter Diagrams

Misunderstanding 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
  Scatter diagrams can only prove "there is a relationship," not "who causes who"

Misunderstanding 2: Relying Solely on the Correlation Coefficient

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

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

Misunderstanding 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 pairs, recommended 50 pairs or more
  The more samples, the more reliable the conclusion

Misunderstanding 4: Too 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

Misunderstanding 5: Ignoring Outliers

× Incorrect Practice:
  Directly delete outliers when seen

✓ Correct Practice:
  Mark outliers, 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
  → Assume N possible causes

Step 2: Verify with Scatter Diagrams
  → Plot a scatter diagram for each assumed cause variable and the result variable
  → Verify whether the relationship is truly significant

Step 3: Screen Key Variables
  → Significant correlation → Include in improvement scope
  → No significant correlation → Exclude to save resources

5.2 Scatter Diagram + Stratification

Combined Use:

Step 1: Stratify data by different dimensions using stratification
Step 2: Plot a scatter diagram for each stratum
Step 3: Compare the scatter diagrams of each stratum for consistency

Example:
  Relationship between temperature and defect rate
  Stratified 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 the relationship between variables
Step 3: Control Chart → Verify the effectiveness of improvements

Note:
  If the process is out of control (affected by special causes),
  the scatter diagram data will be mixed with the influence of special causes,
  leading to unreliable correlation conclusions.

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 the presence of 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 Diagrams

Use different colors/symbols to distinguish different groups on the same chart
This allows you to see: overall trend + differences between groups

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

6.2 Time Series Scatter Diagrams

Label the time sequence on the scatter diagram
This allows you to observe whether the relationship between variables changes over time

Method:
  Label the time sequence next to each point
  Or use a color gradient to represent the time sequence

Value:
  If the distribution of early points differs from later points
  → It indicates that the relationship has undergone a structural change
  → Re-analysis is required

6.3 Multivariate Matrix Scatter Diagrams

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 for DOE (Design of Experiments)

Chapter 7: Evaluation Criteria 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 just to draw a "trendy chart"
  ── But to "replace intuition with data"
  ── To "confirm the direction before taking action"

The greatest value of a scatter diagram is not to "prove there is a relationship," but to "avoid wasting resources on 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.

A scatter diagram is the tool to help you avoid such mistakes.

Next Issue Preview

Issue 5: Scatter Diagram (Scatter Plot)

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