acalculator

What are the statistics of my data?

Paste or type your numbers to get their descriptive statistics at once, with a box plot of how they spread.

Your numbers

Read as: 10; 12; 23; 23; 16; 23; 21; 16
Your numbers are a
Mean
18

For 8 numbers: mean 18, median 18.5, standard deviation 5.237229 (Sample).

Median
18.5
Mode
23
Standard deviation
5.237229
Variance
27.428571
Count (n)
8
Sum
144
Smallest
10
Largest
23
Range
13
First quartile (Q1)
14
Third quartile (Q3)
23
Interquartile range
9
Standard error of the mean
1.85164
Geometric mean
17.252538

Mean: 18. For 8 numbers: mean 18, median 18.5, standard deviation 5.237229 (Sample).

How are your numbers spread?

How to calculate

Computes descriptive statistics of a list of numbers: mean, median, mode, standard deviation, variance, range, quartiles, sum, count, and geometric mean.

Example with the default inputs (Your numbers [10, 12, 23, 23, 16, 23, 21, 16], Your numbers are a Sample): For 8 numbers: mean 18, median 18.5, standard deviation 5.237229 (Sample).

Method: mean = Σx ÷ n; variance = Σ(x − mean)² ÷ (n − 1) for a sample or ÷ n for a population; SD = √variance; Q1 and Q3 = medians of the lower and upper halves; geometric mean = (x₁ × … × xₙ)^(1/n).

  • A sample divides the sum of squares by n − 1 (Bessel’s correction) and needs at least 2 numbers. A population divides by n.
  • Quartiles are the medians of the lower and upper halves of the sorted list, leaving out the median when n is odd (the TI-84 and Moore and McCabe method).
  • All results except the standard deviation, standard error, and geometric mean are exact on the decimals you type, then rounded once.

Machine-readable copies: Markdown, JSON.

Worked examples

Each example is checked against the calculator on every build.

  1. Your numbers 10, 12, 23, 23, 16, 23, 21, 16, Your numbers are a Sample gives Mean 18, Median 18.5, Mode 23, Variance 27.428571, Standard deviation 5.237229, Sum 144, Range 13, First quartile (Q1) 14, Third quartile (Q3) 23, Interquartile range 9, Standard error of the mean 1.85164, Geometric mean 17.252538.Source: NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.5.6 Measures of Scale. https://www.itl.nist.gov/div898/handbook/eda/section3/eda356.htm
  2. Your numbers 10, 12, 23, 23, 16, 23, 21, 16, Your numbers are a Population gives Variance 24, Standard deviation 4.898979, Mean 18.Source: NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.5.6 Measures of Scale. https://www.itl.nist.gov/div898/handbook/eda/section3/eda356.htm
  3. Your numbers 1, 11.5, 6, 7.2, 4, 8, 9, 10, 6.8, 8.3, 2, 2, 10, 1, Your numbers are a Sample gives Median 7, First quartile (Q1) 2, Third quartile (Q3) 9, Interquartile range 7, Mode 1, 2, 10, Count (n) 14.Source: OpenStax, Introductory Statistics 2e, §2.3 Measures of the Location of the Data. https://openstax.org/books/introductory-statistics-2e/pages/2-3-measures-of-the-location-of-the-data (14 values: median 7, Q1 = 2, Q3 = 9, IQR = 7)
  4. Your numbers 0.1, 0.2, 0.3, Your numbers are a Population gives Sum 0.6, Mean 0.2, Variance 0.006667, Median 0.2, Mode No mode.Source: NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.5.1 Measures of Location. https://www.itl.nist.gov/div898/handbook/eda/section3/eda351.htm
  5. Your numbers -5, 4, Your numbers are a Sample gives Mean -0.5, Median -0.5, First quartile (Q1) -5, Third quartile (Q3) 4, Variance 40.5.Source: NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.5.1 Measures of Location. https://www.itl.nist.gov/div898/handbook/eda/section3/eda351.htm

How it works

For a list of n numbers x₁, …, xₙ:

  • Count: n. Sum: Σx.
  • Mean: Σx ÷ n.
  • Median: sort the numbers; the middle one when n is odd, or the mean of the two middle ones when n is even.
  • Mode: every value that occurs the most times, smallest first, separated by a comma and a space. When no value occurs more than once, the result is "No mode". Each mode is written like the other numbers on the page (up to 6 decimal places, halves rounded up) but without thousands separators.
  • Smallest, largest, range: the minimum, the maximum, and maximum − minimum.
  • Variance: Σ(x − mean)² ÷ (n − 1) for a sample, or ÷ n for a population.
  • Standard deviation: √variance.
  • Quartiles: sort the numbers. The lower half is the first ⌊n ÷ 2⌋ numbers and the upper half the last ⌊n ÷ 2⌋ numbers, so the median is left out when n is odd. Q1 is the median of the lower half and Q3 the median of the upper half. Interquartile range = Q3 − Q1. Shown for 2 or more numbers.
  • Standard error of the mean: sample standard deviation ÷ √n, shown for a sample only.
  • Geometric mean: (x₁ × x₂ × … × xₙ)^(1/n), worked out as e^(Σ ln x ÷ n). Shown only when every number is above 0.

Rules

  • The list holds 1 to 1,000 numbers. A sample needs at least 2 numbers; with 1 number and "sample" there is no answer (pick population).
  • Each number must be from −10¹⁰⁰ to 10¹⁰⁰; otherwise there is no answer.
  • The count, sum, mean, median, smallest, largest, range, quartiles, interquartile range, and variance are exact on the decimals as typed (0.1 is exactly one tenth), then rounded once to the nearest 64-bit float. So 0.1, 0.2, 0.3 sum to exactly 0.6.
  • The standard deviation is the square root of that rounded variance; the standard error and the geometric mean are ordinary floating-point results.
  • Results show up to 6 decimal places (6 significant digits below 0.0001), with halves rounded up.

Worked examples by hand

The default list, as a sample: 10, 12, 23, 23, 16, 23, 21, 16.

  • Count 8, sum 144, mean 144 ÷ 8 = 18.
  • Sorted: 10, 12, 16, 16, 21, 23, 23, 23. Median = (16 + 21) ÷ 2 = 18.5. Mode: 23 (three times).
  • Differences from 18: −8, −6, −2, −2, 3, 5, 5, 5; squares add up to 64 + 36 + 4 + 4 + 9 + 25 + 25 + 25 = 192. Sample variance = 192 ÷ 7 = 27.428571; standard deviation = 5.237229. Standard error = 5.237229 ÷ √8 = 1.85164.
  • Lower half 10, 12, 16, 16: Q1 = (12 + 16) ÷ 2 = 14. Upper half 21, 23, 23, 23: Q3 = 23. IQR = 9. Range = 23 − 10 = 13.
  • Geometric mean = (10 × 12 × 23 × 23 × 16 × 23 × 21 × 16)^(1/8) = 17.252538.

The same list as a population: variance = 192 ÷ 8 = 24, standard deviation = √24 = 4.898979.

OpenStax's 14 values 1, 11.5, 6, 7.2, 4, 8, 9, 10, 6.8, 8.3, 2, 2, 10, 1. Sorted: 1, 1, 2, 2, 4, 6, 6.8, 7.2, 8, 8.3, 9, 10, 10, 11.5. Median = (6.8 + 7.2) ÷ 2 = 7. Lower half 1, 1, 2, 2, 4, 6, 6.8: Q1 = 2. Upper half 7.2, 8, 8.3, 9, 10, 10, 11.5: Q3 = 9. IQR = 7. Modes: 1, 2, 10 (each twice).

0.1, 0.2, 0.3 as a population: sum = 0.6, mean = 0.2, median = 0.2, squared differences 0.01 + 0 + 0.01 = 0.02, variance = 0.02 ÷ 3 = 0.006667. No value repeats: No mode.

−5 and 4 as a sample: mean = median = −0.5. Q1 = −5, Q3 = 4. Variance = (4.5² + 4.5²) ÷ 1 = 40.5. No geometric mean, because −5 is not above 0.

Other questions people ask

What does this statistics calculator work out?

From one list of numbers it gives the count, sum, mean, median, mode, smallest and largest value, range, first and third quartiles, interquartile range, variance, standard deviation, standard error of the mean (for a sample), and the geometric mean (when every number is above 0).

Should I choose sample or population?

Choose population when your list holds every member of the group you care about, such as the scores of all 25 students in one class. Choose sample when the list is part of a larger group and you want to estimate that group, such as 25 students surveyed from a whole school. The sample variance divides by n − 1 instead of n, which makes it slightly larger.

What is the difference between the mean and the median?

The mean adds up the numbers and divides by how many there are. The median is the middle number once they are sorted. One very large or very small value moves the mean but barely moves the median, so the median is the better middle for skewed data such as incomes.

What if there is more than one mode, or none?

Every value that occurs most often is a mode, so a list can have several, shown smallest first. If no value repeats, the list has no mode.

How are the quartiles found?

The calculator sorts the numbers and splits them into a lower and an upper half, leaving out the median when the count is odd. Q1 is the median of the lower half and Q3 the median of the upper half. This is the method TI-84 calculators and many textbooks use; other software may interpolate and give slightly different quartiles.

How do I enter my numbers?

Separate them with commas, spaces, semicolons, or new lines. You can paste a column from a spreadsheet. Do not use commas as thousands separators, because a comma starts a new number: type 1200, not 1,200.

When is the geometric mean useful?

Use it for rates of growth and ratios, such as average yearly returns: it is the constant factor that gives the same product. It is only defined here when every number is above 0.