Confidence Intervals Explained: Meaning, Calculation and Real Data

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Below, we explain confidence intervals in plain language and calculate them live on real data from ScholarsColab.

Confidence intervals at a glance

Worked example: which research topics are rising, and how sure can we be?

We estimate the share of papers on five topics over time, each with a 95% confidence interval, notebook-style. Every result is calculated live, and you can run the same code yourself.

The data: 41,792 papers, mostly published between 1998 and 2026, by 967 ScholarsColab mentors (data updated 15 October 2026). The file you will load is anonymised: no paper titles, journal names or researcher identities.

Step 1: Load the data

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

# The ScholarsColab Research Methods Index (papers level): anonymised, updated daily
DATA = "https://scholarscolab.com/wp-json/scholarscolab/v1/research-methods-data?level=papers"
papers = 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(papers.shape)
papers.head()

(41792, 15)

Step 2: Calculate each topic’s share with a confidence interval

For each two-year period and each topic, we calculate the share of papers on that topic and a 95% Wilson interval, which works well for proportions, including small ones. Here are the results for deep learning:

from statsmodels.stats.proportion import proportion_confint

THIS_YEAR = date.today().year
start = THIS_YEAR - 11
recent = papers[papers["year"].between(start, THIS_YEAR)].copy()
recent["period"] = ((recent["year"] - start) // 2 * 2 + start).astype(int)
topics = ["deep_learning", "iot", "blockchain", "covid", "sustainability"]

rows = []
for period, grp in recent.groupby("period"):
    for t in topics:
        k, n = grp[t].sum(), len(grp)
        low, high = proportion_confint(k, n, method="wilson")
        rows.append({"period": f"{period}–{period + 1}", "topic": t,
                     "share %": 100 * k / n, "low %": 100 * low, "high %": 100 * high})
ci = pd.DataFrame(rows)
ci[ci["topic"] == "deep_learning"].round(2)

Each row gives an estimate (share %) and a range of plausible values (low % to high %).

Step 3: Plot the trends with their confidence bands

import matplotlib.pyplot as plt
fig, ax = plt.subplots(figsize=(9, 4.5))
for t in topics:
    d = ci[ci["topic"] == t]
    ax.plot(d["period"], d["share %"], marker="o", label=t)
    ax.fill_between(d["period"], d["low %"], d["high %"], alpha=0.2)
ax.set_ylabel("% of papers")
ax.legend()
plt.show()

Step 4: Read the most recent period

latest = ci[ci["period"] == ci["period"].iloc[-1]]
for _, r in latest.iterrows():
    print(f"{r['topic']:<15} {r['share %']:5.2f}%  (95% CI {r['low %']:.2f}% to {r['high %']:.2f}%)")

deep_learning    9.62%  (95% CI 8.89% to 10.40%)
iot              5.31%  (95% CI 4.76% to 5.91%)
blockchain       2.12%  (95% CI 1.78% to 2.52%)
covid            0.46%  (95% CI 0.32% to 0.67%)
sustainability   4.64%  (95% CI 4.13% to 5.21%)

Confidence intervals vs p-values

A p-value only says whether an effect is likely to be zero. A confidence interval shows the size of the effect and how precisely it is known. That is why many journals now ask authors to report confidence intervals alongside, or instead of, p-values.

How to calculate and report a confidence interval

1. Calculate your estimate (a mean, a proportion, a difference or a regression coefficient).
2. Calculate its standard error, which depends on the variation and the sample size.
3. Multiply the standard error by 1.96 for a 95% interval (for large samples), and add and subtract it from the estimate.
4. For proportions, prefer the Wilson interval (used above), which works better near 0% or 100%.
5. Report it as, for example, “12.5% (95% CI 10.1% to 15.2%)”.

Common mistakes reviewers point out

• Saying there is a 95% chance the true value is in this interval; strictly, the 95% describes the method, not one interval.
• Reporting estimates without any interval.
• Using simple formulas for proportions near 0% or 100%, which can give impossible negative values.
• Ignoring wide intervals when drawing strong conclusions from small samples.

Frequently asked questions

Use this data in your own work

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