Mean, median and mode
The mean is the sum divided by the count. The median is the middle value once the numbers are sorted, or the average of the two middle values when the count is even. The mode is the value that appears most often, and a list can have several or none.
They answer different questions. US household income shows why: the median was about $83,700 in 2024 (Census Bureau), while the mean was over $114,000, pulled up by a small number of very high incomes. When data is skewed like this, the median describes a typical case better.
Standard deviation and variance
variance s² = Σ(x − x̄)² ÷ (n − 1) (sample)
variance σ² = Σ(x − μ)² ÷ N (population)
standard deviation = √variance
The standard deviation measures how far values typically sit from the mean, in the same units as the data. For roughly bell-shaped data, about 68% of values fall within one standard deviation of the mean and 95% within two. The sample formula divides by n − 1 (Bessel's correction) because a sample's own mean sits closer to its values than the true population mean does, which would otherwise understate the spread.
Quartiles, IQR and outliers
Quartiles split sorted data into four equal parts. Q1 is the 25th percentile, Q2 the median and Q3 the 75th percentile. The interquartile range, IQR = Q3 − Q1, covers the middle half of the data. Tukey's rule flags any value below Q1 − 1.5 × IQR or above Q3 + 1.5 × IQR as a possible outlier. This calculator uses the same quartile method as Excel's QUARTILE.INC and NumPy's default (linear interpolation), so textbook answers that use the "median of halves" method can differ slightly.
Statistics and money
Investors use standard deviation as a measure of risk: the S&P 500's yearly returns have averaged around 10% since 1926 with a standard deviation near 20%, which is why a single year can easily fall 20% or rise 30%. The investment calculator uses that spread to simulate thousands of possible futures.