1. Descriptive Intervals

2. Computation
First, edit and summarise the data. Obtain the sample mean (m) and sample standard deviation(sd).
Compute the “short interval” as lower = m-2*sd
And upper = m + 2*sd. Write the interval as [lower,upper].
Compute the “long interval” as lower = m-3*sd
And upper = m + 3*sd. Write the interval as [lower,upper].

3. Interpretive Base – Tchebysheff’s Inequalities
At least 75% of the sample points reside within the short interval…
At least 89% of the sample points reside within the long interval…
In general, at least (1-(1/k2))*100% of the sample points reside within the interval [m-k*sd,m+k*sd].

4. The Bell Curve Assumption
In probability we have a family of populations that follow a Gaussian or Bell Curve Assumption.
These populations have a super-majority of members residing “near” a central value, with population density declining symmetrically as the distance from the central value grows. If one plots population density versus location, the resulting shape resembles a bell.

5. The Gaussian distribution
When many independent random factors act in an additive manner to create variability, data will follow a bell-shaped distribution called the Gaussian distribution, illustrated in the figure below. The left panel shows the distribution of a large sample of data. Each value is shown as a dot, with the points moved horizontally to avoid too much overlap. This is called a column scatter graph. The frequency distribution, or histogram, of the values is shown in the middle panel. It shows the exact distribution of values in this particular sample. The right panel shows an ideal Gaussian distribution.
Ref. link =

6. Center
Center
Center

7. Interpretive Base – Empirical Rule
When the data for our intervals come from a “bell-shaped” population, then:
Approximately 95% of the sample points reside within the short interval…
Approximately 100% of the sample points reside within the long interval…