The problem: you're using the wrong average
Most people know one average: add everything up, divide by the count. That's the arithmetic mean, and it's the right answer for additive quantities — test scores, heights, temperatures. It's the wrong answer for two other common situations.
If you invest $100 and it grows 50% the first year and loses 50% the second, the arithmetic mean of the returns is 0%: (50 + -50) / 2 = 0. You'd think you broke even. But $100 × 1.50 × 0.50 = $75. You lost 25%. The arithmetic mean of growth rates lies about what actually happened to your money. The geometric mean tells the truth: √(1.50 × 0.50) = 0.866, which means an average yearly multiplier of 0.866 — a 13.4% annual loss. That's what your portfolio actually experienced.
If you drive 60 mph to a destination and 40 mph back, the arithmetic mean says your average speed was 50 mph. But the trip took longer at 40 mph than at 60 mph, so the true average is lower. The harmonic mean gives the right answer: 2 / (1/60 + 1/40) = 48 mph. The slow leg weighted more because it took more time.
Three means, three situations, one decision: what kind of quantity are you averaging?
Fastest path
Open the Statistics Calculator, paste your numbers (comma-separated, space-separated, or one per line), and the tool computes all three means, median, mode, quartiles, IQR, variance, standard deviation, outlier detection, skewness, and kurtosis. The tool uses linear interpolation for percentiles and the 1.5×IQR rule for outliers.
The three means and when to use each
Arithmetic mean: for additive quantities
Use when the values combine by addition. Test scores, weights, temperatures, distances, counts. The arithmetic mean is sum divided by count:
Arithmetic mean = (x₁ + x₂ + ... + xₙ) / n
The arithmetic mean is the right answer when the total is meaningful — the sum of all test scores is a meaningful total, so the average score is a meaningful average. It's the wrong answer when the values combine by multiplication (growth rates, return rates) or by rates (speed, throughput).
Geometric mean: for multiplicative quantities
Use when the values combine by multiplication — growth rates, investment returns, population growth, compound interest. The geometric mean is the nth root of the product:
Geometric mean = (x₁ × x₂ × ... × xₙ)^(1/n)
In practice, the tool computes it as exp(mean of logs), which is numerically stable for large datasets. The geometric mean is always less than or equal to the arithmetic mean (this is the AM-GM inequality), and the gap widens when the values are spread out. The 50%/-50% example above shows the gap clearly: arithmetic 0%, geometric -13.4%.
The geometric mean requires all values to be positive. If your data contains zeros or negative numbers (a year with 0% growth, a year with -20% return), the geometric mean is undefined — the product goes to zero or negative, and you can't take the nth root of a negative number. For investment returns that can be negative, add 1 to each return (converting -20% to 0.80, 50% to 1.50), take the geometric mean, then subtract 1. That's what the 0.866 example does: the geometric mean of 1.50 and 0.50 is 0.866, and 0.866 - 1 = -0.134 = -13.4%.
Harmonic mean: for rates and ratios
Use when the values are rates — speed, throughput, price per unit. The harmonic mean is the reciprocal of the mean of reciprocals:
Harmonic mean = n / (1/x₁ + 1/x₂ + ... + 1/xₙ)
The 60 mph / 40 mph example is the canonical case. You're averaging speeds over the same distance, and the harmonic mean accounts for the fact that the slower speed takes more time. The harmonic mean is always less than or equal to the geometric mean, which is less than or equal to the arithmetic mean.
The harmonic mean also requires positive values. If any value is zero, the reciprocal is undefined and the mean breaks. This makes sense — you can't average a speed of 0 mph with a speed of 60 mph over the same distance, because the 0 mph leg takes infinite time.
Mean + SD vs median + IQR: the decision
Once you've picked the right mean, the next question is how to describe the spread. The two pairs are:
- Mean + standard deviation: the standard pair for symmetric, roughly normal data. Mean tells you the center, SD tells you the spread. The empirical rule (68% within 1 SD, 95% within 2) applies if the data is approximately normal.
- Median + IQR: the robust pair for skewed or outlier-prone data. Median is the middle value (unaffected by extremes), IQR is the range of the middle 50% (Q3 - Q1, also unaffected by extremes).
The decision rule: if the mean and median are close (within a few percent of each other), use mean + SD. If they diverge significantly, the data is skewed, and median + IQR is the better description.
Income data is the classic case. US household income has a mean around $97,000 and a median around $75,000. The mean is 29% higher than the median because a small number of very high incomes pull the mean up. Reporting "mean income $97,000 with SD $80,000" gives a misleading picture — the SD is enormous because the outliers inflate it, and the mean is pulled toward the outliers. Reporting "median income $75,000 with IQR $45,000" gives a more accurate picture of what a typical household looks like.
The tool computes both pairs and flags outliers separately, so you can compare. If the outlier list is long or the skewness value is above 1 or below -1, reach for median + IQR.
The 1.5×IQR outlier rule
The tool flags values as outliers using the 1.5×IQR rule: any value below Q1 - 1.5×IQR or above Q3 + 1.5×IQR is an outlier. This is the same rule used in box plots.
Lower fence: Q1 - 1.5 × IQR
Upper fence: Q3 + 1.5 × IQR
The 1.5 multiplier is not arbitrary — for a normal distribution, it flags approximately 0.7% of values as outliers, which is about one in 140. That's rare enough to be meaningful but not so rare that you miss real anomalies. A value beyond 3×IQR (the "far fence") is an extreme outlier — roughly 1 in 425,000 for normal data. If you see one, it's either a genuine anomaly or a data error.
The rule is robust because IQR is based on quartiles, which don't move when the tails of the distribution move. A single extreme value can double the standard deviation but barely changes the IQR. That's why IQR-based outlier detection works on skewed data where SD-based methods (like "flag anything beyond 2 standard deviations") fail — the SD itself is inflated by the outliers it's trying to detect.
Skewness and kurtosis: what shape tells you
Skewness measures asymmetry. Zero is perfectly symmetric. Positive means the right tail is longer (a few large values pull the distribution right — income, house prices). Negative means the left tail is longer (a few small values pull it left — age at death, where most people live to 70+ but infant mortality creates a left tail). The tool classifies |skewness| < 0.5 as approximately symmetric, > 0.5 as skewed.
Skewness tells you which mean is being pulled where. In right-skewed data, the mean is above the median. In left-skewed data, the mean is below. The direction is consistent: the mean follows the tail.
Kurtosis (excess kurtosis) measures tail weight relative to a normal distribution. Zero means normal tails. Positive means heavier tails (more outliers than a normal distribution would produce — financial returns, insurance claims). Negative means lighter tails (fewer outliers — heights, which are tightly bounded by biology). The tool computes excess kurtosis, which subtracts 3 so that the normal distribution is the zero point.
Kurtosis is the statistic people misunderstand most. High kurtosis does not mean "more peaked" — it means "heavier tails." The center of the distribution can be flat or peaked; kurtosis is about what happens at the extremes. A distribution with high kurtosis has more values far from the mean than a normal distribution would predict. This matters for risk assessment: if your loss data has high kurtosis, extreme losses happen more often than a normal model would suggest.
Gotchas
- The geometric mean requires positive values. The tool returns null for geometric and harmonic means if any value is zero or negative. This is correct behavior — those means are undefined for non-positive data. If you need to average growth rates that include negative values, convert to multipliers (1 + rate) first, take the geometric mean, then convert back.
- Quartile calculation methods differ. The tool uses linear interpolation between observed values (the same method as NumPy's default and R's type 7). Excel's QUARTILE function uses a different method, and TI-84 calculators use yet another. The Q1 and Q3 values may differ by a few percent across tools. None is wrong — the methods make different assumptions about where the quartile boundary falls when n is even. If you're comparing results across tools, check which method each uses.
- Skewness and kurtosis need n ≥ 3. The tool requires at least three values and a non-zero standard deviation to compute shape measures. With n=2, skewness and kurtosis are undefined (the third and fourth moments need more data to be meaningful). Even at n=10, skewness estimates are noisy — the confidence interval is wide. Treat shape measures as rough guides for small samples.
- The coefficient of variation is unitless but not always meaningful. CV = (SD / mean) × 100. It's useful for comparing spread across datasets with different units (is stock A more volatile than stock B?) but meaningless for data that passes through zero. Temperature in Celsius has an arbitrary zero, so CV of temperature is not interpretable. Use CV only for ratio-scale data (counts, lengths, money) where zero means "none of the quantity."
- Mode is unstable for continuous data. With floating-point values, most data points are unique, so the mode is "no mode" or a single value that happened to repeat. Mode is useful for categorical or integer data (survey responses, test scores on a fixed scale) but unreliable for continuous measurements. For continuous data, use the histogram to find the peak instead.
Summary
- Three means for three situations: arithmetic for additive data (scores, weights), geometric for multiplicative data (returns, growth rates), harmonic for rates and ratios (speeds, prices per unit).
- Use mean + SD for symmetric data (mean ≈ median). Use median + IQR for skewed data (mean ≠ median). The gap between mean and median tells you which pair to report.
- The 1.5×IQR rule flags outliers robustly because IQR is unaffected by the tails. SD-based outlier detection fails on skewed data because the outliers inflate the SD.
- Skewness measures asymmetry (mean follows the tail). Kurtosis measures tail weight, not peakedness. High kurtosis means more extreme values than a normal model predicts.
- Use the Statistics Calculator for the full suite of descriptive statistics, the Standard Deviation Calculator for the focused SD computation with a step-by-step deviation table, and the Percentage Calculator for ratio-based comparisons.