Calculate mean, median, mode, quartiles, and sample or population variance and standard deviation for a list of values, with worked steps.
Direct solving is free and needs no account. Have a word problem instead? Submit it as text.
Worked examples
These are the solver's own reference problems — the answers come from published sources or independent hand computation, never from the solver itself. The solution below is the live solver output for each.
Example 1
classic set [2,4,4,4,5,5,7,9]: population σ = 2 exactly
Given
data
1.2
2.4
3.4
4.4
5.5
6.5
7.7
8.9
Assumptions
Quartiles use linear interpolation between order statistics (textbook methods differ slightly at small n).
Solution steps
Center of the data
The mean is the arithmetic average; the median is the middle of the sorted values.
x̄ = Σxᵢ / n
n = 8; x̄ = 5; median = 4.5
= 5
Spread of the data
Sample statistics divide by n − 1 (estimating from a sample); population statistics divide by n (the data IS the whole population). Exams usually want the SAMPLE version unless told otherwise.
s² = Σ(xᵢ − x̄)²/(n − 1); σ² = Σ(xᵢ − x̄)²/n
Σ(xᵢ − x̄)² = 32; sample s = 2.1381; population σ = 2
= 2.1381
Quartiles
Q1 and Q3 bracket the middle half of the data; their difference is the interquartile range.
IQR = Q3 − Q1
Q1 = 4; Q3 = 5.5; IQR = 1.5; range = 7
Mode
The most frequent value.
Mode = 4 (appears 3 times)
Results
Number of observations n
8
Mean x̄
5
Median
4.5
Sample standard deviation s
2.1381
Population standard deviation σ
2
Sample variance s²
4.5714
Population variance σ²
4
Range
7
First quartile Q1
4
Third quartile Q3
5.5
Interquartile range
1.5
Coefficient of variation s/x̄
0.4276
Mode (most frequent value)
4
Where this answer was checked
source
Standard worked example for population standard deviation — n=8, μ=5, Σ(x−μ)² = 32
Quartiles use linear interpolation between order statistics (textbook methods differ slightly at small n).
Solution steps
Center of the data
The mean is the arithmetic average; the median is the middle of the sorted values.
x̄ = Σxᵢ / n
n = 5; x̄ = 3; median = 3
= 3
Spread of the data
Sample statistics divide by n − 1 (estimating from a sample); population statistics divide by n (the data IS the whole population). Exams usually want the SAMPLE version unless told otherwise.
s² = Σ(xᵢ − x̄)²/(n − 1); σ² = Σ(xᵢ − x̄)²/n
Σ(xᵢ − x̄)² = 10; sample s = 1.5811; population σ = 1.4142
= 1.5811
Quartiles
Q1 and Q3 bracket the middle half of the data; their difference is the interquartile range.
Σ(x−3)² = 4+1+0+1+4 = 10; σ² = 2; s² = 2.5. Q1 at index 1 → 2; Q3 at index 3 → 4; IQR 2; range 4.
Example 3
two values: sample vs population spread differ by √2
Given
data
1.10
2.20
Assumptions
Quartiles use linear interpolation between order statistics (textbook methods differ slightly at small n).
Solution steps
Center of the data
The mean is the arithmetic average; the median is the middle of the sorted values.
x̄ = Σxᵢ / n
n = 2; x̄ = 15; median = 15
= 15
Spread of the data
Sample statistics divide by n − 1 (estimating from a sample); population statistics divide by n (the data IS the whole population). Exams usually want the SAMPLE version unless told otherwise.
s² = Σ(xᵢ − x̄)²/(n − 1); σ² = Σ(xᵢ − x̄)²/n
Σ(xᵢ − x̄)² = 50; sample s = 7.0711; population σ = 5
= 7.0711
Quartiles
Q1 and Q3 bracket the middle half of the data; their difference is the interquartile range.
IQR = Q3 − Q1
Q1 = 12.5; Q3 = 17.5; IQR = 5; range = 10
Results
Number of observations n
2
Mean x̄
15
Median
15
Sample standard deviation s
7.0711
Population standard deviation σ
5
Sample variance s²
50
Population variance σ²
25
Range
10
First quartile Q1
12.5
Third quartile Q3
17.5
Interquartile range
5
Coefficient of variation s/x̄
0.4714
Where this answer was checked
source
n=2 boundary case — [10, 20]
verified by
hand-recomputed
derivation
mean 15; Σd² = 25+25 = 50; s = √50 = 7.0711; σ = √25 = 5.