# Using standard deviation when distribution doesn't follow normal distribution? Chebyshev's Theorem?

I have a 36 data points of monthly sales from the past 3-years. Now I need to use those data to predict or estimate the variations of sales for one year ahead. My approach is to calculate the mean value and standard deviation of the historical data, then adjust the standard deviation with the new mean value, which is the sales target for the next year. This estimate of variations in monthly sales will serve for determining the safety stock I should keep in my inventory.

When I look at the historical data for the first time I wasn't suspecting much. However, when I tried to make a frequency histogram out of it, the data doesn't seem to fit a normal distribution bell curve at all. In fact, the data is train wreck compared to the ideal bell curve.

the histogram of one of the inventory item

My question is, can I still use standard deviation to calculate safety stock knowing that the data doesn't fit a normal distribution curve?

Standard deviation is the only way to take uncertainty into account in determining inventory policy, as far as I know:

Safety Stock = z σL

with z being the number of standard deviations to achieve a specific service level.

I was planning to write a reasoning like "Assuming the fluctuations in sales are following normal distribution curve, we can use standard deviation to calculate the amount of safety stock needed in our inventory." to justify my analysis, but now I'm confused. Looking at that histogram I can't justify the normal distribution assumption, can I?

EDIT: After some more research I found about Chebyshev's Theorem. According to my understanding, based on Chebyshev's Theorem, I can still use Standard Deviation with any non-normal distribution. However, while normal distribution has z-score, Chebyshev's Theorem have "k", which is more loose than the z-score.

For example when in normal distribution 3σ guarantees 99.86% service level, with Chebyshev's Theorem 3σ only guarantees 88.89% service level.

That way no modification is needed in safety stock formula, but the service level goes considerably lower because I have to substitute z with k. Am I right? Please correct me if I'm wrong.

• Questions: (a) You say you want to est the var in sales, but you show a non-normal graph of inventory. What is the connection btw sales and inventory? (b) If mean sales can change year-to-year, what makes you think the var of sales will be stable? (c) Are you allowed to send the 36 monthly sales numbers so we can judge what meth to use to est var? OK to move decimal pt. Best format is sales = (372, 589, ..., ..., 552). Along a row, sep. by commas; not in a col. // And remember: "Prediction is hard, especially about the future." N Bohr, A Einstein, or Yogi Berra (depending who you believe). – BruceET Sep 26 '17 at 15:57