Pearson Correlation Coefficient (Pearson’s r): Meaning, Examples and Real Data

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Below, we explain Pearson’s r in plain language and calculate it live on real data from ScholarsColab, where it behaves in a surprising and very instructive way.

Reading Pearson’s r at a glance

Worked example: do researchers who publish more have a higher h-index?

We work through a full correlation analysis, notebook-style, on real researcher data. Every result is calculated live, and you can run the same code yourself.

The data: one row per ScholarsColab mentor with at least five dated papers, giving their discipline, career length, number of papers, h-index and total citations. The file is anonymised: no names or identifiers, rows are shuffled and citation totals are rounded.

Step 1: Load the data

import numpy as np
import pandas as pd
from datetime import date

# The ScholarsColab Research Methods Index (mentors level): anonymised, updated daily
DATA = "https://scholarscolab.com/wp-json/scholarscolab/v1/research-methods-data?level=mentors"
mentors = pd.read_csv(DATA, comment="#")   # lines starting with # hold the citation

def p_text(p):
    return "< 0.001" if p < 0.001 else f"= {p:.3f}"

print(mentors.shape)
mentors.head()

(891, 6)

The h-index is a common measure of research impact: an h-index of 10 means a researcher has 10 papers with at least 10 citations each.

Step 2: Plot the two variables first

Always start with a scatter plot. It shows whether the relationship looks like a straight line, and whether a few extreme points might dominate.

import matplotlib.pyplot as plt
from scipy import stats

fit = stats.linregress(mentors["papers"], mentors["h_index"])

fig, ax = plt.subplots(figsize=(8, 4.5))
ax.scatter(mentors["papers"], mentors["h_index"], s=12, alpha=0.4)
x = np.linspace(0, 300, 50)
ax.plot(x, fit.intercept + fit.slope * x, color="green", linewidth=3)
ax.set_xlim(0, 300)
ax.set_xlabel("Number of papers")
ax.set_ylabel("h-index")
plt.show()

Step 3: Calculate Pearson’s r and Spearman’s rho

r, p = stats.pearsonr(mentors["papers"], mentors["h_index"])
rho, p_s = stats.spearmanr(mentors["papers"], mentors["h_index"])
print(f"Pearson r    = {r:.2f}, p {p_text(p)}")
print(f"Spearman rho = {rho:.2f}, p {p_text(p_s)}")

Pearson r    = 0.62, p < 0.001
Spearman rho = 0.82, p < 0.001

Step 4: See how extreme values and curves affect Pearson

r_log, _ = stats.pearsonr(np.log(mentors["papers"]), np.log(mentors["h_index"] + 1))
top = mentors["papers"] > mentors["papers"].quantile(0.99)
r_trim, _ = stats.pearsonr(mentors.loc[~top, "papers"], mentors.loc[~top, "h_index"])
print(f"Pearson r on log scales:                {r_log:.2f}")
print(f"Pearson r without the top 1% by papers: {r_trim:.2f}")

Pearson r on log scales:                0.79
Pearson r without the top 1% by papers: 0.69

When to use Pearson or Spearman

How to calculate and report Pearson’s r

1. Plot the two variables in a scatter plot and look for a straight-line pattern and extreme values.
2. Check the assumptions: both variables numeric, roughly normal, and a linear relationship.
3. Calculate r in SPSS, R, Python or Excel (CORREL function).
4. Test significance to get a p-value, and report a confidence interval if you can.
5. Report it fully, for example: “There was a moderate positive correlation between sleep and test scores, r(98) = 0.45, p < 0.001.”

Common mistakes reviewers point out

• Using Pearson on heavily skewed data, as our raw citation example shows.
• Not plotting the data, so curved relationships or outliers go unnoticed.
• Reading correlation as causation. See our guide to correlation vs causation.
• Reporting r without the sample size or p-value.

Frequently asked questions

Use this data in your own work

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