Empirical Rule

An empirical rule in statistics states almost all (95%) of the observations in a normal distribution lie within three standard deviations from the mean. It is a very important rule and helps in forecasting. However, one should always note that these are approximations. There are always chances of outliers that don’t fall in the distribution.

It is a statistical concept that helps portray the probability of observations and is very useful when finding an approximation of a huge population. So, the findings are not accurate and precautionary measures should be taken when acting as per the forecast.

The empirical rule helps estimate the outcome and assess the extent to which the same would vary. It helps to have three levels of standard deviation to check the expected variations in the estimated outcome.

Data is like gold in the modern world. Huge data flow from different sources is used for different approximations or forecasts. For example, if a data set follows a normal distribution, it shows a bell shaped curve. Then, one can use the empirical rule. Finally, it is applied to observations to create an approximation for the population.

Once the observations show a normal distribution structure, one may follow an empirical rule to find several observations' probabilities. The rule is extremely useful for many statistical forecasts.

The formula shows the predicted percentage of observations within each standard deviation from the mean.

The empirical rule formula works on the following specifications:

• 68% of the observations will lie within +/- 1 standard deviation from the mean
• 95% of the observations will lie within +/- 2 standard deviations from the mean
• 99.7% of the observations will lie within +/- 3 standard deviations from the mean

Let us check the example below to see how to calculate empirical rule possibilities:

Mr. X is trying to find the average number of years a person survives after retirement, considering the retirement age to be 60. If the mean survival years of 50 random observations are 20 years and SD is 3, then determine the probability that a person will draw a pension for more than 23 years.

Solution

The empirical rule states that 68% of the observations will lie within 1 standard deviation from the mean. Here, the mean of the observations is 20.

68% of the observations will lie within 20 +/- 1 (standard deviation), which is 20 +/- 3. So, the range is 17 to 23.

There is a 68% chance that the minimum number of years a person survives after retirement lies between 17 to 23. The percentage lying outside this range is (100 – 68) = 32%. Therefore, 32 is distributed equally on both sides, which means a 16% chance that the minimum years will be below 17 and a 16% chance that the minimum years will be greater than 23.

So, the probability that the person will draw more than 23 years of pension is 16%.

One may use it in the forecasting trend of a data set. When the data set is extensive, it gets challenging to study the entire population. For example, one can apply the empirical rule to the sample to get an estimation of how the data in the population will react if asked to find the average salary of all the accountants in the US. Then that is a difficult task to perform as the population set is enormous. So, in that case, you can select, say, 90 observations randomly from the entire population.

So now you will have 90 salaries. Next, you need to find the mean and standard deviation of the observations. If the observation follows a normal distribution, one can apply this, and one can estimate the salary of all accountants in the US.

Say the mean salary of the sample comes out to be $90,000. And the standard deviation is $5,000. Out of the entire population, 68% of the accountants are drawing a salary ranging between +/- 1 standard deviation from the mean. The mean is $90,000, and the standard deviation is $5,000. So as per the empirical rule in a normal distribution, 68% of all the accountants in the US paid in the range of $90,000 +/- (1*$5,000). That is within $85,000 to $95,000.

If we spread a bit more, then 95% of all the accountants in the US are being paid in the range of mean +/- 2 standard deviations. $90,000 +/- (2*5000). So the range is $80,000 to $100,000.

In a broader range, 99.7% of all accountants draw salaries ranging from mean +/- 3 standard deviations. That is 90,000 +/- (3*5000). The range is $75,000 to $105,000.

You can see that without studying the entire population, one could estimate the population. For example, if someone plans to work as an accountant in the US, he can easily expect his salary to range from $75,000 to $105,000.

This kind of estimation helps to ease work and make forecasts regarding the future.

The empirical rule applies to data sets that follow a normal distribution. That means bell-shaped. In a normal distribution, both sides have a 50% probability each.

Chebyshev's theorem applies another approximation or rule to all types of data sets if the data set is distributed not normally. It says three things:

• At least 3/4^th of all the observations will lie within 2 standard deviations from the mean. It is a strong approximation. For example, it means if there are 100 observations, then 3/4^th of the observations that are 75 observations will lie within +/- 2 standard deviations from the mean.
• At least 8/9^th of all observations will lie within 3 standard deviations from the mean.
• At least 1 – 1/k^2 of all the observations lie within K standard deviations from the mean. Here, K is any whole number.