CivilSolve

Statistics

Descriptive Statistics

Mean, median, mode, sample AND population standard deviation/variance (both reported — know which your course wants), coefficient of variation, range, quartiles and IQR for a list of values. Use for: summarizing lab/test data, 'find the standard deviation' problems. Not for: probability distributions or hypothesis tests.

Direct solving is free and needs no account. Have a word problem instead? Submit it as text.

Inputs

Load a sample problem:
dataThe data values, in any order, consistent units assumed (e.g. test scores, concrete strengths, daily flows)

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. 1.2
  2. 2.4
  3. 3.4
  4. 4.4
  5. 5.5
  6. 6.5
  7. 7.7
  8. 8.9

Assumptions

  • Quartiles use linear interpolation between order statistics (textbook methods differ slightly at small n).

Solution steps

  1. 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

  2. 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

  3. 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

  4. 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
verified by
hand-recomputed
derivation
mean = 40/8 = 5. Deviations² = 9+1+1+1+0+0+4+16 = 32. σ² = 32/8 = 4 → σ = 2. s² = 32/7 = 4.5714 → s = 2.1381. median = (4+5)/2 = 4.5; mode = 4 (three times).

Example 2

1..5: exact simple values

Given

data
  1. 1.1
  2. 2.2
  3. 3.3
  4. 4.4
  5. 5.5

Assumptions

  • Quartiles use linear interpolation between order statistics (textbook methods differ slightly at small n).

Solution steps

  1. 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

  2. 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

  3. Quartiles

    Q1 and Q3 bracket the middle half of the data; their difference is the interquartile range.

    IQR = Q3 − Q1

    Q1 = 2; Q3 = 4; IQR = 2; range = 4

Results

Number of observations n

5

Mean x̄

3

Median

3

Sample standard deviation s

1.5811

Population standard deviation σ

1.4142

Sample variance s²

2.5

Population variance σ²

2

Range

4

First quartile Q1

2

Third quartile Q3

4

Interquartile range

2

Coefficient of variation s/x̄

0.527

Where this answer was checked
source
Hand-derivable uniform sequence — mean/median 3, popVar 2, sampleVar 2.5
verified by
hand-recomputed
derivation
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. 1.10
  2. 2.20

Assumptions

  • Quartiles use linear interpolation between order statistics (textbook methods differ slightly at small n).

Solution steps

  1. 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

  2. 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

  3. 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.