# No Correlation: Definition, Graphs and Examples

No correlation means two variables have no linear relationship: as one changes, the other does not move in any consistent direction. In a scatterplot the points form a shapeless cloud with no upward or downward trend [1]. The correlation coefficient sits near zero, and the best-fitting straight line is nearly flat.

## Quick Answer

- No correlation means one variable has no linear effect on the other [1].
- A no correlation graph shows points scattered everywhere with no upward or downward sloping line [1].
- The Pearson correlation coefficient $r$ ranges from -1 to +1, and 0 indicates no linear association [2].
- A small $r$ is not proof of independence. It only rules out a straight-line relationship.
- Always check the scatterplot. Curved or clustered patterns can hide behind a near-zero $r$.

## What No Correlation Means

In plain terms, no correlation describes two variables that move independently of each other. Knowing the value of one tells you nothing useful about the value of the other. If you compare people's height to their exam scores, tall people are not systematically better or worse scorers, so the two variables show no correlation [1].

The precise statistical definition is narrower. Correlation measures a linear association between two continuous variables [3]. No correlation means the linear association is absent, so the Pearson correlation coefficient equals zero in the population. The coefficient is scaled from -1 to +1, where 0 indicates no linear or monotonic association and the relationship strengthens as the value approaches either end [2]. A sample estimate near zero is consistent with no correlation, though sampling noise means it rarely lands exactly on 0.000.

One distinction matters. Zero linear correlation does not mean the variables are unrelated in every way. A perfect U-shaped relationship can produce $r$ near zero because the linear measure cannot see it. That is why the scatterplot and the coefficient belong together.

## How It Works

The Pearson correlation coefficient standardizes the covariance between two variables. The formula is:

$$r = \frac{\text{cov}(x, y)}{s_x \, s_y}$$

Each symbol has a specific job:

- $\text{cov}(x, y)$ is the covariance, the average tendency of $x$ and $y$ to deviate from their means together.
- $s_x$ is the standard deviation of $x$.
- $s_y$ is the standard deviation of $y$.

Dividing by both standard deviations removes the units, which is why $r$ is a pure number between -1 and +1. When positive deviations in $x$ pair equally often with positive and negative deviations in $y$, the covariance cancels toward zero and so does $r$. That cancellation is the arithmetic signature of no correlation.

The coefficient is symmetric, so $r$ between $x$ and $y$ equals $r$ between $y$ and $x$. It is also unaffected by changing the units of either variable, since the standard deviations absorb any rescaling.

## Worked Example

This example uses 40 independent (study hours, exam score) pairs generated with a fixed random seed. The first 10 rows are shown below.

| study_hours | exam_score |
|---|---|
| 4.4 | 47.3 |
| 9.6 | 69.7 |
| 7.6 | 42.1 |
| 6.4 | 94.6 |
| 2.4 | 55.5 |
| 2.4 | 79.8 |
| 1.5 | 58.7 |
| 8.8 | 71.2 |
| 6.4 | 72.8 |
| 7.4 | 51.1 |

The steps follow the formula directly.

- Sample size: $n = 40$
- Mean study hours: $\bar{x} = 5.1075$
- Mean exam score: $\bar{y} = 68.5325$
- Standard deviation of study hours: $s_x = 2.6650$
- Standard deviation of exam scores: $s_y = 19.1306$
- Covariance: $\text{cov} = 7.7826$
- Pearson $r$ by hand: $r = \dfrac{7.7826}{2.6650 \times 19.1306} = 0.1526$
- Pearson $r$ from software: $r = 0.1526$, $p = 0.3470$
- Regression slope: $1.0958$
- Regression intercept: $62.9358$

The covariance is small relative to the spread of the two variables, so the ratio collapses to 0.1526. That is close to zero and far from the strong values near 1 or -1. The p-value of 0.3470 is well above the usual 0.05 threshold, so you cannot reject the null hypothesis of no linear association. The regression line has a slight positive slope of 1.0958, but with $r = 0.1526$ it explains almost none of the variation in exam scores.

```python
import numpy as np
from scipy import stats
np.random.seed(42)
x = np.round(np.random.uniform(1, 10, 40), 1)
y = np.round(np.random.uniform(40, 100, 40), 1)
r, p = stats.pearsonr(x, y)
print(f"r = {r:.4f} (p = {p:.4f})")
```

Output:

```
r = 0.1526 (p = 0.3470)
```

You can reproduce these numbers with the [Correlation Coefficient Calculator](/tools/correlation-coefficient-calculator) if you prefer not to run code.

## How to Interpret It

Read the scatterplot first, then the coefficient. A no correlation graph has no upward or downward sloping line, just points scattered across the plot [1]. If you see a clear curve, a fan shape, or two separate clusters, the coefficient alone will mislead you.

For the coefficient itself, treat the value as a sliding scale. Values near 0 indicate no linear association, and the relationship strengthens as $r$ moves toward -1 or +1 [2]. There is no universal cutoff where "weak" becomes "strong," and different researchers label the same magnitude differently, so report the number and the direction explicitly instead of relying on a word like "weak" [4]. A common rule of thumb exists for interpreting the size of a correlation coefficient, but it is a guide, not a law [3].

The p-value answers a separate question. It tests whether the observed $r$ could plausibly arise from a population with zero correlation. A large p-value like 0.3470 supports no correlation. A small p-value with a small $r$ means the association is detectable but still weak in practice.

## When to Use It (and when not to)

Use a correlation coefficient when both variables are continuous and you want to quantify a straight-line association [3]. It is a fast, simple summary that is easy to calculate and interpret [3]. It also works well as a screening step before fitting a regression, since weak correlations suggest a linear fit may not be worthwhile [5].

Do not use Pearson correlation when the relationship is clearly curved, when the data contain influential outliers, or when the variables are ordinal. For nonnormal continuous data, ordinal data, or data with relevant outliers, a Spearman rank correlation measures a monotonic association instead [2]. Do not use correlation to claim that one variable causes another. A near-zero coefficient also does not prove two variables are unrelated, only that no straight-line pattern exists.

## No Correlation vs Zero Correlation

The two phrases describe the same idea, and analysts use them interchangeably. "No correlation" is the everyday phrasing. "Zero correlation" emphasizes the population value of the coefficient. The practical difference is only in emphasis, not in meaning.

| Feature | No correlation | Zero correlation |
|---|---|---|
| Meaning | No linear relationship between variables | Population coefficient equals 0 |
| Typical sample $r$ | Near 0, rarely exactly 0 | Exactly 0 in theory |
| Scatterplot | Shapeless cloud, no trend [1] | Same pattern |
| Common phrasing | Everyday description | Statistical description |

If you want the fuller treatment of the exact-zero case, see [what zero correlation means](/blog/data-analysis/zero-correlation-definition-examples).

## Common Mistakes

- **Reading a small $r$ as proof of independence.** A near-zero coefficient rules out a linear pattern only. The fix is to plot the data and check for curves or clusters.
- **Judging correlation from the coefficient alone.** Two very different datasets can share the same $r$. The fix is to always inspect the scatterplot before interpreting the number [5].
- **Ignoring outliers.** A single extreme point can push $r$ toward zero or away from it. The fix is to check for influential points and consider a rank-based coefficient [2].
- **Calling a weak correlation "no correlation" without context.** A small but real association can matter in some fields. The fix is to report $r$ and the p-value together [4].
- **Using Pearson correlation on ordinal data.** The coefficient assumes a linear relationship between continuous variables [3]. The fix is to use Spearman rank correlation for ordinal or nonnormal data [2].
- **Treating no correlation as no relationship of any kind.** Variables can be strongly linked in a curved way while $r$ stays near zero. The fix is to fit a curve or bin the data and compare group means.

## Limitations

A correlation coefficient cannot detect nonlinear relationships. A perfect parabola can yield $r$ near zero, which looks identical to genuine independence if you only read the number. The coefficient also says nothing about causation, and it is sensitive to the range of the data. Restrict the range of $x$ and a real association can shrink toward zero.

Sampling adds another layer of uncertainty. A sample $r$ of 0.1526 from 40 pairs is an estimate, and a different sample would give a different value. The p-value helps here, but it depends on assumptions about the data. For nonnormal data or data with outliers, the standard Pearson test can mislead, which is why the rank-based alternative exists [2]. Correlation also assumes each pair is independent of the others, so repeated measurements on the same subject violate the method.

## Frequently Asked Questions

### What does no correlation look like on a graph?

The points form a shapeless cloud with no upward or downward sloping line [1]. They scatter everywhere across the plot instead of clustering near a line. You may still see random clumps, but no consistent direction.

### What correlation coefficient means no correlation?

A value of 0 indicates no linear association, and the coefficient ranges from -1 to +1 [2]. In practice, sample values near zero, like the 0.1526 in the worked example, are consistent with no correlation. There is no sharp cutoff, so report the number and its p-value.

### Can you have no correlation but still have a relationship?

Yes. A strong curved relationship, such as a U-shape, can produce $r$ near zero because Pearson correlation only measures straight-line association [3]. Plotting the data reveals the pattern that the coefficient misses.

### Is no correlation the same as no causation?

They are different questions. No correlation means there is no linear association to explain. Correlation never proves causation in any case, so a zero coefficient simply removes one possible line of evidence.

### Does a p-value above 0.05 prove no correlation?

No. A large p-value means the data are consistent with zero correlation, not that zero is proven. Small samples have low power, so a real association can go undetected. Report the coefficient, the p-value, and the sample size together [4].

For the broader picture of how correlation fits into regression, start with [what correlation is](/blog/data-analysis/what-is-correlation) and [correlation vs covariance](/blog/data-analysis/correlation-vs-covariance).

## References

1. [What is the difference between positive correlation, negative correlation, and no correlation? - CGTC FAQs](https://libanswers.centralgatech.edu/AcademicSupport/faq/420629)
2. [Schober P, Boer C, Schwarte LA. (2018). Correlation Coefficients: Appropriate Use and Interpretation. Anesthesia and analgesia](https://pubmed.ncbi.nlm.nih.gov/29481436/)
3. [Mukaka MM. (2012). Statistics corner: A guide to appropriate use of correlation coefficient in medical research. Malawi medical journal : the journal of Medical Association of Malawi](https://pmc.ncbi.nlm.nih.gov/articles/PMC3576830/)
4. [Akoglu H. (2018). User's guide to correlation coefficients. Turkish journal of emergency medicine](https://pmc.ncbi.nlm.nih.gov/articles/PMC6107969/)
5. [1.3.3.16. Linear Correlation Plot](https://www.itl.nist.gov/div898/handbook/eda/section3/linecorr.htm)

## Further Reading

- [NIST/SEMATECH e-Handbook of Statistical Methods](https://www.itl.nist.gov/div898/handbook/index.htm)

## Related Articles

- [What Is Zero Correlation? Definition and Examples](/blog/data-analysis/zero-correlation-definition-examples)
- [Correlation Examples: Positive, Negative and Zero Relationships](/blog/data-analysis/correlation-examples-positive-negative)
- [Negative Correlation Examples: Definition and Real Data Cases](/blog/data-analysis/negative-correlation-examples-definition)
- [Spurious Correlation: Definition, Examples and How to Spot It](/blog/data-analysis/spurious-correlation-definition-examples)
- [Statistical Synonyms: A Guide to Terminology in Statistics](/blog/guides/statistical-synonyms-a-guide-to-terminology-in-statistics)
- [Correlational Research](/blog/guides/correlational-research)
- [Correlation Analysis in SPSS](/knowledge/bioinformatics/correlation-analysis-in-spss-how-to-run-interpret-and-report-pearson-and-spearman-coefficients)