Dataset Analyzer: Mean, Median, Std Deviation, Quartiles, and Outliers
Raw numbers become meaningful through descriptive statistics. Whether you are reviewing test scores, server latency logs, sales figures, or scientific measurements, a few key statistics immediately reveal the shape of your data: where it centers (mean, median), how spread out it is (standard deviation, IQR), and whether any values are outliers. Paste any list of numbers and get the full descriptive stats in under a second.
Use sample variance (÷n−1) when your dataset is a sample from a larger population. Use population variance (÷n) when you have the complete dataset.
Mean vs Median — When Each Is Better
The mean sums all values and divides by count — it is sensitive to outliers. One extreme value (a $500K salary in a list of $50K salaries) pulls the mean far from where most values cluster. The median is the middle value when sorted — it ignores outliers and better represents the typical experience. Use median for income, house prices, and any distribution you expect to be skewed. Use mean when the distribution is symmetric.
Standard Deviation as Spread from Center
Standard deviation measures how much values typically differ from the mean. A small std dev means values are clustered tightly; a large one means they are spread wide. In a normal distribution: ~68% of values fall within 1 std dev of the mean, ~95% within 2, and ~99.7% within 3 (the 68-95-99.7 rule). Std dev is in the same units as your data, making it more interpretable than variance.
Detecting Outliers with the Tukey Fence Method
The Tukey fence defines outliers as values below Q1 − 1.5 × IQR or above Q3 + 1.5 × IQR. This is the same method used by box plots. IQR (Interquartile Range = Q3 − Q1) measures the spread of the middle 50% of data, making it robust against the very outliers it helps detect. Values outside the fences warrant investigation before analysis.
Mean vs Median Decision Guide
- Use mean: symmetric data, no extreme outliers (heights, temperatures)
- Use median: skewed data or outliers present (incomes, prices, latency)
- Both equal: data is symmetric and normally distributed
- Mean >> Median: right-skewed distribution (positive outliers pulling up)
- Mean << Median: left-skewed distribution (negative outliers pulling down)
Five-Number Summary Quick Reference
- Min: smallest value in the dataset
- Q1: 25th percentile — 25% of values fall below this
- Median (Q2): 50th percentile — middle of the dataset
- Q3: 75th percentile — 75% of values fall below this
- Max: largest value in the dataset
Frequently Asked Questions
What is the difference between mean and median?
Mean is the sum of all values divided by the count — it weights every value equally, including outliers. Median is the middle value when sorted — it is unaffected by extreme values. For skewed distributions (income, house prices), the median better represents the typical case. For symmetric distributions, they are equal.
When should I use the median instead of the mean?
Use median when your data is skewed or contains outliers. If a dataset of 9 salaries is $50K each and one person earns $1M, the mean ($140K) is misleading while the median ($50K) accurately represents the group. Any distribution with a long tail (income, prices, response times) benefits from median reporting.
What is standard deviation?
Standard deviation measures how spread out values are around the mean. A small std dev means values cluster tightly; a large one means they are dispersed. It is the square root of variance, expressed in the same units as the original data. In a normal distribution, ~68% of values fall within 1 std dev of the mean.
What is an outlier in statistics?
An outlier is a data point that falls unusually far from the rest of the dataset. The Tukey fence method (used by box plots) defines outliers as values below Q1 − 1.5×IQR or above Q3 + 1.5×IQR. Outliers may indicate measurement errors, data entry mistakes, or genuinely exceptional cases worth investigating.
What is the difference between population and sample standard deviation?
Population std dev (divided by N) is used when you have data for every member of the group. Sample std dev (divided by N−1) is used when your data is a sample from a larger population — the N−1 correction (Bessel's correction) reduces bias in the estimate. Use sample std dev unless you are certain you have the complete population.
What is the interquartile range?
IQR = Q3 − Q1, the spread of the middle 50% of your data. It is robust against outliers because it ignores the bottom and top 25% of values. The IQR is used by the Tukey fence to define outlier thresholds and by box plots to define whisker length.
What does the five-number summary tell me?
The five-number summary — Min, Q1, Median, Q3, Max — gives a complete picture of a dataset's distribution. It shows the range (Max − Min), the spread of the middle half (IQR = Q3 − Q1), and whether the distribution is symmetric (Q2 centered between Q1 and Q3) or skewed.