# Service Level Z-Score and Loss Function Table

> Source: https://netray.co/tools/service-level-z-score-and-loss-function-table
> Publisher: Netray (netray.co), AI and integration engineering for ERP and manufacturing systems.
> Section: Free Tools

## Short answer

Look up the z-score and standard normal loss function value for any cycle service level, for safety stock and fill rate calculations.

## Category

supply-chain

## Audience

- Materials planner
- Demand planner
- Supply chain analyst

## Tool

**Kind:** worksheet

**Mode:** reference

**Columns:**

#### Cycle service level

**Key:** svc

**Type:** text

#### Z score

**Key:** z

**Type:** text

#### Loss function L(z)

**Key:** lz

**Type:** text

#### Typical use

**Key:** note

**Type:** text

**Rows:**

**Svc:** 50%

**Z:** 0.0000

**Lz:** 0.3989

**Note:** No safety stock, order at expected demand

**Svc:** 60%

**Z:** 0.2533

**Lz:** 0.2850

**Note:** Very low priority parts

**Svc:** 70%

**Z:** 0.5244

**Lz:** 0.1904

**Note:** Low value, easily substituted items

**Svc:** 75%

**Z:** 0.6745

**Lz:** 0.1492

**Note:** Low criticality C items

**Svc:** 80%

**Z:** 0.8416

**Lz:** 0.1116

**Note:** Common for C items

**Svc:** 85%

**Z:** 1.0364

**Lz:** 0.0777

**Note:** Moderate service target

**Svc:** 88%

**Z:** 1.1750

**Lz:** 0.0590

**Note:** Between B and C targets

**Svc:** 90%

**Z:** 1.2816

**Lz:** 0.0473

**Note:** Common for B items

**Svc:** 92%

**Z:** 1.4051

**Lz:** 0.0363

**Note:** Higher B items

**Svc:** 93%

**Z:** 1.4758

**Lz:** 0.0310

**Note:** Higher B items

**Svc:** 94%

**Z:** 1.5548

**Lz:** 0.0258

**Note:** Between B and A targets

**Svc:** 95%

**Z:** 1.6449

**Lz:** 0.0209

**Note:** Common default for A items

**Svc:** 96%

**Z:** 1.7507

**Lz:** 0.0161

**Note:** Important A items

**Svc:** 97%

**Z:** 1.8808

**Lz:** 0.0116

**Note:** Important A items

**Svc:** 98%

**Z:** 2.0537

**Lz:** 0.0073

**Note:** Critical items

**Svc:** 99%

**Z:** 2.3263

**Lz:** 0.0034

**Note:** Critical or line-down parts

**Svc:** 99.5%

**Z:** 2.5758

**Lz:** 0.0016

**Note:** Very critical parts, check cost of buffer

**Svc:** 99.7%

**Z:** 2.7478

**Lz:** 0.0009

**Note:** Rarely justified, check carrying cost

**Svc:** 99.9%

**Z:** 3.0902

**Lz:** 0.0003

**Note:** Extreme target, buffer grows quickly

**Notes:** Safety stock = z x sigma_L, where sigma_L is the standard deviation of demand over the lead time (or review plus lead time). Expected units short per cycle = sigma_L x L(z). Fill rate = 1 - sigma_L x L(z) / Q for order quantity Q. Values assume normally distributed demand and are rounded to four decimals.

## Content

**Intro:** Every safety stock formula starts with a z-score for the service level you want. This table gives the z-score and the loss function value, so you can also convert between cycle service level and fill rate without a spreadsheet.

**Sections:**

#### How to use the table

**Body:** Search for your target service level and read the z-score. Multiply by the standard deviation of demand over the lead time to get safety stock. To estimate fill rate, take the loss function value L(z) in the same row, multiply by that standard deviation to get expected units short per cycle, divide by order quantity and subtract from one. Use the notes column as a prompt for choosing targets by class, not as a rule.

#### Cycle service level versus fill rate

**Body:** Cycle service level is the probability of getting through a replenishment cycle without a stockout. Fill rate is the share of demand served from stock. A 95% cycle service level on an item ordered in large lots can mean a fill rate well above 99%, because stockouts only occur at the end of the cycle and are small relative to the lot. Mixing the two leads to buffers that are too large. State which one your policy uses.

**Bullets:**

- Cycle service level uses z
- Fill rate uses the loss function L(z)
- Larger order quantities raise fill rate at the same z

#### Worked example

**Body:** Demand over the lead time has a standard deviation of 120 units and the order quantity is 1,000. At a 95% cycle service level, z is 1.6449 and safety stock is about 197 units. L(z) is 0.0209, so expected shortage per cycle is 120 x 0.0209, about 2.5 units, and the fill rate is 1 - 2.5 / 1,000, about 99.75%. Moving to 99% adds 77 units of safety stock for a gain of about 0.2 points of fill rate, a figure worth weighing against carrying cost.

#### Limits of the table

**Body:** The values assume a normal distribution, which suits steady demand with a reasonably high average. For low volume, intermittent or highly skewed demand, a normal approximation can understate the risk of a long stockout or give negative demand tails. In those cases use a Poisson, negative binomial or empirical method and treat these numbers as a rough starting point only.

**Cta:** If you want service targets set by item class and the resulting safety stock checked against what is loaded in your ERP, Netray can build that review over your ERP data, read-only.

**Faq:**

#### What z-score do I use for 95% service level?

**Answer:** Use 1.6449 for a one-sided 95% cycle service level, usually rounded to 1.65. This is the value most inventory formulas need, not the 1.96 used for two-sided confidence intervals.

#### Why does 99.9% cost so much more than 99%?

**Answer:** Safety stock rises with z, and z grows faster as you approach 100%. Going from 99% to 99.9% moves z from 2.33 to 3.09, about 33% more buffer for a tiny gain in service. Weigh it against the cost of a stockout.

#### What is the loss function used for?

**Answer:** It gives the expected shortage per replenishment cycle in standard deviations. Multiplying by the demand standard deviation yields expected units short, which feeds fill rate and stockout cost estimates.

#### Can I use these values for monthly forecast error?

**Answer:** Yes, if you use the standard deviation of forecast error over the lead time plus review period, not the standard deviation of demand itself. Make sure the units and time basis match.

## Search terms

service level z score table, safety stock z value 95%, z score for 99 service level, standard normal loss function table, fill rate vs service level, expected units short formula, safety stock z table, cycle service level vs fill rate

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