About Statistics Calculator
This statistics calculator is free, needs no sign-up and runs in your browser, so the numbers you paste are never uploaded. Type or paste a list separated by commas, spaces or new lines and it returns the count, sum, mean, median, mode, minimum, maximum, range, variance, standard deviation, standard error, quartiles, interquartile range, skewness and excess kurtosis in one go.
Under the results you get the working: the mean written out as a sum, the sum of squared deviations, the variance and the standard deviation, so you can compare each line with a textbook or your own homework. A switch between sample (divide by n - 1) and population (divide by n) changes the variance, the standard deviation, the skewness and the kurtosis together, which is the most common source of "my answer is different" questions.
The same page also covers the next questions people ask about a data set. It gives the z-score of any value against your list, any percentile, a confidence interval for the mean at 90%, 95% or 99% using the t distribution, and the Pearson correlation and least-squares line between your list and a second list. A box plot, a histogram with labelled axes, a frequency table and an outlier check based on the 1.5 times IQR rule sit under the numbers. A second mode takes grouped data (class intervals with frequencies) and gives the mean, variance and an interpolated median.
It is a descriptive tool. It does not run hypothesis tests, so there is no p-value, t-test, sample size calculation or probability table here. If you need those, use the results below as inputs to a package that has them.
How to use Statistics Calculator
- Paste or type your numbers in the data box. Commas, semicolons, spaces and line breaks all work as separators, and the line under the box tells you how many values were read.
- Choose Sample (n - 1) if your numbers are a sample of a larger group, or Population (n) if they are every value you care about.
- Pick the quartile method if you need to match a textbook, Excel or R. The quartiles, the IQR, the fences, the box plot and the percentile box all follow the same choice.
- For grouped data, switch Input type to Grouped data and enter one class per line, such as 10-20 5, meaning the class 10 to 20 with a frequency of 5.
- Read the result cards. The card for the page you opened is outlined, for example the standard deviation on the standard deviation page.
- For a z-score or percentile, type a value or a percentage in the matching box and read the line below it.
- For a confidence interval, pick 90%, 95% or 99%. For correlation and regression, paste a second list of the same length in the Y box.
- Open Steps and formulas at the bottom to see the calculation written out with your numbers.
Features
- Count, sum, mean, median, mode (all tied values), minimum, maximum and range.
- Sample or population variance and standard deviation with one switch, plus the standard error of the mean.
- Quartiles by a method you choose (inclusive, exclusive or linear), the same method for the percentile box, the interquartile range, outlier fences and a list of flagged outliers.
- Grouped data: class intervals with frequencies give the mean, variance, SD and the median by interpolation inside the median class.
- A box plot with whiskers and outlier dots, and a frequency table with relative and cumulative columns.
- A warning that lists every item that was left out because it is not a number.
- Skewness and excess kurtosis, using the adjusted forms for a sample and the plain moment forms for a population.
- z-score of any value against your data and any percentile by linear interpolation.
- Confidence interval for the mean at 90%, 95% or 99%, using the t distribution with n - 1 degrees of freedom.
- Pearson r, R squared, slope and intercept for paired X and Y lists.
- A histogram with axis titles and a values table for screen readers, and a written-out working for the mean, variance, standard deviation, range and IQR.
- Fifteen focused pages (standard deviation, variance, quartiles, five-number summary, grouped data, z-score and more) that open the same tool with the matching result outlined.
The formulas behind the numbers
The mean is the sum divided by the count. The variance adds up the squared distances from the mean and divides by n for a population or n - 1 for a sample. The standard deviation is its square root, and the standard error of the mean is the sample standard deviation divided by the square root of n.
A z-score is how many standard deviations a value sits from the mean. The confidence interval for the mean is the mean plus or minus a critical t value times the standard error. The Pearson correlation is the covariance of X and Y divided by the product of their standard deviations.
mean x̄ = Σx / n
variance s² = Σ(x − x̄)² / (n − 1) σ² = Σ(x − x̄)² / n
std. dev. s = √s² σ = √σ²
std. error SE = s / √n
z-score z = (x − mean) / SD
conf. int. x̄ ± t × SE, df = n − 1
correlation r = Σ(x − x̄)(y − ȳ) / √(Σ(x − x̄)² × Σ(y − ȳ)²)
regression slope = Σ(x − x̄)(y − ȳ) / Σ(x − x̄)², intercept = ȳ − slope × x̄Worked examples
2, 4, 4, 4, 5, 5, 7, 9 gives mean 5, sum of squares 32. Population: variance 4, SD 2. Sample: variance 4.5714, SD 2.1381.
The same eight numbers give two different standard deviations. The sample version is larger because it divides 32 by 7 instead of 8.
4, 8, 15, 16, 23, 42 (sample) gives n 6, sum 108, mean 18, median 15.5, no mode, variance 182, SD 13.4907, standard error 5.5076, Q1 9.75, Q3 21.25, IQR 11.5.
This is the list the calculator opens with. Every value is different, so there is no mode.
3, 5, 7, 8, 9, 10, 11, 40 (sample) gives mean 11.625, median 8.5, SD 11.7587, Q1 6.5, Q3 10.25, IQR 3.75, fences 0.875 and 15.875, outlier 40.
Without the 40 the mean would be 7.5714. One extreme value moves the mean a long way and the median hardly at all.
Sample or population: which one to pick
| Result | Sample (n - 1) | Population (n) | Use it when |
|---|---|---|---|
| Variance | s² = SS / (n - 1) | σ² = SS / n | Sample: your data is a subset. Population: it is every value. |
| Standard deviation | s = √s² | σ = √σ² | Same rule as the variance. |
| Skewness | adjusted (needs n > 2) | plain moment form | Sample is the default in most software. |
| Excess kurtosis | adjusted (needs n > 3) | plain moment form | Zero for a normal distribution. |
Critical t values used for the confidence interval
| Degrees of freedom | 90% | 95% | 99% |
|---|---|---|---|
| 2 | 2.920 | 4.303 | 9.925 |
| 4 | 2.132 | 2.776 | 4.604 |
| 9 | 1.833 | 2.262 | 3.250 |
| 19 | 1.729 | 2.093 | 2.861 |
| 29 | 1.699 | 2.045 | 2.756 |
Good to know
How the page reads your numbers
The box accepts any mix of commas, semicolons, spaces and line breaks. Anything that is not a number is left out, and an amber warning lists exactly which items were dropped, so a typo such as 1O (letter O) or 1,5 for 1.5 is easy to spot.
Use a dot as the decimal mark. A value written as 1,5 is read as the two numbers 1 and 5, because the comma is a separator.
Two quartile methods, two answers
You choose the method, and one choice drives the quartiles, the IQR, the fences, the box plot and the percentile box, so Q1 is always the 25th percentile. Inclusive uses position (n - 1)p + 1 like Excel QUARTILE.INC, R type 7 and NumPy. Exclusive uses (n + 1)p like Excel QUARTILE.EXC. Linear is the textbook percentile by interpolation, which gives the same numbers as Inclusive for quartiles. For 1, 3, 5, 7, 9, 11, 13, Q1 is 4 inclusive and 3 exclusive.
Excel, R and Python each have their own default quartile method, so a textbook, a spreadsheet and this page can all disagree by a little on small lists without any of them being wrong.
What this calculator does not do
It does not calculate p-values, run t-tests or other hypothesis tests, find a required sample size, or look up probabilities from a normal or t table. It has no field for a known mean and standard deviation: the z-score always uses the mean and SD of the list you paste. The confidence interval is for a mean only, and it always uses the sample standard deviation, whichever switch is selected.
Privacy: how it works
- Everything is calculated by JavaScript running in your own browser tab. Your numbers are not sent to Ulkaa or to any other server, and the page makes no network requests while you type.
- There is no account and nothing is stored. Closing or reloading the tab clears what you entered.
Tips
- If two sources give different standard deviations for the same data, check whether one divides by n and the other by n - 1 before assuming a mistake.
- Compare the mean with the median. A mean far above the median usually means a few large values are pulling it up, and the outlier fences will often show them.
- The confidence interval gets narrower with more data and wider with a higher confidence level. Try 90%, 95% and 99% on the same list to see how much.
- Correlation only measures a straight-line relationship. Two lists can have an r near 0 and still be tightly related by a curve, so look at the histogram and your data before you rely on r.
- Pair the X and Y lists by position. If they differ in length, only the first n pairs, where n is the shorter list, are used.
Frequently asked questions
Is it really free?
Yes. There is no price, no trial and no credit card. Ulkaa Tool does the work with your own device, so there is no server bill to pass on to you.
Do I need an account?
No. There is no sign-up and no login. Open the page and use it. An Ulkaa account is not needed for anything on this tool.
Does it run in my browser?
Yes. Your numbers are processed by code running in your browser tab, on your own device.
Is my data uploaded?
No. Your numbers are never sent to Ulkaa or to anyone else. You can check this yourself by watching the Network tab in your browser developer tools while you use the tool.
Should I use sample or population standard deviation?
Use the sample version (divide by n - 1) when your numbers are a sample drawn from a larger group, which is the usual case in coursework and research. Use the population version (divide by n) only when your list is every value in the group. When in doubt, sample is the safer default.
Why is the standard deviation of 2, 4, 4, 4, 5, 5, 7, 9 equal to 2?
The mean is 5 and the squared distances add up to 32. Dividing by n = 8 gives a variance of 4 and a standard deviation of 2. Dividing by n - 1 = 7 gives 4.5714 and 2.1381, which is the sample answer.
How is the median calculated for an even number of values?
The values are sorted and the two middle ones are averaged. For 1, 3, 7, 9 the middle pair is 3 and 7, so the median is 5.
What does it show when a list has no mode?
It shows "none" when every value appears only once. When several values tie for the highest count, it lists all of them, so 1, 2, 2, 3, 3, 4 has modes 2 and 3.
How do I read a z-score?
A z-score of 1.5 means the value is 1.5 standard deviations above the mean, and -1 means one standard deviation below. In roughly bell-shaped data about 68% of values lie within 1 and about 95% within 2. The calculator reports the z-score but does not convert it to a probability.
What does the 95% confidence interval mean?
It is a range built so that, if you repeated the sampling many times, about 95% of intervals made this way would contain the true mean. It is not a 95% chance that this one interval does. It assumes the values are a random sample and are roughly normal, and it is wide when n is small.
How are outliers detected?
With the 1.5 times IQR rule. A value below Q1 - 1.5 × IQR or above Q3 + 1.5 × IQR is flagged. It is a screening rule, not proof that a value is an error.
Can it handle grouped data?
Yes. Switch to Grouped data and list each class as lower-upper frequency. The mean and variance treat every value in a class as its midpoint, and the median is interpolated inside the median class, so all three are estimates. A table shows the midpoints, f times x and the cumulative frequency.
Which quartile method should I pick?
Use the one your course or software uses. Inclusive matches Excel QUARTILE.INC, R and NumPy, and is the default here. Exclusive matches Excel QUARTILE.EXC and many textbooks that use (n + 1)p. The two can differ on short lists.
Can it calculate a p-value or run a t-test?
No. It is a descriptive statistics calculator: summaries, quartiles, a box plot, grouped data, z-scores, percentiles, a confidence interval for the mean, correlation and a simple regression line. Hypothesis tests are not built.
How many numbers can I paste?
There is no fixed limit in the page. Very long lists are handled by your device, so speed depends on how much you paste.
Popular Statistics Calculator pages
- Standard Deviation CalculatorPaste your numbers and get the sample and population standard deviation with every step.
- Mean, Median and Mode CalculatorMean, median, mode and range of any list, in one view.
- Mean CalculatorThe average of a list of numbers, with the sum and count.
- Median CalculatorThe middle value of a list, with the quartiles alongside.
- Mode CalculatorThe most frequent value in a list, with ties handled.
- Variance CalculatorSample or population variance, with the sum of squares.
- Range CalculatorMinimum, maximum and range of a list of numbers.
- Z-Score CalculatorHow many standard deviations a value is from the mean of your data.
- Confidence Interval CalculatorA t-based confidence interval for the mean of your data.
- Correlation CalculatorPearson r and R squared for two paired lists.
- Linear Regression CalculatorThe least-squares line y = mx + b for paired data.
- Quartile and IQR CalculatorQ1, median, Q3, IQR and outlier fences for a list of numbers.
- Interquartile Range (IQR) CalculatorThe IQR of your data, with Q1, Q3 and the outlier fences.
- Five-Number Summary CalculatorMinimum, Q1, median, Q3 and maximum, with a box plot.
- Grouped Data Mean CalculatorMean, variance, SD and median from class intervals and frequencies.