Statistics Calculator

Math

Perform full descriptive statistics on numerical datasets: mean, median, mode, sample & population standard deviation, variance, quartiles, IQR, outliers, and frequency distribution.

Presets:

Descriptive Statistics Summary (N = 10)

Arithmetic Mean (x̄) 25.1

Median (Q₂) 23.5

Mode 22 (freq: 2)

Dispersion & Spread s & σ

Sample SD (s):9.291

Population SD (σ):8.8142

Sample Variance (s²):86.3222

Standard Error (SE):2.9381

Mean Absolute Dev (MAD):7.32

Five-Number Summary Quartiles

Minimum:12

Q₁ (25th Percentile):18

Median (Q₂ / 50th):23.5

Q₃ (75th Percentile):31

Maximum:42

Interquartile Range (IQR):13

Outlier Detection (1.5×IQR)

Lower Fence:-1.50

Upper Fence:50.50

Outliers:None

Skewness:0.410

Sums & Other Means

Sum (∑x):251

Sum of Squares (∑x²):7,077

Geometric Mean:23.5202

Harmonic Mean:21.9497

Range:30

📊 Frequency Distribution Table

Value	Frequency	Relative %	Cumulative
12	1	10.0%	1
15	1	10.0%	2
18	1	10.0%	3
22	2	20.0%	5
25	1	10.0%	6
29	1	10.0%	7
31	1	10.0%	8
35	1	10.0%	9
42	1	10.0%	10

Step-by-Step Statistical Derivations

1 1. Measures of Central Tendency

Mean x̄ = 25.1000, Median = 23.5000, Mode = 22

Arithmetic mean balances the sum of values; median identifies the 50th percentile rank.

2 2. Five-Number Summary & Quartiles

Min = 12, Q₁ = 18.00, Median = 23.50, Q₃ = 31.00, Max = 42

The fundamental distribution summary dividing observations into four equal quarters.

3 3. Dispersion & Spread

IQR = Q₃ - Q₁ = 13.00, s = 9.2910, σ = 8.8142, Range = 30

Interquartile range and standard deviation quantify cluster tightness and dispersion.

4 4. Outlier Analysis (Tukey 1.5×IQR Rule)

Lower Fence = -1.50, Upper Fence = 50.50 → Outliers: None detected

Observations falling beyond fences are flagged as potential outliers.

How This Calculator Works

Descriptive statistics summarize and characterize features of a dataset without drawing conclusions about a larger population. They provide quantitative descriptions across central tendency (mean, median, mode) and dispersion (variance, standard deviation, interquartile range).

📐 Descriptive Statistics Formulas

Mean: x̄ = ∑x/n | Median: Q₂ = 50th percentile | IQR: Q₃ - Q₁ | Variance: s² = ∑(x - x̄)²/(n - 1)

x̄: Sample mean

Q₁, Q₃: First quartile (25th percentile) and third quartile (75th percentile)

IQR: Interquartile range = Q₃ - Q₁, measuring the spread of the middle 50% of data

MAD: Mean absolute deviation = (1/n) ∑|x_i - x̄|

Skewness: Measure of distribution asymmetry around the mean (0 = symmetric, >0 = right skew)

Key Mathematical Rules & Properties

Mean vs Median

The mean is sensitive to extreme values and outliers. The median is a robust measure that remains unaffected by extreme skewness.

Tukey's Outlier Rule

Data points smaller than Q₁ - 1.5×IQR or greater than Q₃ + 1.5×IQR are classified as potential outliers.

✎

Worked Example: Five-Number Summary

Calculate the five-number summary and IQR for: 3, 5, 7, 8, 9, 11, 15, 16, 20, 21.

1. Ordered values: 3, 5, 7, 8, 9, 11, 15, 16, 20, 21 (n = 10).
2. Minimum = 3, Maximum = 21.
3. Median (middle 2 values: 9 and 11): (9 + 11) / 2 = 10.
4. First quartile Q₁ (median of lower 5 values: 3, 5, 7, 8, 9) = 7.
5. Third quartile Q₃ (median of upper 5 values: 11, 15, 16, 20, 21) = 16.
6. Interquartile Range IQR = 16 - 7 = 9.

Final Answer: Min=3, Q₁=7, Med=10, Q₃=16, Max=21, IQR=9

Frequently Asked Questions

Can a dataset have multiple modes or no mode?

Yes. If every number appears with equal frequency, there is no mode. If two or more values share the highest frequency, the dataset is multimodal (e.g. bimodal).

What is the difference between standard deviation and IQR?

Standard deviation measures spread based on squared differences from the mean and is sensitive to outliers. IQR measures the spread of the central 50% between Q₁ and Q₃ and is highly resistant to outliers.

Understanding your calculation

Method, example and limitations

How the calculation works

Descriptive statistics summarize the entered observations using counts, averages and measures of spread. Their meaning depends on the selected sample or population convention.

Example to check

For 2, 4 and 6, the count is 3, sum is 12, mean is 4 and range is 4.

Before using the result

A summary does not establish causation or representativeness. Check missing values, duplicate entries, outliers and whether all observations use the same units.

Calculation methodology · Report a result that needs review