Descriptive Statistics Calculator
Paste or type your data set (numbers separated by commas, spaces, or line breaks) and get the complete picture: central tendency (mean, median, mode), spread (standard deviation, variance, range, IQR), shape (skewness, kurtosis), quartiles, outliers, and more. Choose between sample and population formulas. Results update instantly as you edit.
What descriptive statistics measure
Descriptive statistics are a set of numbers that summarize and describe the key properties of a data set without making inferences about a wider population. They fall into three groups. Central tendency measures - mean, median, and mode - tell you where the middle of your data sits. Spread measures - standard deviation, variance, range, and interquartile range - tell you how far the values fan out from the center. Shape measures - skewness and kurtosis - tell you whether the distribution is symmetric and how heavy its tails are. Together these statistics paint a complete first portrait of any data set.
Mean vs. median: which to report
The mean (arithmetic average) is the sum of all values divided by the count. It is the most commonly cited measure of center, but it is sensitive to extreme values. A single very large or very small number can pull the mean far from where most of the data sits. The median is the middle value when the data is sorted - exactly half of values fall below it and half above. Because it depends only on rank and not magnitude, the median is much more robust to outliers. When your data is skewed or contains outliers, the median is usually the more honest single-number summary. When the mean and median are close together, the distribution is roughly symmetric and either measure works.
Standard deviation, variance, and the IQR
Variance is the average squared deviation from the mean. Standard deviation (SD) is its square root, which brings it back to the same units as the data. The sample formula divides by n - 1 (Bessel's correction) to give an unbiased estimate when you have a subset of a larger population; the population formula divides by n when you have the entire group. The interquartile range (IQR) is Q3 minus Q1 - the spread of the middle 50% of values - and is resistant to outliers for the same reason the median is. Tukey's method flags a value as a potential outlier when it falls more than 1.5 times the IQR below Q1 or above Q3.
Skewness, kurtosis, and distribution shape
Skewness measures the asymmetry of the distribution. A positive value means the right tail is longer (a few very high values inflate the mean relative to the median); a negative value means the left tail is longer. A value between -0.5 and 0.5 is considered approximately symmetric. Excess kurtosis measures tail weight relative to a normal distribution. A value near zero is mesokurtic (normal-like). Positive excess kurtosis (leptokurtic) means heavier-than-normal tails and a sharper peak - common in financial returns. Negative excess kurtosis (platykurtic) means lighter tails and a flatter peak. The coefficient of variation (CV) expresses the standard deviation as a percentage of the mean, making it possible to compare spread across data sets with different units or scales.
Skewness and kurtosis interpretation
| Statistic | Value range | Interpretation |
|---|---|---|
| Skewness | < -1 | Strongly left-skewed |
| Skewness | -1 to -0.5 | Moderately left-skewed |
| Skewness | -0.5 to 0.5 | Approximately symmetric |
| Skewness | 0.5 to 1 | Moderately right-skewed |
| Skewness | > 1 | Strongly right-skewed |
| Excess kurtosis | < -1 | Platykurtic (thin tails) |
| Excess kurtosis | -1 to 1 | Mesokurtic (normal-like tails) |
| Excess kurtosis | > 1 | Leptokurtic (heavy tails) |
Common thresholds for characterizing distribution shape.
Frequently asked questions
What is the difference between sample and population standard deviation?
The population standard deviation divides the sum of squared deviations by n (the total count), and is used when your data contains every member of the group you are studying. The sample standard deviation divides by n - 1 (Bessel's correction), which compensates for the fact that a sample tends to underestimate the spread of the full population. In almost all real-world analysis, where you have a sample rather than a complete census, you should use the sample formula.
How does this calculator detect outliers?
Outliers are identified using Tukey's fences. The lower fence is Q1 - 1.5 times the IQR, and the upper fence is Q3 + 1.5 times the IQR. Any value outside those fences is flagged as a potential outlier. This method does not require you to assume the data is normally distributed, which makes it a robust general-purpose rule. Extreme outliers are sometimes defined using a factor of 3 instead of 1.5.
When should I use the median instead of the mean?
Use the median when your data is skewed or contains outliers, or when you are working with ordinal data (ranked categories). Household income is a classic example: a few very high earners pull the mean well above what most households earn, so the median gives a more representative picture. When your data is roughly symmetric and free of extreme values, the mean and median will be close together and either is appropriate.
What does the coefficient of variation tell me?
The coefficient of variation (CV) is the standard deviation expressed as a percentage of the mean. It answers the question: how large is the spread relative to the typical value? A CV below about 15% is considered low variability; above 35% is considered high. Because it is a ratio, it lets you compare the relative spread of two data sets measured in different units - for example, comparing the variability of heights (cm) with the variability of weights (kg).
What is the interquartile range (IQR) used for?
The IQR is the distance between the 25th percentile (Q1) and the 75th percentile (Q3), covering the middle 50% of your data. It is used as a robust measure of spread when the data has outliers or is skewed, because it ignores the extreme values entirely. The IQR also forms the basis for Tukey's outlier detection rule and is the spread measure shown in a box-and-whisker plot.
How is skewness interpreted?
Skewness measures how asymmetric the distribution is. A value between -0.5 and 0.5 is usually treated as approximately symmetric. Values between 0.5 and 1 (or -0.5 and -1) indicate moderate skew; values beyond 1 in either direction indicate strong skew. Positive skew means the right tail extends further - a few large values pull the mean above the median. Negative skew means the left tail extends further. Highly skewed data often benefits from a log or square-root transformation before further analysis.