Standard Deviation and Variance: Meaning, Formula and Real Data

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

Standard deviation vs variance at a glance

Worked example: how spread out, and how unequal, are citations?

We measure the spread of real citation data, 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 the mean, median, standard deviation and variance

We use papers from the same publication window, so that all of them have had time to collect citations.

THIS_YEAR = date.today().year
window = papers[papers["year"].between(THIS_YEAR - 16, THIS_YEAR - 4)]
c = window["citations"]
print(f"mean = {c.mean():.1f}, median = {c.median():.0f}")
print(f"standard deviation = {c.std():.1f}, variance = {c.var():,.0f}")

mean = 34.8, median = 7
standard deviation = 306.1, variance = 93,723

Step 3: See the shape with histograms

import matplotlib.pyplot as plt
fig, axes = plt.subplots(1, 2, figsize=(11, 4))
axes[0].hist(c.clip(upper=100), bins=range(0, 105, 5), color="lightgreen", edgecolor="green")
axes[0].set_title("Citations (values above 100 grouped at 100)")
axes[1].hist(np.log10(c + 1), bins=np.linspace(0, 4, 41), color="lightgreen", edgecolor="green")
axes[1].set_title("log10(citations + 1)")
plt.tight_layout()
plt.show()

The first histogram shows the problem: a tall bar at zero and a long thin tail. The second, on a log scale, is far closer to a bell shape. That is why skewed data are often described with the median, or analysed after taking logs.

Step 4: Compare the spread between groups

window.groupby("venue_group")["citations"].agg(["mean", "median", "std", "var"]).round(1)

Step 5: Measure how unequal the citations are

s = c.sort_values(ascending=False)
top1 = s.head(int(len(s) * 0.01)).sum() / s.sum() * 100
top10 = s.head(int(len(s) * 0.10)).sum() / s.sum() * 100
print(f"Top 1% of papers hold {top1:.0f}% of all citations")
print(f"Top 10% of papers hold {top10:.0f}% of all citations")
print(f"Papers never cited: {(c == 0).mean() * 100:.0f}%")

Top 1% of papers hold 40% of all citations
Top 10% of papers hold 71% of all citations
Papers never cited: 21%

How to calculate standard deviation

1. Calculate the mean of your values.
2. Subtract the mean from each value and square the result.
3. Add the squared values and divide by n − 1 for a sample (or n for a whole population). This is the variance.
4. Take the square root to get the standard deviation.
5. In Excel, use STDEV.S for a sample and VAR.S for its variance.

Common mistakes reviewers point out

• Reporting mean ± SD for skewed data, where a result such as “35 ± 306 citations” wrongly suggests that negative values are possible.
• Confusing standard deviation with standard error; standard error measures uncertainty in the mean, not spread of the data.
• Using the population formula (divide by n) for a sample.
• Not stating which measure of spread is shown in tables and error bars.

Frequently asked questions

Use this data in your own work

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