Statistics Calculator
Full descriptive statistics for a dataset: centre, spread, quartiles, skew and outliers.
Inputs
Separate values with commas or spaces.
Use 'Sample' unless your data IS the full population (e.g. census).
Mean
6.8000
Count (n)
10
Sum
68.0000
Median
7.0000
Mode
7
Range
14.0000
Minimum
1.0000
Maximum
15.0000
Variance
21.0667
Std Deviation
4.5898
Std Error
1.4514
Q1 (25th pct)
3.2500
Q2 / Median
7.0000
Q3 (75th pct)
8.5000
IQR
5.2500
Lower Fence
-4.6250
Upper Fence
16.3750
Outliers
None
Skewness
0.6002
Positive = right tail; negative = left tail.
Excess Kurtosis
-0.4273
0 for normal; positive = heavy tails.
Step by step
Dataset
[4, 7, 13, 2, 1, 7, 9, 15, …]
= n = 10 values
Mean
(4 + 7 + 13 + 2 + 1 + …) ÷ 10
= 6.8000
Variance
Σ(xᵢ − 6.800)² ÷ 9
= 21.0667
Std deviation
√21.0667
= 4.5898
Quartiles
Q1=3.250, Q2=7.000, Q3=8.500
= IQR = 5.2500
Tukey fences
[-4.625, 16.375]
= 0 outlier(s)
How it works
Descriptive statistics summarise a dataset without assuming any distribution. The mean is the arithmetic average; the median is the middle value; the mode is the most frequent value (or no mode when all values are unique). Spread is captured by the range, variance and standard deviation. Quartiles divide the sorted data into four equal parts, and the IQR (Q3−Q1) is the middle 50 %. Tukey fences sit at Q1−1.5×IQR and Q3+1.5×IQR; any point outside them is flagged as an outlier. Skewness measures distributional asymmetry; excess kurtosis measures tail heaviness relative to a normal distribution.
Formulas
Mean
Mean = (sum of all values) / n
- n
- Number of values
- x_i
- Each value
Sample variance
Sample variance = Σ(xᵢ − x̄)² / (n−1)
- x̄
- Mean
- n
- Count
Tukey fences
Lower fence = Q1 − 1.5·IQR; Upper fence = Q3 + 1.5·IQR
- IQR
- Q3 − Q1
Frequently Asked Questions
When should I choose population vs. sample variance?
Choose sample (÷ n−1, Bessel's correction) when your data is a subset drawn from a larger population — the typical research scenario. Choose population (÷ n) only when your dataset contains every element of the population you care about, such as the grades of all 30 students in a specific class.
What does it mean if there is no mode?
When every value appears exactly once, no value is more frequent than any other, so the dataset has no mode. Some textbook definitions force a mode in that case, but it conveys no useful information. This calculator honestly reports 'No mode' rather than inventing one.
How are outliers detected?
This calculator uses Tukey fences: any data point below Q1−1.5×IQR or above Q3+1.5×IQR is classified as an outlier. This is the standard method used by box-and-whisker plots. For datasets with heavier tails you might apply a stricter 3×IQR rule, but 1.5 is the default.
What does excess kurtosis tell me?
Excess kurtosis compares the tail weight of your distribution to a normal distribution (which has excess kurtosis = 0). A positive value means heavier tails and more extreme values are likely; a negative value means lighter tails. It requires at least 4 data points to compute.