LoadingLoading
The rule of thumb, also known as the three sigma rule or the 68-95-99.7 rule, is a statistical rule that states that in a normal distribution, almost all observed data will fall within ± three standard deviations (denoted by the Greek letter σ) of the mean ...
The rule of thumb, also known as the three sigma rule or the 68-95-99.7 rule, is a statistical rule that states that in a normal distribution, almost all observed data will fall within ± three standard deviations (denoted by the Greek letter σ) of the mean (denoted by the Greek letter μ).
Specifically, the rule of thumb predicts that in a normal distribution, 68% of observations will fall within the first standard deviation (μ ± σ), 95% will fall within the first two standard deviations (μ ± 2σ), and 99.7% will fall within the first three standard deviations (μ ± 3σ).
Rules of thumb are often used in statistics to predict final outcomes. After the standard deviation has been calculated, but before the complete data is collected, this rule can be used as a rough estimate of the results of the data that will be collected and analyzed.
This probability distribution can serve as an evaluation technique because in some cases collecting appropriate data may be time-consuming or even impossible. Such considerations are particularly important when a company reviews its quality control measures or assesses risk exposure. For example, the commonly used risk tool "Value at Risk (VaR)" assumes that the probability distribution of risk events is a normal distribution.
Rules of thumb are also used as a rough way to test the "normality" of a distribution. If too many data points fall outside the boundary of three standard deviations, it indicates that the distribution is not normal, may be skewed, or follow another distribution.
The rule of thumb is also known as the three sigma rule because "three sigma" refers to the statistical distribution of data with mean ± three standard deviations in a normal distribution (bell curve), as shown in the figure below.
Assume that the life spans of animals in a zoo are normally distributed. The average lifespan of each animal is 13.1 years (mean), and the standard deviation of lifespan is 1.5 years. If someone wanted to know the probability of an animal living past 14.6 years, they could use a rule of thumb. Knowing that the mean of the distribution is 13.1 years old, the following are the age ranges corresponding to each standard deviation:
Someone solving this problem would need to calculate the overall probability that an animal would live to be 14.6 years old or older. A rule of thumb shows that 68% of the distribution lies within one standard deviation, which is 11.6 to 14.6 years old. Therefore, the remaining 32% of the distribution lies outside this range. Half were over 14.6 years old and the other half were under 11.6 years old. Therefore, the probability of an animal living past 14.6 years is 16% (calculated as 32% divided by 2).
Most market data is not normally distributed, so the 68-95-99.7 rule generally does not apply to investing. However, many analysts still use some aspects of it - such as standard deviation - to estimate volatility.
You can calculate the standard deviation of a portfolio, index, or other investment and use it to evaluate volatility. Calculating the standard deviation of an investment is simple if you have access to a spreadsheet and the prices or returns of the selected investments.
Market analysts generally express standard deviation as a percentage. For example, from May 2 to June 2, 2023, the daily standard deviation (annualized) of the S&P 500 (using daily closing prices) was 13.29%.
In a spreadsheet, you can paste returns, prices, or values, calculate the percent change from the previous trading day, and use the standard deviation function: 1
IMPORTANT: Using more than one month of transaction data, such as three years or more, will give more accurate results. The following example uses the daily values of the index over a month and annualizes the standard deviation to limit the table size.
To annualize the standard deviation, multiply it by the square root of the number of trading days in a yearusually 252.
| S&P 500 Standard Deviation (Annualized) |---|---|---|---| ||A|B|C|D |1|Date|Closing|Day Change|Formula |2|05/02/23|4119.58|-|- |3|05/03/23|4090.75|-0.70%|- |4|05/04/23|4061.22|-0.72%|- |5|05/05/23|4136.25|1.85%|- |6|05/08/23|4138.12|0.05%|- |7|05/09/23|4119.17|-0.46%|- |8|05/10/23|4137.64|0.45%|- |9|05/11/23|4130.62|-0.17%|- |10|05/12/23|4124.08|-0.16%|- |11|05/15/23|4136.28|0.30%|- |12|05/16/23|4109.9|-0.64%|- |13|05/17/23|4158.77|1.19%|- |14|05/18/23|4198.05|0.94%|- |15|05/19/23|4191.98|-0.14%|- |16|05/22/23|4192.63|0.02%|- |17|05/23/23|4145.58|-1.12%|- |18|05/24/23|4115.24|-0.73%|- |19|05/25/23|4151.28|0.88%|- |20|05/26/23|4205.45|1.30%|- |21|05/30/23|4205.52|0.00%|- |22|05/31/23|4179.83|-0.61%|- |23|06/01/23|4221.02|0.99%|- |24|06/02/23|4282.37|1.45%|- |25|-|Daily standard deviation|0.84%|=stdev(B2:B24) |26|-|Annualized standard deviation|13.29%|=sqrt(252)*C25
Therefore, the annualized volatility based on the data in the table is 13.29%. The higher the standard deviation, the riskier analysts believe the investment is.
Alternatively, you can find the standard deviation of an investment on popular investment websites. For example, Morningstar displays the three-, five-, and 10-year standard deviations for the S&P 500. [2]
In statistics, the rule of thumb states that in a normal distribution, 99.7% of observations will fall within ± three standard deviations of the mean. Specifically, 68% of observations will be within one standard deviation, 95% will be within two standard deviations, and 97.5% will be within three standard deviations.
Rules of thumb should be used to predict possible outcomes in a normal distribution. For example, statisticians can use it to estimate the percentage of cases within each standard deviation range. Assume the standard deviation is 3.1 and the mean is 10. In this case, the first standard deviation will range between (10+3.2)=13.2 and (10-3.2)=6.8. The second standard deviation will be between 10 + (2 × 3.2) = 16.4 and 10 - (2 × 3.2) = 3.6, and so on.
The benefit of a rule of thumb is that it can be used as a means of predicting data. This is especially important when working with large data sets and data with unknown variables.
Analysts use rules of thumb to observe how much data in a data set falls within a specific interval from the mean. Investment analysts can use it to estimate the volatility of a specific investment, portfolio, or fund.
[1] Google Docs Editors Help. "STDEV."
[2] Morningstar. "S&P 500 PR."
TheQuantHackers is a cutting-edge community composed of a group of people from the world's top universities and quantitative finance practitioners, dedicated to exploring the infinite possibilities in the fields of artificial intelligence (AI) and quantification (Quant). Our team members come from world-renowned universities such as the University of Cambridge, Oxford University, Harvard University, ETH Zurich, Peking University, and the University of Science and Technology of China. Our external consultants come from first-class companies such as Microsoft, HSBC, Citadel, Man Group, Citi, Jump Trading, and top domestic private equity companies.
Upgrade to unlock all institutional-grade algorithms, derivative pricing engines, factor backtesting frameworks, and live QuantLab execution.
Explore Membership Access
Pre-order · £490Quantitative analytics platform
Waitlist closed — signal locked. Pre-order TQH TERMINAL now at £490 with launch-day delivery.