Statistikrechner
Mittelwert, Median, Modus, Standardabweichung und mehr — alles auf einmal
Statistics Calculator
Calculate mean, median, mode, and standard deviation
Enter comma-separated numbers
Mean = sum / n, StdDev = sqrt(sum((xi - mean)^2) / n)What is a Statistics Calculator?
A Statistics Calculator is a math tool that analyzes a set of numbers and computes common statistical measures such as mean (average), median, mode, range, and often variance and standard deviation. These measures help summarize data so you can understand patterns, compare groups, and make decisions based on numbers.
Statistics is used in school assignments, business reporting, scientific research, finance, sports analytics, and everyday life (like tracking budgets or comparing test scores). Instead of manually calculating multiple values—especially for larger datasets—a statistics calculator gives results instantly and reduces mistakes.
This calculator is helpful whenever you have a list of values and want quick insights about the center of the data (typical value), the spread (how scattered values are), and whether there are values that stand out from the rest.
How to Use This Statistics Calculator
- Enter your data values -- Input numbers into the data field (numbers only)
- Separate values correctly -- Use commas, spaces, or new lines—depending on the calculator’s input format
- Click 'Calculate' -- to analyze the dataset
- Review the results -- such as mean, median, mode, range, and standard deviation (if shown)
- Adjust your list -- add or remove values and recalculate to compare different datasets
Tips:
- Make sure you don’t include extra symbols (like $ or %) unless the calculator supports them
- If your dataset contains decimals, enter them exactly (example: 12.5)
- If the calculator offers a choice between population vs sample statistics, pick the one that matches your situation (explained below)
Statistics Formulas
Let your dataset be: x₁, x₂, x₃, …, xₙ where n is the number of values.
Mean (Average)
Mean = (x₁ + x₂ + … + xₙ) / n
Sum all values, then divide by the count
Median
The middle value after sorting the numbers:
- If n is odd: median is the middle value
- If n is even: median is the average of the two middle values
Mode
The most frequently occurring value(s):
- Unimodal (one mode), bimodal (two), or multimodal
- No mode if all values occur equally often
Range
Range = max − min
The difference between the largest and smallest values
Variance and Standard Deviation
Variance measures spread by looking at how far values are from the mean.
Population formulas
- σ² = [ Σ(xᵢ − μ)² ] / n
- σ = √σ²
μ = population mean
Sample formulas
- s² = [ Σ(xᵢ − x̄)² ] / (n − 1)
- s = √s²
x̄ = sample mean
Example Calculations
Example 1: Mean, Median, Mode
Data: 2, 4, 4, 7, 9
Mean: (2 + 4 + 4 + 7 + 9) / 5 = 26 / 5 = 5.2
Median: sorted list is 2, 4, 4, 7, 9 → middle value is 4
Mode: 4 occurs most often → mode = 4
Results: Mean = 5.2, Median = 4, Mode = 4
Example 2: Range
Data: 12, 15, 19, 22, 30
Max: 30, Min: 12
Range: 30 − 12 = 18
Result: Range = 18
Example 3: Population Standard Deviation
Data: 1, 2, 3
Mean (μ): (1 + 2 + 3) / 3 = 2
Differences: (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 4: Sample Standard Deviation
Data: 1, 2, 3
Mean (x̄): 2
Squared differences sum: 2 (same as above)
s²: 2 / (3 − 1) = 2 / 2 = 1
s: √1 = 1
Result: Sample SD = 1
Frequently Asked Questions
What’s the difference between mean, median, and mode?
The mean is the average (sum divided by count). The median is the middle value when sorted. The mode is the most frequent value. They can differ, especially if the data has outliers.
What are outliers, and how do they affect statistics?
Outliers are values far from the rest of the data. They can strongly affect the mean and standard deviation, but the median is usually more resistant to outliers.
What’s the difference between population and sample standard deviation?
Use population formulas when your dataset includes every member of the group you’re studying. Use sample formulas when your dataset is a subset (sample) of a larger population. Sample formulas divide by (n − 1) to reduce bias.
Can a dataset have more than one mode?
Yes. If two values tie for most frequent, it’s bimodal. If more than two values tie, it’s multimodal. If all values occur equally often, it may have no mode.
Why is standard deviation useful?
Standard deviation shows how spread out the data is. A low standard deviation means values are close to the mean; a high standard deviation means values vary widely.
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Was sind deskriptive Statistiken?
Statistik ist die Wissenschaft des Sammelns, Analysierens und Interpretierens numerischer Daten. Deskriptive Statistiken fassen die wichtigsten Merkmale eines Datensatzes zusammen — zentrale Tendenz (Mittelwert, Median, Modus) und Streuung (Spannweite, Varianz, Standardabweichung). Anstatt Schlussfolgerungen über eine groessere Grundgesamtheit zu ziehen, beschreiben sie einfach, was in deinen Daten steckt.
Deskriptive Statistiken werden ueberall eingesetzt — von Schulnoten und Sportanalysen bis hin zur medizinischen Forschung und Unternehmens-KPIs. Dieser Rechner nimmt eine beliebige Zahlenreihe und liefert sofort alle wichtigen Kennzahlen: zentrale Tendenz, Streuung, Quartile und mehr. Keine Formeln auswendig lernen, keine Tabellenkalkulation erforderlich.
So verwendest du den Statistikrechner
- Gib deine Zahlen durch Kommas getrennt ein (z. B. 2, 4, 6, 8, 10).
- Klicke auf Berechnen, um die Analyse zu starten.
- Sieh dir alle Statistiken im Ergebnisbereich an: Mittelwert, Median, Modus, Spannweite, Varianz, Standardabweichung, Q1, Q3 und IQR.
- Verwende die einzelnen Kennzahlen fuer deinen Bericht, deine Hausaufgabe oder Datenanalyse.
Verwendete Formeln
Mittelwert (μ): Σx / n
Median: mittlerer Wert nach Sortierung (oder Durchschnitt der zwei mittleren Werte)
Modus: haeufigster Wert
Spannweite: Max − Min
Varianz: Σ(x − μ)² / n
Standardabweichung: √Varianz
Q1, Q3: 25. und 75. Perzentil
IQR: Q3 − Q1Die Populationsstandardabweichung teilt durch n; die Stichprobenstandardabweichung teilt durch (n − 1). Verwende die Populationsvariante, wenn dein Datensatz die gesamte Gruppe darstellt; verwende die Stichprobenvariante, wenn es eine Teilmenge einer groesseren Grundgesamtheit ist.
Durchgerechnete Beispiele
Beispiel 1: Datensatz {2, 4, 4, 4, 5, 5, 7, 9}
Mittelwert = (2+4+4+4+5+5+7+9) / 8 = 5,00. Median = Durchschnitt des 4. und 5. Werts = (4+5)/2 = 4,50. Modus = 4 (erscheint 3-mal). Spannweite = 9 − 2 = 7. Populationsstandardabweichung ≈ 2,00.
Beispiel 2: Testergebnisse {70, 80, 90, 100}
Mittelwert = (70+80+90+100) / 4 = 85,00. Median = (80+90)/2 = 85,00. Modus = keiner (alle Werte erscheinen einmal). Spannweite = 100 − 70 = 30. Standardabweichung = 11,18.
Beispiel 3: Symmetrischer Datensatz {1, 2, 3, 4, 5}
Mittelwert = 3,00. Median = 3,00. Modus = keiner. Spannweite = 4. Wenn ein Datensatz perfekt symmetrisch ist, sind Mittelwert und Median gleich — nuetzlich zur Erkennung von Schiefe in realen Daten.