Standardabweichungsrechner
Berechne Populations- und Stichproben-Standardabweichung mit Schritt-für-Schritt-Ergebnissen
Standard Deviation Calculator
Calculate standard deviation and variance
Enter comma-separated numbers
sigma = sqrt(sum((xi - mean)^2) / N)What is a Standard Deviation Calculator?
A Standard Deviation Calculator is a statistics tool that measures how spread out a set of numbers is. Standard deviation tells you, on average, how far the values in a dataset are from the mean (average). A small standard deviation means the numbers are close together and close to the mean. A large standard deviation means the numbers vary widely.
Standard deviation is one of the most important ideas in statistics because it describes variability. It's used in many real-world areas such as finance (volatility of returns), education (spread of test scores), science (measurement error and consistency), and business analytics (variation in sales or performance metrics).
This calculator helps you compute standard deviation quickly and accurately—especially for larger datasets—without manually doing multiple steps like finding the mean, subtracting values, squaring differences, and taking square roots.
This Calculator Outputs Both Modes
- Population StdDev -- use when your data includes every member of the group
- Sample StdDev -- use when your data is a subset (sample) of a larger population
- Variance -- the square of the standard deviation (population variance)
How to Use This Standard Deviation Calculator
- Enter your data values -- input numbers into the data field (numbers only)
- Separate values using commas -- for example: 5, 10, 15, 20, 25
- Click 'Calculate' -- to compute the standard deviation
- Review the results -- both Population StdDev and Sample StdDev are displayed, along with Variance
- Interpret the spread -- compare the standard deviation relative to the mean to understand how consistent or varied the data is
Tips:
- Use Population when your data includes every member of the group you're measuring
- Use Sample when your data is a subset (sample) taken from a larger population
- Standard deviation is in the same units as your data (unlike variance, which is in squared units)
Standard Deviation Formulas
Let your dataset be: x₁, x₂, x₃, …, xₙ where n is the number of values.
Mean (Average)
x̄ = (x₁ + x₂ + … + xₙ) / n
Sum all values, then divide by the count
Population Standard Deviation
Variance:
σ² = [ Σ(xᵢ − μ)² ] / n
Standard deviation:
σ = √σ²
μ = population mean; divide by n
Sample Standard Deviation
Variance:
s² = [ Σ(xᵢ − x̄)² ] / (n − 1)
Standard deviation:
s = √s²
x̄ = sample mean; divide by (n − 1)
Why divide by (n − 1) for a sample?
Using (n − 1) instead of n corrects bias when estimating population variability from a sample. This adjustment is called Bessel's correction and produces a more accurate estimate of the true population standard deviation.
Example Calculations
Example 1: Population Standard Deviation
Data: 1, 2, 3
Mean (μ): (1 + 2 + 3) / 3 = 2
Differences from mean: (1−2) = −1, (2−2) = 0, (3−2) = 1
Squared → Sum: 1 + 0 + 1 = 2
σ²: 2 / 3 = 0.6667
σ: √0.6667 ≈ 0.8165
Result: Population SD ≈ 0.8165
Example 2: Sample Standard Deviation
Data: 1, 2, 3
Mean: 2, squared differences sum: 2
s²: 2 / (3 − 1) = 2 / 2 = 1
s: √1 = 1
Result: Sample SD = 1
Example 3: Low vs High Variability Comparison
Dataset A: 9, 10, 10, 11 (values close together)
Dataset B: 2, 6, 14, 18 (values spread out)
Both sets have a mean of 10, but Dataset B has a much larger standard deviation because values are farther from the mean.
Result: Higher spread → higher standard deviation
Example 4: Real-World Example (Test Scores)
Scores: 78, 80, 82, 85, 95
The mean is around the low-to-mid 80s. The score 95 is farther from the mean and increases the spread.
Result: The standard deviation helps quantify how consistent (or inconsistent) the scores are.
Frequently Asked Questions
What does standard deviation tell me?
It tells you how much values typically vary from the mean. Lower standard deviation means the data is clustered; higher standard deviation means the data is more spread out.
What's the difference between variance and standard deviation?
Variance is the average of squared differences from the mean. Standard deviation is the square root of variance. Standard deviation is preferred because it's in the same units as the original data.
Should I use sample or population standard deviation?
Use population if you have the full group you care about. Use sample if your data is only part of a larger group and you're estimating the population variability.
Can standard deviation be zero?
Yes. If all numbers are the same (e.g., 5, 5, 5, 5), there is no variation, so the standard deviation is 0.
How do outliers affect standard deviation?
Outliers (very high or very low values) usually increase standard deviation because they are far from the mean and contribute large squared differences.
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Was ist die Standardabweichung?
Die Standardabweichung misst, wie weit die Werte eines Datensatzes von ihrem Mittelwert entfernt sind. Eine niedrige Standardabweichung bedeutet, dass die Werte eng um den Durchschnitt gruppiert sind; eine hohe zeigt an, dass sie weit gestreut sind. Sie ist eine der meistgenutzten Statistiken der Welt — in der Finanzwelt (Aktienkursvolatilität), in der Wissenschaft (Fehlermargen bei Experimenten), im Bildungswesen (Notenverteilung) und in der industriellen Qualitätskontrolle.
Es gibt zwei Varianten: die Populations-Standardabweichung (σ), wenn der Datensatz alle Mitglieder der Grundgesamtheit enthält, und die Stichproben-Standardabweichung (s), wenn du mit einer Teilmenge aus einer größeren Population arbeitest. Dieser Rechner berechnet beide Werte sowie Varianz und Mittelwert aus jeder beliebigen Zahlenreihe, die du eingibst.
So verwendest du den Standardabweichungsrechner
- Gib deine Zahlen kommagetrennt in das Eingabefeld ein oder füge sie ein (z. B.: 4, 7, 13, 2, 1).
- Wähle, ob du die Populations- oder Stichproben-Standardabweichung möchtest — oder lass den Rechner beide anzeigen.
- Klicke auf Berechnen, um die Berechnung zu starten.
- Lies die Standardabweichung, Varianz und den Mittelwert im Ergebnisbereich ab.
Formel der Standardabweichung
Mittelwert: μ = Σx / n
Populations-σ: √(Σ(x − μ)² / n)
Stichproben-s: √(Σ(x − μ)² / (n − 1))
Varianz (Popul.): σ² = Σ(x − μ)² / n
Varianz (Stichpr.): s² = Σ(x − μ)² / (n − 1)Verwende die Populations-σ, wenn deine Daten DIE gesamte Grundgesamtheit darstellen (z. B. alle Noten einer Klasse). Verwende die Stichproben-s, wenn deine Daten eine Teilmenge einer größeren Population sind (z. B. eine Umfrage mit 500 Personen, die Millionen repräsentiert). Die Stichprobenformel teilt durch (n − 1) statt durch n — die sogenannte Bessel-Korrektur — um eine unverzerrte Schätzung der tatsächlichen Streuung der Grundgesamtheit zu erhalten.
Ausgearbeitete Beispiele
Beispiel 1 — Klassischer Datensatz: {2, 4, 4, 4, 5, 5, 7, 9}
Mittelwert = (2+4+4+4+5+5+7+9) / 8 = 40 / 8 = 5. Die Summe der quadratischen Abweichungen von 5 beträgt 32. Populations-Standardabweichung σ = √(32/8) = √4 = 2,00. Dies ist ein klassischer Datensatz, der σ = 2 exakt veranschaulicht.
Beispiel 2 — Gleichmäßig verteilte Werte: {10, 20, 30, 40, 50}
Mittelwert = 150 / 5 = 30. Summe der quadratischen Abweichungen = (20²+10²+0²+10²+20²) = 1000. Populations-σ = √(1000/5) = √200 ≈ 14,14. Stichproben-s = √(1000/4) = √250 ≈ 15,81.
Beispiel 3 — Keine Streuung: {100, 100, 100}
Mittelwert = 100. Jeder Wert ist gleich dem Mittelwert, daher ist jede quadratische Abweichung 0. Standardabweichung = 0 — in diesem Datensatz gibt es keinerlei Streuung. Dies tritt immer auf, wenn alle Werte identisch sind.