# Statistical Sampling for Process Improvement using Python

Use sample data to estimate the average lead time to process customer orders in the customer service of an elevator parts supplier.

Use sample data to estimate the average lead time to process customer orders in the customer service of an elevator parts supplier.

*Article originally published on Medium. *

As a critical component of Supply Chain Management, ** Customer Service** is where your company gives your customers a feel of the products and the business that you are selling.

An important performance indicator is the average lead time between the reception of a customer order and its transmission to the warehouse for preparation.

In this article, we will introduce a methodology using ** statistical sampling **to

**using**

**estimate this overall average lead time****.**

**a sample of 200 observations**

**SUMMARY**

**I. Scenario**

**Problem Statement**You are the

**that produce and deliver engine parts for elevators.**

**Customer Service Manager of an elevator parts supplier**

**Question**Can you estimate the average processing time with a confidence interval of 90% using your sample data?

**II. Statistical Sampling**1. Notations

2. Application of the Central Limit Theorem

3. Confidence Interval

4. Final estimation

**III. Conclusion**## I. Scenario

Problem Statement

You are the ** Customer Service Manager of an elevator parts supplier **that produces and delivers engine parts for elevators.

Your team is in charge of order processing:

- A Customer sends an order by phone or email with a requested delivery time
*(e.g: Customer order 5 units of SKU X and would like to be delivered the same day at 10:00)* - Your team confirms the order and allocates it to the closest warehouse for preparation and shipment.
- The order is prepared and shipped from the warehouse using an express courier company.

You recently received many complaints from your customers because of late deliveries. According to the warehouse manager, it is mainly due to the delays during order processing by customer service.

During 3 months, you measured the order processing time of randomly selected operators and gathered 200 observations.

### Question

Can you estimate the average processing time with a confidence interval of 90% using your sample data?

## II. Statistical Sampling

As we are not able to measure the average processing time of all your operators for every order we would like to estimate the total population average using these sample records.

### 1. Notations

To simplify the comprehension, letâ€™s introduce some notations:

### 2. Application of the Central Limit Theorem

In a previous article (Link), we have been using the Central Limit Theorem (CLT) to estimate the probability of a random variable P(Xâ‰¥k) assuming that X was following a normal distribution.

The CLT also tells us:

### 3. Confidence Interval

Our objective is to know the population mean a range [Âµ-b, Âµ+b] with a confidence of 90%.

And we know by the construction of the unit normal distribution that for P(-zâ‰¤Zâ‰¤z) = 0.9 we have z = 1.64

Finally, we can get our estimate range or the population mean

### 4. Final estimation

mean 22.705

std 6.81

min 4.0

25% 18.0

50% 23.0

75% 27.0

max 41.0

We have,

xÌ„ = 22.705 (min)

s = 6.81 (min)

The confidence interval is [21.96, 23.54]

## III. Conclusion

For a confidence level of 90% and with a moderate effort of experimentation, we have a very good estimation of the average lead time for order processing.

This approach can be used when process performance measurement is expensive and take effort and time.

However, you need to put effort into the experimental protocol to ensure that your sample data have been built based on a random selection of operators.

## References

[1] Central Limit Theorem for Process Improvement with Python, Samir Saci