Descriptive Statistics Calculator
Free Descriptive statistics Calculator for descriptive & distributions. Enter values to get step-by-step solutions with formulas and graphs.
Descriptive Statistics Calculator
Calculator
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Formula: Mean = Sum / n | Variance = Sum((xi - mean)^2) / (n-1) | Std Dev = sqrt(Variance)
Worked example — Mean: 82.5 | Median: 83.5 | Std Dev: 8.79 | IQR: 14
Formula
Mean = Sum / n | Variance = Sum((xi - mean)^2) / (n-1) | Std Dev = sqrt(Variance)
The mean is the sum of all values divided by the count. Sample variance uses n-1 (Bessel's correction) for an unbiased estimate. Standard deviation is the square root of variance. Quartiles divide the sorted data into four equal parts.
Worked Examples
Example 1: Test Scores Analysis
Problem:A class of 10 students scored: 72, 85, 90, 68, 95, 78, 82, 88, 91, 76. Calculate descriptive statistics.
Solution:Sorted: 68, 72, 76, 78, 82, 85, 88, 90, 91, 95 Mean: 82.5 | Median: 83.5 Std Dev: 8.79 | Variance: 77.17 Q1: 76 | Q3: 90 | IQR: 14 Range: 27 (68 to 95)
Result:Mean: 82.5 | Median: 83.5 | Std Dev: 8.79 | IQR: 14
Example 2: Salary Distribution
Problem:Salaries (in thousands): 45, 50, 55, 55, 60, 65, 70, 75, 80, 120. Note the outlier at 120.
Solution:Mean: 67.5 | Median: 62.5 The mean is pulled up by the outlier (120). Median is more representative of the typical salary. Skewness is positive, confirming right-skewed distribution.
Result:Mean: 67.5 | Median: 62.5 | Positive skew due to outlier
Frequently Asked Questions
What are descriptive statistics?
Descriptive statistics summarize and describe the main features of a dataset. They include measures of central tendency (mean, median, mode), measures of spread (range, variance, standard deviation, IQR), and measures of shape (skewness, kurtosis). Unlike inferential statistics, descriptive statistics do not draw conclusions beyond the data at hand — they simply describe what is in the data.
When should I use mean vs. median?
Use the mean when your data is roughly symmetrical and has no extreme outliers. Use the median when your data is skewed or contains outliers — the median is robust to extreme values. For example, median household income is preferred over mean income because a few billionaires can drastically inflate the mean. As a rule of thumb, if the mean and median are close, the data is approximately symmetric.
What do skewness and kurtosis tell you?
Skewness measures asymmetry. Positive skew means the right tail is longer (data piled up on the left). Negative skew means the left tail is longer. A skewness near 0 indicates symmetry. Kurtosis (excess) measures tail heaviness compared to a normal distribution. Positive kurtosis (leptokurtic) means heavier tails and more outliers. Negative kurtosis (platykurtic) means lighter tails. The normal distribution has excess kurtosis of 0.
What is the interquartile range (IQR)?
The IQR is the difference between the third quartile (Q3, 75th percentile) and the first quartile (Q1, 25th percentile). It represents the middle 50% of the data and is a robust measure of spread that is not affected by outliers. The IQR is used in box plots and for detecting outliers: values below Q1 - 1.5 × IQR or above Q3 + 1.5 × IQR are typically considered outliers.
Why does sample variance divide by n - 1 instead of n?
Deviations are measured from the sample mean, which is itself pulled toward the data, so squared deviations come out systematically smaller than they would around the true population mean. Dividing by n - 1 rather than n, known as Bessel's correction, inflates the estimate just enough to remove that bias. The effect is large on small samples — with n = 5 it raises the variance by 25 percent — and negligible once n runs into the hundreds.
When should I use the population standard deviation formula instead?
Use the population form, which divides the sum of squared deviations by n, only when the numbers in front of you are the complete group you want to describe: every employee in a department, every item in a finished production run. If the data are a sample drawn from a larger population and you intend to generalise, the n - 1 sample form is correct. Choosing the population divisor for sample data understates the spread and makes downstream intervals too narrow.
Can a dataset have more than one mode, and when is the mode the right summary?
Yes. A dataset with two equally frequent peaks is bimodal and one with several is multimodal; if every value appears once, no mode exists. The mode is the only measure of central tendency that works for nominal categories such as colours or product types, where a mean is undefined. Bimodality also carries information a mean would hide, often signalling that two different subpopulations have been pooled into one dataset.
What is the difference between range and standard deviation as measures of spread?
Range is simply maximum minus minimum, so it uses only two observations and ignores everything between them. That makes it fragile: a single mistyped value can double it, and it tends to grow with sample size because larger samples reach further into the tails. Standard deviation uses every deviation from the mean, so it describes typical spread rather than worst case and is stable as n increases. Report range for context, standard deviation for analysis.
How do I turn a raw value into a z-score using these results?
A z-score rescales an observation as z = (x - mean) / standard deviation, giving the number of standard deviations the value sits above or below the mean. A score of 91 in a set with mean 82.5 and standard deviation 8.79 gives z = 8.5 / 8.79 = 0.97, just under one standard deviation above average. Because z-scores are unitless, they let you compare positions across datasets measured on entirely different scales.
What is the standard error of the mean and how does it differ from standard deviation?
Standard deviation describes how spread out the individual observations are. The standard error of the mean, s / sqrt(n), describes how much the sample mean itself would bounce around if you repeatedly drew fresh samples of the same size. It shrinks as n grows, while the standard deviation converges to a fixed population value. Use the standard deviation to describe the data and the standard error when quantifying uncertainty about the average.
References
Reviewed for accuracy by Daniel Agrici, Founder & Lead Developer · Editorial policy
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