The Probability That a Measurement Falls within a Range of 𝑛 Standard Deviations from an Estimate of the Mean

Abstract

We derive a general equation for the probability that a measurement falls within a range of n standard deviations from an estimate of the mean. So, we provide a format that is compatible with a confidence interval centered about the mean that is naturally independent of the sample size. The equation is derived by interpolating theoretical results for extreme sample sizes. The intermediate value of the equation is confirmed with a computational test.

1. Introduction

A confidence interval is an interval in which a measurement or trial falls corresponding to a given probability [1, 2]. In statistics, confidence intervals centered about an estimate of the mean target the mean [1]. Because these confidence intervals rely on the standard error of the estimate, they increase as sample size decreases and they decrease as sample size increases. Consequently, the confidence interval, in statistics, is often a margin of error for the estimate of the mean [3].

We are interested in finding the probability of a confidence interval centered about an estimate of the mean that targets an arbitrary measurement and is independent of the sample size. In this way, we provide a format that is compatible with a confidence interval centered about the mean that is naturally independent of sample size, but has a width of 𝑛 standard deviations. We approach this problem by considering the known case of a confidence interval centered about the mean and we derive the associated probability. We correlate this result to the case of infinite sample size. The next step is to calculate the probability associated with the minimum sample size of one. This requires that we average the probability over different possible sample values. Finally, we propose an equation for probability that naturally interpolates our results and we show that the equation is consistent with intermediate probability values by comparing the equation to estimates of probability produced by computer simulation.

2. Expected Probability

In order to make the necessary confidence interval calculations, we will have to determine an expected or average probability. This can be examined by considering the cumulative distribution function [4, 5], 𝐹𝑥;𝜇,𝜎²=1√2𝜋𝜎^𝑥−∞𝑒^{−(𝑡−𝜇)2/2𝜎2}𝑑𝑡, (2.1) where 𝑥 is a measurement, 𝜇 is the mean, and 𝜎 is the standard deviation.

Let 𝑣=(𝑡−𝜇)/√2𝜎. So, 𝑑𝑣=𝑑𝑡/√2𝜎. Therefore, (2.1) becomes 𝐹𝑥;𝜇,𝜎²=1√𝜋^{√(𝑥−𝜇)/2𝜎}_−∞𝑒^−𝑣2=1𝑑𝑣√𝜋⁰_−∞𝑒^−𝑣21𝑑𝑣+√𝜋^{√(𝑥−𝜇)/2𝜎}_−∞𝑒^−𝑣2=1𝑑𝑣2+12erf𝑥−𝜇√,2𝜎 (2.2) where erf(𝑥) is the so-called erf function [6, 7].

Now, we know that 𝐹𝑥;𝜇,𝜎²=𝑃(𝑋≤𝑥). (2.3) Rather than specifying a value of 𝑥, we would like to compute the probability averaged over all possible values of 𝑥. So we have 𝐸𝐹𝑥;𝜇,𝜎²=𝐸(𝑃(𝑋≤𝑥)). (2.4) This yields 𝐸𝐹𝑥;𝜇,𝜎²=1√2𝜋𝜎^∞−∞12𝑒^{−(𝑥−𝜇)2/2𝜎2}1𝑑𝑥+√2𝜋𝜎^∞−∞12erf𝑥−𝜇√𝑒2𝜎^{−(𝑥−𝜇)2/2𝜎2}𝑑𝑥=12+1√2𝜋𝜎^∞−∞12erf𝑥−𝜇√𝑒2𝜎^{−(𝑥−𝜇)2/2𝜎2}𝑑𝑥. (2.5)(2.6) Let 𝑧=(𝑥−𝜇)/√2𝜎. So, 𝑑𝑧=𝑑𝑥/√2𝜎. Therefore, (2.6) becomes 𝐸𝐹𝑥;𝜇,𝜎²=12+12√𝜋^∞_−∞erf(𝑧)𝑒^−𝑧2𝑑𝑧. (2.7) Since erf(𝑧) is an odd function, we have 𝐸𝐹𝑥;𝜇,𝜎²=12. (2.8) This can be easily confirmed by an alternative calculation. The probability density 𝑓(𝑥) is 𝑓(𝑥)=1√𝑒2𝜋𝜎^{−(𝑥−𝜇)2/2𝜎2}. (2.9) Therefore, we can write [8] 𝑓=𝑑𝐹.𝑑𝑥 (2.10) The expected value [9] of the cumulative distribution function is thus 𝐸(𝐹)=^∞_−∞𝐹𝑓𝑑𝑥. (2.11) Note that when 𝑥→∞, 𝐹=1 and when 𝑥→−∞, 𝐹=0. Therefore, we can write 𝐸(𝐹)=¹₀=1𝐹𝑑𝐹2. (2.12) Consequently, as per (2.4), we can write the expected probability as 𝐸(𝑃(=1𝑋≤𝑥))2. (2.13)

3. The Probability of a Confidence Interval

We want to determine the probability given as 𝑃(𝑥−𝑛𝜎≤𝑥≤𝑥+𝑛𝜎), (3.1) where 1𝑥=𝑁^𝑁_𝑖=1𝑥_𝑖 (3.2) is the estimate of the mean, 𝜇. At the maximal value of 𝑁, we find that lim_𝑁→∞1𝑁^𝑁_𝑖=1𝑥_𝑖≈𝜇. (3.3) For a normal distribution, the probability that a measurement falls within 𝑛 standard deviations (𝑛𝜎) of the mean 𝜇 (i.e., within the interval [𝜇−𝑛𝜎,𝜇+𝑛𝜎]) is given by 𝑃=1𝜇−𝑛𝜎≤𝑥≤𝜇+𝑛𝜎𝜎√2𝜋^{𝜇+𝑛𝜎𝜇−𝑛𝜎}𝑒^{−(𝑥−𝜇)2/(2𝜎2)}=2𝑑𝑥𝜎√2𝜋^{𝜇𝜇+𝑛𝜎}𝑒^{−(𝑥−𝜇)2/(2𝜎2)}𝑑𝑥. (3.4) Now, let 𝑢=(𝑥−𝜇)/√2𝜎, so 𝑑𝑢=𝑑𝑥/√2𝜎. Then 𝑃=2𝜇−𝑛𝜎≤𝑥≤𝜇+𝑛𝜎𝜎√√2𝜋2𝜎^√𝑛/2₀𝑒^−𝑢2𝑑𝑢=2√𝜋^√𝑛/2₀𝑒^−𝑢2𝑑𝑢𝑛=erf√2. (3.5)(3.6)(3.7) Observe that we can express (3.7) in terms of the cumulative probability distribution function 𝐹. Thus, we can write 𝑃𝜇−𝑛𝜎≤𝑥≤𝜇+𝑛𝜎=𝐹𝜇+𝑛𝜎;𝜇,𝜎²−𝐹𝜇−𝑛𝜎;𝜇,𝜎². (3.8) The minimal value of 𝑁 is one, so that 𝑥=𝑥₁.

In this case, we have 𝑃𝑥₁−𝑛𝜎≤𝑥≤𝑥₁+𝑛𝜎=𝐹𝑥₁+𝑛𝜎;𝜇,𝜎²−𝐹𝑥₁−𝑛𝜎;𝜇,𝜎². (3.9) We find that (3.9) reduces into 𝑃𝑥₁−𝑛𝜎≤𝑥≤𝑥₁=1+𝑛𝜎2𝑥erf¹+𝑛𝜎−𝜇𝜎√2+12erf𝑛𝜎+𝜇−𝑥₁𝜎√2. (3.10) The probability in (3.10) is written in terms of the value 𝑥₁. Consequently, we can derive an expected probability by computing 𝐸_𝑥1(𝑃)=1𝜎√2𝜋^∞−∞12𝑥erf¹+𝑛𝜎−𝜇𝜎√2+12erf𝑛𝜎+𝜇−𝑥₁𝜎√2𝑒^{−(𝑥1−𝜇)2/2𝜎2}𝑑𝑥₁=1𝜎√2𝜋^∞−∞12𝑥erf¹−𝜇𝜎√2+𝑛√2+12𝑛erf√2−𝑥₁−𝜇𝜎√2𝑒^{−(𝑥1−𝜇)2/2𝜎2}𝑑𝑥₁. (3.11)(3.12) Let 𝑧=(𝑥₁−𝜇)/𝜎√2 which implies that 𝑑𝑧=𝑑𝑥₁/𝜎√2. This reduces (3.12) into 𝐸_𝑥1(𝑃)=1√𝜋^∞_−∞12𝑛erf𝑧+√2+12𝑛erf√2−𝑧𝑒^−𝑧2𝑑𝑧=1√𝜋^∞_−∞𝑛erf𝑧+√2𝑒^−𝑧2𝑑𝑧. (3.13)(3.14) Equation (3.14) can be evaluated numerically to yield the following: 𝐸_𝑥1𝑃𝑥₁−𝑛𝜎≤𝑥≤𝑥₁𝑛+𝑛𝜎=erf2. (3.15) A portion of the numerical results is presented in Table 1.

Table 1 

Comparisons of numerical integration of (1/√𝜋)∫^∞_−∞erf(𝑧+(𝑛/√2))𝑒^−𝑧2𝑑𝑧 and erf(𝑛/2).

Based on (3.7) and (3.15), we can propose the equation 𝑃(𝑥−𝑛𝜎≤𝑥≤)𝑛𝑥+𝑛𝜎=erf√2√2^1/𝑁, (3.16) in which we are by default referring to the expected probability.

Equation (3.16) clearly converges for the extreme estimates of the mean, since lim_𝑁→∞𝑛erf√2√2^1/𝑁𝑛=erf√2,lim_𝑁→1𝑛erf√2√2^1/𝑁𝑛=erf2. (3.17) A plot of (3.16) for four values of 𝑁 is shown in Figure 1. Observe that the curve for 𝑁=2 is intermediate to the other curves. Consequently, we can have confidence in (3.16) if we can show that it is valid for 𝑁=2.

Figure 1 

A plot of the equation 𝑃(𝑥−𝑛𝜎≤𝑥≤𝑥+𝑛𝜎)=erf(𝑛/√2√21/𝑁) for four different values of 𝑁.

4. Computational Test

We can estimate 𝑃(𝑥−𝑛𝜎≤𝑥≤𝑥+𝑛𝜎) computationally. Simulate the normal, independent random variables 𝑋 and {𝑋_𝑖}, for which 𝑥∈𝑋,𝑥_𝑖∈𝑋_𝑖. (4.1) Let the condition 𝛽 be 𝛽∶(𝑥−𝑛𝜎≤𝑥≤𝑥+𝑛𝜎). (4.2) If 𝑀(𝛽) is the number of trials in which the condition 𝛽 is met and 𝑀 is the total number of trials, then an estimate of 𝑃 is given as 𝑀𝛽𝑃=𝑀. (4.3) Figure 2 shows a plot of 𝑃 versus 𝑃 for 𝑁=2 and 𝑀=2000.

Figure 2 

A plot of 𝑃 versus 𝑃 for 𝑁=2 and 𝑀=2000.

5. Conclusions

We derive a general equation for the probability that a measurement falls within a range of 𝑛 standard deviations from an estimate of the mean. So, we provide a format that is compatible with a confidence interval centered about the mean that is naturally independent of the sample size. It is consistent with our equation that probability reduces with sample size. However, for samples greater than ten, the value of probability begins to converge. The equation for probability is derived by considering the minimal and maximal sample sizes and producing an equation which naturally interpolates the results. Computer simulation is used to estimate probability for the sample size 𝑁=2 that produces intermediate results that are in strong agreement with the general equation.

Acknowledgment

Discussions with Maxwell Lueckenhoff are appreciated.