Paste a list of numbers to get a clear descriptive statistics summary, including quartiles, spread, possible outliers, and sample or population results.
Advanced options
Display
Quartiles (Q1 and Q3)
Extra outputs
How to use our Statistics Calculator
- Enter your numbers in Data values. You can separate values with commas, spaces, or line breaks.
- Choose Data type: use Sample if your list is part of a bigger group, or Population if your list is the whole group.
- If needed, open Advanced options and choose a Quartile method, Decimal places for results, and whether to Show frequency table.
- Click Calculate to generate the full summary.
- First check Count and Sorted data to make sure every value was read correctly and in the order you expect.
- Use Mean, Median, and Mode to compare the center of the data, then use Range, Standard deviation, and Interquartile range (IQR) to judge spread.
- Review Lower outlier fence, Upper outlier fence, and Possible outliers to spot unusual values worth checking.
- If results do not match your class notes, compare the selected Data type and Quartile method, because both can change the answers.
Definitions
Data type: A choice between sample data and population data. This changes how Variance and Standard deviation are computed.
Count: The number of valid values read from Data values.
Sorted data: Your values arranged from smallest to largest so you can check the list, median, and quartiles.
Mean: The arithmetic average.
Median: The middle value in sorted order, or the average of the two middle values when there is an even number of values.
Mode: The value, or values, that appear most often.
Variance: A measure of spread based on squared distance from the mean.
Standard deviation: The square root of variance, so it describes spread in the same units as the data.
First quartile (Q1): A value near the 25th percentile of the sorted data.
Third quartile (Q3): A value near the 75th percentile of the sorted data.
Interquartile range (IQR): Q3 minus Q1. It measures the spread of the middle half of the data.
Lower outlier fence / Upper outlier fence: Cutoff values based on Q1, Q3, and 1.5 times the IQR. Values outside these fences are possible outliers [1].
Frequency table: A list of each unique value and how many times it appears.
Common mistakes and quick fixes
Mistake: Leaving Data values blank or pasting text like words, units, or symbols mixed with numbers.
Fix: Enter only numeric values in Data values , separated by commas, spaces, or line breaks.
Mistake: Choosing the wrong Data type and then wondering why Variance or Standard deviation changed.
Fix: Use Sample when your list is only part of a larger group, and Population when your list is the full group you want to describe.
Mistake: Thinking a low Count means the calculator lost values, when the real issue is a formatting problem in Data values .
Fix: Check Count and Sorted data right away to confirm every value was parsed.
Mistake: Expecting your textbook's quartiles when a different Quartile method is selected.
Fix: Match the Quartile method to your class rule before comparing First quartile (Q1) , Third quartile (Q3) , or Interquartile range (IQR) .
Mistake: Treating Possible outliers as automatic mistakes.
Fix: Use Lower outlier fence and Upper outlier fence as review guides, then check whether those values are real but unusual.
Mistake: Reading too much into Coefficient of variation when the Mean is zero or very close to zero.
Fix: If the Mean is zero, treat Coefficient of variation as not defined; if the mean is tiny, interpret it carefully because the percent can look misleadingly large.
Limitations & Key Assumptions / Boundary Conditions
- The calculator is for descriptive statistics only. It summarizes the numbers you enter but does not explain why they happened.
- Variance, Standard deviation, and Standard error of the mean depend on whether Data type is set to Sample or Population.
- Standard error of the mean is shown only for sample data with at least 2 values. In population mode, it is not applicable here.
- Coefficient of variation is not defined when the Mean is zero, and it can be hard to interpret when the mean is very close to zero or negative.
- First quartile (Q1) and Third quartile (Q3) can differ across textbooks and software because quartile rules vary, especially for small data sets.
- Possible outliers are based on Tukey fences, so they are flags for review, not proof that a value is wrong [1].
- If your list is very long, the calculator may shorten the Sorted data preview for display, but all valid values are still used in the calculations.
Methodology
Parsing and cleaning the data
The calculator reads Data values as numbers separated by commas, spaces, tabs, or line breaks. Repeated values, decimals, and negative numbers are allowed. If a token is not numeric, calculation stops and shows an error that points to the bad entry.
Core calculations
mean = (sum of x_i) / n
median = middle sorted value, or average of the two middle sorted values
mode = value(s) with highest frequency
range = max - min
midrange = (min + max) / 2
SS = sum((x_i - mean)^2)
variance_pop = SS / n
variance_samp = SS / (n - 1)
standard deviation = sqrt(variance)
MAD = sum(abs(x_i - mean)) / n
IQR = Q3 - Q1
CV = (standard deviation / mean) x 100
SE = sample standard deviation / sqrt(n)
lower fence = Q1 - 1.5 x IQR
upper fence = Q3 + 1.5 x IQR
Sample vs population
If Data type is Sample, variance uses n - 1. If Data type is Population, variance uses n. That is why Variance, Standard deviation, and Coefficient of variation can change when you switch the mode.
Quartiles and outliers
First quartile (Q1) and Third quartile (Q3) come from the selected Quartile method. The calculator then finds Interquartile range (IQR) and uses Tukey's 1.5 x IQR rule to flag possible outliers [1]. Values outside the fences are listed as unusual values to review, not guaranteed mistakes.
Worked mini-example
Using the sample data 4, 8, 6, 5, 3, 8, 9, the sorted list is 3, 4, 5, 6, 8, 8, 9.
n = 7
sum = 43
mean = 43 / 7 = 6.142857...
median = 6
Q1 = 4, Q3 = 8, so IQR = 4
lower fence = 4 - 1.5 x 4 = -2
upper fence = 8 + 1.5 x 4 = 14
Since all values lie between -2 and 14, the outlier list is empty for this example.
Assumptions and limits
Results are descriptive summaries of the entered list only. For sample mode with just one value, sample-only spread outputs such as Variance, Standard deviation, and Standard error of the mean are not available because they require at least 2 values. If the Mean is zero, Coefficient of variation is not defined. Different quartile conventions can produce different Q1 and Q3 values on the same small dataset.