# What are the statistics of my data?

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

- Page: https://www.acalculator.org/statistics/statistics-calculator
- JSON spec: https://www.acalculator.org/statistics/statistics-calculator.json
- Version: 40857b844111

## Default answer

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).

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| data | Your numbers | The numbers, separated by commas, spaces, semicolons, or new lines. |
| type | Your numbers are a | A sample from a larger group (the variance divides by n − 1) or the whole population (divides by n). Pick sample when unsure. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| mean | Mean | The sum divided by the count: the average. |
| median | Median | The middle number when sorted, or the mean of the two middle numbers. |
| mode | Mode | The value or values that occur most often, or No mode when no value repeats. |
| sd | Standard deviation | The square root of the variance: how far the numbers typically are from the mean. |
| variance | Variance | The sum of squared differences from the mean, divided by n − 1 (sample) or n (population). |
| count | Count (n) | How many numbers are in the list. |
| sum | Sum | All the numbers added up. |
| min | Smallest | The smallest number. |
| max | Largest | The largest number. |
| range | Range | The largest number minus the smallest. |
| q1 | First quartile (Q1) | The median of the lower half of the sorted numbers. Shown for 2 or more numbers. |
| q3 | Third quartile (Q3) | The median of the upper half of the sorted numbers. Shown for 2 or more numbers. |
| iqr | Interquartile range | Q3 minus Q1: the spread of the middle half. Shown for 2 or more numbers. |
| sem | Standard error of the mean | The sample standard deviation divided by √n. Shown for a sample only. |
| geometricMean | Geometric mean | The nth root of the product of the numbers. Shown only when every number is above 0. |

## 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).

## Assumptions

- 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.

## Worked examples

1. data = 10 or 12, type = sample gives mean = 18, median = 18.5, mode = 23, variance = 27.428571, sd = 5.237229, sum = 144, range = 13, q1 = 14, q3 = 23, iqr = 9, sem = 1.85164, geometricMean = 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. data = 10 or 12, type = population gives variance = 24, sd = 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. data = 1 or 11.5, type = sample gives median = 7, q1 = 2, q3 = 9, iqr = 7, mode = 1, 2, 10, count = 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. data = 0.1 or 0.2, type = 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. data = -5 or 4, type = sample gives mean = -0.5, median = -0.5, q1 = -5, 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.

## FAQ

### 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.

## Sources

- 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
- 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
- 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