DETERMINING STANDARD DEVIATION USING THE EMPIRICAL RULE: A CASE STUDY

In this article, we discuss a case study in which several descriptive statistical measures are calculated, including the mean, deviation, squared deviation, sample variance, standard deviation, and the empirical rule.

In descriptive statistics, there are several commonly used mathematical symbols, which are presented as follows:

symbols, meanings and uses

Let’s go straight to the case study :

Here, we have a dataset representing daily visitor attendance at an institution, consisting of 30 observations :

Visitor Data

From the data above, how many chairs are needed to accommodate 68% of visitors ?

1. Number of rows of visitor data ?

n = 30

2. Total Visitor Data Amount ?

∑xi = 193

3. Mean (Average)

x̄ = Σx / n

x̄=193/30 = 6,43333333333333

4. Calculate Deviation (dᵢ)

dᵢ = xi — x̄

dᵢ = xi — x̄

In Excel, the table above can use the following formula :

=B2-6,43333333333333

5. Calculate Square Deviation (dᵢ²)

dᵢ² = (xi​−x̄)²

dᵢ² = (xi​−x̄)²

In Excel, the table above can use the following formula :

=C2^2

Sum of Squared Deviations (Σdᵢ² = (xi​−x̄)²)= 133,366666666667

6. Calculate Sample Variance

s² = Σdᵢ² = (xi​−x̄)²/n-1

= 133,366666666667/29 = 4,59885057471264

7. Calculate Standard Deviation

s = √(s²)

s = √(4,59885057471264)

s = 2,14449308105963

In Excel, the table above can use the following formula :

=SQRT(Cell_s²)

8. Getting the Empirical Rule Value

From the question above, it is stated that the seating can accommodate 68% of the maximum. So to determine the calculation, pay attention to the following diagram :

Since we’re looking for the maximum or upper limit, we use +1. The overall formula is as follows :

x̄ ± k . s

6,43333333333333 + 1 . 2,14449308105963 = 8,57782641439297

Rounded Up = 9

Conclusion :

9 chairs are needed to accommodate 68% of visitors.

Date Created : December 15, 2025 Author : Satria Bagaskara