Do the values add? Use arithmetic mean
Scores, counts, distances, and temperatures combine by addition. The arithmetic mean preserves the total: AM × n = sum of all values.
Compound growth & annualized returns
Finance Growth Calculators: CAGR and XIRR
CAGR (compound annual growth rate) and XIRR are the two standard annualized return metrics in finance. CAGR uses the geometric mean formula to convert start-to-end growth into a single annual rate. XIRR extends this to irregular, dated cash flows such as SIP investments and partial withdrawals. This hub links to both calculators and explains when each metric applies.
Calculators in the Finance section
Each tool below applies a distinct formula. Read the eyebrow label to confirm the formula matches your data before opening the calculator.
Geometric mean of annual growth CAGR Calculator — Compound Annual Growth Rate Formula and Examples
CAGR (compound annual growth rate) is the single annual rate that connects a beginning value to an ending value as if growth were perfectly smooth each year. Formula: CAGR = (EV / BV)^(1/n) − 1. Example: $1,000 growing to $1,750 over 5 years → CAGR = (1,750/1,000)^(0.20) − 1 = 11.84% per year. This calculator returns CAGR, total return %, and the ending value multiple.
Timing-adjusted annualized return CAGR to XIRR Converter — Annualized Return for Dated Cash FlowsXIRR (extended internal rate of return) is the annualized rate that makes the net present value of all your dated cash flows equal zero: NPV = Σ [CFᵢ / (1 + r)^(dᵢ/365)] = 0. Unlike CAGR, it handles multiple investment dates, SIP contributions, partial withdrawals, and dividend reinvestment. This calculator solves for r using Newton’s method and shows how XIRR compares with a simple CAGR on the same cash flows.
Three decision rules for choosing the right average
Each mean is correct for a specific data structure. Using the wrong one produces a number that looks reasonable but answers the wrong question.
Do the values multiply? Use geometric mean
Growth factors, return indexes, and ratios combine by multiplication. Geometric mean gives the single factor that, applied n times, reproduces the same product.
Are values rates over equal units? Use harmonic mean
Speeds over equal distances, prices per equal budget, and output rates per equal workload require harmonic mean. It weights slower rates more because they consume more time or resource.
Match the formula to the situation
Each row below shows a real situation, the correct metric, and why a different metric would give a wrong answer.
| Situation | Input | Best next move | Why |
|---|---|---|---|
| Average exam score across a class | 72, 80, 88, 96 | Use arithmetic mean → 84 | Scores are additive; AM preserves the class total and each student's equal weight. |
| Average annual investment return over 4 years | Return factors: 1.08, 0.92, 1.15, 1.06 | Use geometric mean → 1.0494 (4.94%/yr) | Returns compound; AM would give 5.25%/yr but overstates the actual ending value. |
| Average speed on a round trip | 30 mph going, 60 mph returning, same distance | Use harmonic mean → 40 mph | Equal distance at unequal speeds; AM gives 45 mph but would predict the wrong travel time. |
Step-by-step: how to use this calculator correctly
Identify whether you have a single start-to-end growth story (use CAGR) or multiple dated contributions (use XIRR).
For CAGR: enter beginning value, ending value, and the number of years — the calculator applies the geometric mean formula.
For XIRR: enter each dated cash flow on its own row, with investments as negative numbers and the final portfolio value as positive.
Compare CAGR with total return percentage to understand both the pace and the absolute gain.
Do not use CAGR as a forecast — it summarizes past or assumed growth and ignores intermediate volatility.
CAGR: $1,000 portfolio growing to $1,750 in 5 years
Given
Beginning value = $1,000 | Ending value = $1,750 | Years = 5
Work
CAGR = (1,750 / 1,000)^(1/5) − 1 = 1.75^0.20 − 1 = 1.1184 − 1
Result
CAGR = 11.84% per year. Total return = 75%. Each compounding year multiplies the value by 1.1184.
Takeaway
Two portfolios at the same 11.84% CAGR could have completely different year-by-year paths. Pair CAGR with drawdown or yearly returns to understand risk.
Which average fits your data: a decision guide
The same dataset produces different values for arithmetic, geometric, and harmonic mean. Choosing the wrong one gives a numerically plausible but meaningless result.
| Method | Best for | Watch for | Example |
|---|---|---|---|
| Arithmetic mean | Additive values: test scores, temperatures, counts where each observation contributes equally. | One outlier shifts the result sharply. For 4, 9, 16, 100: AM = 32.25, far above most values. | Class scores 72, 80, 88, 96 → AM = 84. |
| Geometric mean | Multiplicative data: growth factors, index ratios, and return factors that compound across periods. | All values must be positive (or percentage returns above −100%). Zero collapses the product. | Annual return factors 1.06, 0.92, 1.15 → GM = 1.038, meaning 3.8% per year. |
| Harmonic mean | Rate data where the denominator is fixed: speeds over equal distances, prices per equal budget. | Wrong for equal-time rates. For 30 mph and 60 mph over equal distance: HM = 40, AM = 45. | 30 mph out, 60 mph back over equal distance → average speed = 40 mph. |
HM ≤ GM ≤ AM: the same four numbers, three averages
For dataset 4, 9, 16, 25: harmonic mean = 8.63, geometric mean = 10.95, arithmetic mean = 13.50. The gap signals whether data is additive, multiplicative, or rate-based.
Correct calculation flow for any mean
The same four steps apply regardless of which mean you use. The formula in step 3 is the only thing that changes.
Determine whether values add, multiply, or are rates over equal units.
Check for zeros, negatives, mixed units, or missing values that would break the formula.
Apply AM = sum/n, GM = (product)^(1/n), or HM = n / Σ(1/xᵢ).
State what the result means in plain terms, including its unit and what it is compared against.
Key facts before you calculate
Why investment returns require geometric mean, not arithmetic mean
Consider annual returns of 6%, 7%, 8%, −35%, 10%. Arithmetic mean = 0.40% — but this implies no gain. The actual ending value on a $1,000 investment is $923.60, a loss. Geometric mean = −1.58% per year, which correctly reproduces the ending value. CAGR is the same geometric mean calculation applied between two endpoint values.
CAGR versus XIRR: which metric to use
CAGR requires exactly one beginning value, one ending value, and one time span. Use it for business revenue, market size, or lump-sum portfolio growth. XIRR handles any number of dated cash flows — monthly SIP contributions, dividend reinvestments, or mid-period withdrawals. XIRR solves for the discount rate that sets the net present value of all dated flows to zero.
Common mistakes that produce wrong averages
Most errors come from applying an additive formula to multiplicative data, or vice versa. The result looks plausible but is numerically incorrect.
Averaging investment returns with arithmetic mean
Returns of 50% and −50% give AM = 0%, implying no change — but the actual result is a 25% loss. Use geometric mean: GM = √(1.5 × 0.5) − 1 = −13.4%.
Averaging round-trip speed with arithmetic mean
30 mph and 60 mph gives AM = 45 mph, but the actual average speed is 40 mph (harmonic mean), because more time is spent at the slower pace.
Including a zero in geometric or harmonic mean
A zero makes GM = 0 (product collapses) and HM undefined (division by zero). Treat zero returns as a factor of 1.00, not as the number 0.
Frequently asked questions
Is CAGR the same as the arithmetic average of annual returns?
No. CAGR is the geometric mean of growth multipliers minus 1. It is always less than or equal to the arithmetic average of annual percentage returns. For 6%, 7%, 8%, −35%, 10%: arithmetic average = 0.40%, CAGR = −1.58%. Only CAGR reproduces the correct ending value.
When does XIRR give a higher return than CAGR?
When later contributions are smaller or earlier withdrawals are larger, XIRR can exceed CAGR. Example: if you invest $1,000 early and only $100 later, more capital has longer to compound — XIRR reflects that timing advantage.
Why is HM ≤ GM ≤ AM for positive values?
This is the AM-GM-HM inequality, proven from the Cauchy-Schwarz inequality. For dataset 4, 9, 16, 25: HM = 8.63, GM = 10.95, AM = 13.50. The three are equal only when all values are identical.
Are the calculators on this site usable offline?
Yes. Once the page has loaded, all calculations run in your browser using JavaScript. No network connection is required to compute results.