To sum up, Tolerance stackup analysis is a very important part of product design. It also helps in reducing part cost. Various tolerance stackup analysis methods are available.

If you look at the results from Worst case and RSS method. In the RSS method, the tolerance band is reduced. Therefore designers can give more flexibility to manufacturers.

Production of closely mating parts without tolerances is economically unfeasible. Tight tolerances can result in an increased rejection rate and manufacturing cost. Whereas loose tolerances can affect product function. Therefore it is always recommended to define optimized tolerances. Part tolerance optimization increases the part tolerance band. It also helps in reducing part manufacturing costs.

Various tolerance stack up analysis methods are used to calculate optimized tolerance values. Each method has its own advantages, limitation, and application. Out of these Worst Case and Statistical Tolerance Analysis methods are more popular. Let’s discuss these methods in detail.

Statistical tolerance analysis is based on the Bell curve and normal distribution. For example, when a thousand disks of the same size are manufactured. Disk thickness will vary from upper limit to lower limit. But all disks will not measure the same because their thickness will be distributed.

®/™ Mastercard, World Mastercard, World Elite and the circles are registered trademarks of Mastercard International Incorporated.

Let’s consider an example of five different size disks with different tolerances stacked on one another. We will calculate the overall maximum and minimum total stacked disk height using Worst Case Tolerance Stackup Analysis method.

*NO PURCHASE NECESSARY. Contest begins at 12:00 p.m. Eastern Time (“ET”) on September 19, 2024 and ends at 11:59 p.m. ET on October 3, 2024 after which time the Contest will be closed and no further entries shall be accepted. Open to legal residents of Canada who are a registered Triangle® Loyalty member and who are 18 years old or older. There is one grand prize that consists of a trip for two to attend Game 3 and Game 4 of the MLB World Series with a total approximate retail value (“ARV”) of $25,449.96CAD and one secondary prize consist of a PWR POD gift basket each with an ARV of $449.96 CAD . Winners will be selected through a random electric draw on October 4, 2024. Odds of winning depend upon total number of eligible entries received. Correct answer to mathematical skill-testing question required. Visit

If I calculate the total deviation (not std deviation) I get 1.4832… that is +- 0.74162 (the same as in your calculation). My formula that I used: (0,5+0,5)^2+(0,1+0,1)^2 .. (0,4+0,4)^2)^-0,5 ~ 1.48324.

Tolerance Stackup analysis is a design tool used to analyse and optimize product design for assembly. It is used to calculate the cumulative effects of part tolerances in an assembly. Therefore tolerance stackup analysis ensures smooth part assembly on the production floor. In this article, we will discuss the worst case and RSS methods to solve tolerance stack up problems.

Worst Case tolerance stackup analysis method utilizes simple arithmetic (addition and subtraction) operations to calculate optimized tolerances. In this method, all dimensions are assumed at the extreme limit. For low production volumes, worst case tolerance analysis is recommended.

One question is troubling me. “Standard deviation is calculated by considering manufacturing process is 3 sigma capable.” -> Std deviation = total tolerance/ (2×3). This one I understand.

in above formula denominator (6) defines the distribution of dimensions. Denominator “6” indicates process is 3σ capable with cpk value equal to 1. If your manufacturing process is 6σ capable value ‘6’ need to be changes to 12.

Informative log Clears my all confusions of tolarrlence stack up analysis… Language used is very easy to understand and satal.

The first step in doing tolerance stackup analysis is to create the dimensional chain. It is used to determine the direction of tolerance. In the above example, all dimensions are in a positive direction.

Why do we need to determine the process capability index in the first place? Is the answer that first we need to determine the manufacturing process capability (e.g. 3 sigma) – and second we need to determine the process capability for the assembly “event” (that can differ from 3 sigma)?

*The Triangle Rewards Program is owned and operated by Canadian Tire Corporation, Limited. The Triangle® credit cards are issued by Canadian Tire Bank. Rewards are in the form of electronic Canadian Tire Money® (CT Money®). To collect bonus CT Money you must present a Triangle Rewards card/key fob, or use any approved Cardless method, at time of purchase or pay with a Triangle credit card. You cannot collect paper CT Money on bonus offers. Minimum $150 spend is on the pre-tax amount on the same tender and for a single transaction. $150 must be spent on items that earn CT Money. Not all items sold at Canadian Tire are eligible to earn CT Money or to be redeemed for. Conditions apply. Visit triangle.com for full program rules and Partner location information. The offered rate is exclusive of any bonus or promotional offers or redemption transactions. CT Money is collected on the pre-tax amount of the purchase. Bonus CT Money may take up to 5 weeks to be applied to the Member’s Triangle Rewards Account. No rainchecks. Single use only per Triangle Rewards Account.

Let’s consider an example of five different size disks with different tolerances stacked on one another. We will calculate the overall maximum and minimum total stacked disk height using RSS Tolerance Stack up Analysis method. RSS tolerance stackup analysis for a given problem can be done in the following steps:

For a good understanding of tolerance stack up. We suggest you first read this article on What are Limit, Fits, and Tolerance?

I do not have much knowledge of the sampling theory approach. But my understanding is we can use it since it is based on statics only.

Root sum square (RSS) tolerance stack up analysis works on a statistical approach. It is assumed that most of the parts fall to the middle of the tolerance zone.

Because otherwise, if the there is only one and the same process capability value to be used after it’s chosen – there is no need the include it to the calculation. Or am I missing something?

You mentioned in your calculation the direction of the stack. if you have a 6th step, lets say an etching step, that removed a portion of the last disk, would you include the statistical tolerance calculation of that 6th subtractive step into the same way you did for the RSS of the first 5? or would you RSS the 5 additive steps and then get a tolerance analysis done, followed by an RSS of the 5 steps with the subtractive step?

Sir, Can you help in calculate pin diameter and their distance tolerance for manufacturing tolerance for gauge design ? Assuming three holes are diameter 10+/- 0.1,diameter 15+/- 0.2 and diameter 25+/-.1 these holes distance is 120+/-1,100+/-0.5 and 80+/-1.5 respectively. Please share with formula to my Email

In RSS, the tolerace band is reduced. How it will help the supplier to produce parts. reduced tolerance band is the constraint for supplier, right?

Your spreadsheet is incorrect. For – 1.05 +0/-.01, the two values that are calculated are -1.06 and -1.05. The two values should be -1.05 and -1.04.

Hi, I like using your stack up calculator spread sheet. Can you share the password so I can name my project? Thanks Sean

In this example, maximum and minimum disk stack-up height is calculated by adding nominal dimensions with upper and lower tolerance respectively.

Standard deviation is calculated by considering manufacturing process is 3σ capable. Mathematically standard deviation is equal to two times of process capability because the process is distributed on both sides. Therefore standard deviation for 3σ process is given by:

According to the worst case tolerance stack up analysis disk stack height can vary from 65.5 mm to 68.5 mm. Whereas according to the RSS method disk height can vary from 66.2584 mm to 67.74162 mm.

In the above example, the total stacked disk height/thickness will vary from 66.258 mm to 67.74 mm. We suggest you also use this calculator to calculate total disk height using worst case tolerance analysis.

Can you provide a presentation or resources to guide in creating tol stack analysis stack diagrams, to include any assembly of 3 or more parts with a fixed fastener screw rectangular pattern? Simple disc stackup is clear but I am not clear how to handle more difficult aforementioned multipart ‘fixed fastener’ assy stack up.

Mathematically tolerance zone is equal to the multiple of required process capability and total standard deviation for the assembly. Lower the value of process capability, narrow will be the tolerance zone and higher will be the rejection.

I’ve been told if there is a sequence of just 2 dimensions then RSS is not required. I can’t find where this rule originates. Can you confirm?

Hello, I have the same question asked by Nagaraju. So how does the worst case analysis has disadvantage over RSS method? Compared to Worst case, in RSS, you allow some parts rejection. But what about the cost of those rejected parts? How exactly RSS method is advantageous over worst case?

We suggest you read this article on Process Capability Analysis Cp and Cpk for better understanding on Statistical Tolerance Analysis.

CANADIAN TIRE® and the CANADIAN TIRE Triangle Design are registered trade-marks of Canadian Tire Corporation, Limited. ±Was price reflects the last national regular price this product was sold for. **Online prices and sale effective dates may differ from those in-store and may vary by region. Dealers may sell for less. * Financing available is “Equal payments, no interest” for 24 months (unless otherwise stated) and is only available on request, on approved credit and on purchases of $150 (unless otherwise stated) or more (Gift Cards excluded) made with your Triangle credit card at Canadian Tire, Sport Chek, Mark’s, L’Équipeur, Atmosphere, Sports Rousseau, Hockey Experts, L’Entrepôt du Hockey and participating Sports Experts. Interest does not accrue during the period of the plan. However, if we do not receive the full minimum due on a statement within 59 days of the date of that statement, or any event of default (other than a payment default) occurs under your Cardmember Agreement, all special payment plans on your account will terminate and (i) you will then be charged interest on the balances outstanding on such plans at the applicable regular annual rate from the day after the date of your next statement, and (ii) the balances outstanding will form part of the balance due on that statement. There is no administration fee charged for entering into a special payments plan. Each month during an equal payments plan you are required to pay in full by the due date that month’s equal payments plan instalment. Any unpaid portion not received by the due date will no longer form part of the equal payments plan and interest will accrue on that amount from the day after the date of your next statement at the applicable regular annual rate. Offer subject to change without notice. Additional information for residents of Quebec only: The regular annual rate for persons applying for the Triangle Mastercard and the Triangle World Elite Mastercard is 21.99% for cash transactions and related fees and 21.99% for all other charges. Some applicants may receive a higher or lower regular annual rate depending on a credit evaluation. The minimum payment is the sum of (a) the greater of: (i) interest and fees shown on your statement + $10; or (ii) 5% of the New Balance, excluding amounts on special payment plans, (b) any balance over your credit limit, (c) any amounts past due not included in (b) above, and (d) the amount of any equal payments plan instalments then due. Balances under $10 are due in full. For residents of Quebec, the period between the statement date and the due date for payment is 26 days. The billing period covered by each statement can be from 28-33 days. The Triangle Mastercard and the Triangle World Elite Mastercard do not have an annual fee. Examples of borrowing costs (rounded to the nearest cent) assuming that all charges are purchases bearing interest at the regular annual rate of 21.99%, a 30 day month, no charges made on special payment plans and no other fees, additional payments or other changes are:

*$9.99 + tax delivery fee.  Select your store and enter postal code on the product page while shopping to determine if same day delivery is available.  Products must meet certain cube/weight dimensions and distance from store must be less than 10km.  For full terms/conditions, visit https://www.canadiantire.ca/en/customer-service/online-ordering.html  *Free at the majority of stores. Some stores may require a minimum order value (before taxes). Orders that do not meet the minimum order value will be subject to a fee.  Orders ready typically within 24 hours. Wait for Ready for Pick Up email before proceeding to store. **$9.99 + tax delivery fee. Select your store and enter postal code on the product page while shopping to determine if same day delivery is available. Products must meet certain cube/weight dimensions and distance from store must be less than 10km. For full terms/conditions, visit https://www.canadiantire.ca/en/customer-service/online-ordering.html

How should we deal with uneven tolerances (e.g. +0.1/-0.3)? Do we “normalize” it to +0.2/-0.2 (& adjust the nominal value accordingly)?

In your bell curve normal distribution curve there is an error – for 6 sigma, instead of 99.72% you have marked 95.72%, and as a newbie I spent 20 minutes to re-affirm that… just wanted to let you know. Thanks for this article and calculator, it’s really helpful!

*The Triangle Rewards Program is owned and operated by Canadian Tire Corporation, Limited. The Triangle® credit cards are issued by Canadian Tire Bank. Rewards are in the form of electronic Canadian Tire Money® (CT Money®). To collect bonus CT Money you must pay with a Triangle credit card. You cannot collect paper Canadian Tire Money on bonus offers. Any bonus multiplier is based on the base rate of collecting CT Money (0.4%), and will be added to whatever the Member would otherwise collect, without the bonus. Example: On a $100 (pre-tax) purchase with a 20x bonus multiplier a Member would earn a bonus $8 in CT Money (20 X 0.4% X $100). The 30x for Triangle® Mastercard®, World Mastercard® and World Elite® Mastercard® customers consists of the 10x everyday plus a 20x bonus. Not all items sold are eligible to earn CT Money or to be redeemed for. Conditions apply. Visit triangle.com for full program rules and Partner location information. The offered rate is exclusive of any bonus or promotional offers or redemption transactions. CT Money is collected on the pre-tax amount of the purchase. Bonus CT Money may take up to 5 weeks to be applied to the Member’s Triangle Rewards Account. No rainchecks. Must pay with a Triangle credit card. Gift cards and other items excluded.

How do you calculate if an element in the stack-up has unbalanced tolerance? For example X +0.5/-0.1 or X +1/-0? Do you shift the “nominal” and then balance the tolerance, or is there a more accurate way to calculate the final tolerance stackup?

Nice writeup. Please note that there is a typo for the 3 sigma area under the curve of the normal distribution bell curve. It should read 99.72% rather than 95.72%. Regards.

I don’t understand the RSS scaling of the total tolerance band by 1/(2×3) to get the total standard deviation. If we assume the errors are uniform distributed – the standard deviation should = tolerance band/sqrt(12) = tolerance band/3.46. This is close to the factor of 3 used – but I don’t see where the additional factor of 2 comes in….

Yes; It wont help machining supplier ; However this may help to casting supplier in case wall thickness calculation. If there is a defect on casting surface and it could lead to reduction in wall thickness. Due to RSS method machining tolerance will be reduced and in result increase in wall thickness.

Thanks Edward for your feedback. This is a linear tolerance calculator, that simply works by adding the tolerance. Here -ve sign shows the tolerance direction. So ideally output shall be irrespective of sign. So i agree with you and we will improve on this point. But at the same time, i believe it has no impact on end results as this is calculated based on cumulative values. We recommend you to also check our online tolerance calculator where we tried to over come the limitations of excel: https://www.smlease.com/entries/product-design/online-tolerance-stackup-calculator/

Tolerance stackup analysis is used to calculate optimized part tolerances. It can be done manually or using stackup analysis software.

Statistical tolerance Analysis does not focus on the extreme dimensional limits. Because variation in manufactured part dimensions is not linear. Each dimension has a unique distribution based on the part manufacturing process, machines, and other parameters.

Total nominal disk thickness is calculated similarly to done during worst case tolerance stackup analysis by adding the nominal thickness of all disks. Therefore

In RSS tolerance band is not reduced, you actually allow some rejection compared to worst case tolerance. And the advantage of this allowable rejection can be given to supplier in term of tolerance.

If you think we missed Something?  You can add to this article by sending a message in the comment box. We will do our best to add it to this post.

In the above example, the total stacked disks height / thickness will vary from 65.5 mm to 68.5 mm. We suggest you also use this calculator to calculate total disk height using worst case tolerance analysis.

Similar to worst case tolerance analysis, the first step in RSS tolerance stackup analysis is to create the dimensional chain. It is used to determine the direction of tolerance. In the above example, all dimensions are in a positive direction.

As you said, Mathematically standard deviation is equal to two times of process capability because the process is distributed on both sides. based on above, for ‘3 sigma’ the total tolerance of each dimension is divide by 6 (i.e. 2×3) to calculate standard deviation. but at the time of total tolerance zone calculation ‘sigma assembly’ multiplied by only 3 for ‘3 sigma’ capacity. my concern is, just like at the time of total tolerance calculation we divide each total dimension tolerance value by 6. like wise we need to multiply by 6 to ‘sigma assembly’ to get total tolerance zone. because process is distributed on both sides of mean value. please provide your feedback.

You can also use our worst-case and RSS method tolerance stack-up calculator for tolerance stack-up calculations. You can Download Tolerance stack-up calculation sheet here.

† In the form of electronic Canadian Tire Money® (CT Money®). You cannot collect paper Canadian Tire Money on bonus offers. Any bonus multiplier is based on the base rate of collecting CT Money. Not all items sold at Canadian Tire earn CT Money. The offered rate is exclusive of any bonus or promotional offers or redemption transactions. CT Money is collected on the pre-tax. Bonus CT Money collected from online orders will be applied to the member’s Triangle Rewards™ account within 5 weeks of the purchase date. Terms and Conditions apply. Visit Triangle Rewards: Sign Up & Earn Today for more information. ®/TMMark’s is a registered trademark of Mark’s Work Warehouse Ltd, used under licence. ®/TMSport Chek is a registered trademark of FGL Sports Ltd, used under licence. ◊Off our original prices. Pricing, selection, and availability of clearance items are determined by each store. Applicable on select styles/sizes. Clearance items may be display models or not exactly as shown. Quantities may be limited and may vary by store. Rainchecks unavailable. We reserve the right to limit quantities. Shop early for greatest selection. Contact your local store for more information. ‡‡Shipping fees apply. Shipping fees and delivery times vary depending on location, size and weight of the item(s) and is only available within the province of the Canadian Tire retail location (“Store”) from which the item(s) was purchased. Bulk items will only be delivered within a 100km radius of the Store. Not available in Recontre East, NL. Conditions and restrictions apply. Visit Online Ordering for more information. ®/™ Mastercard, World Mastercard and World Elite are registered trademarks, and the circles design is a trademark of Mastercard International Incorporated. ®/TM Unless otherwise noted, all trademarks are owned by Canadian Tire Corporation, Limited and are used under license. ◊Off our original price. Quantities may vary by store. Some clearance product and price points may not be available at all locations. ± Was price reflects the last national regular price this product was sold for *The Triangle Rewards Program is owned and operated by Canadian Tire Corporation, Limited. Triangle® credit cards are issued by Canadian Tire Bank. Rewards are in the form of electronic Canadian Tire Money® (CT Money®). To collect bonus CT Money you must present a Triangle Rewards card/key fob, or use any approved Cardless method, at time of purchase or pay with a Triangle credit card. You cannot collect paper Canadian Tire Money on bonus offers. Any bonus multiplier is based on the base rate of collecting CT Money (0.4%), and will be added to whatever the Member would otherwise collect, without the bonus. Example: On a $100 (pre-tax) purchase with a 20x bonus multiplier a Member would earn a bonus $8 in CT Money (20 X .4% X $100). Not all items sold are eligible to earn CT Money or to be redeemed for. Conditions apply. Visit triangle.com for full program rules and Partner location information. The offered rate is exclusive of any bonus or promotional offers or redemption transactions. CT Money is collected on the pre-tax amount of the purchase. Bonus CT Money may take up to 5 weeks to be applied to the Member’s Triangle Rewards Account. No rainchecks. 1 Participating retailers: Canadian Tire, Sport Chek, Mark’s, L’Équipeur, Atmosphere, Party City, Sports Rousseau, Hockey Experts, L’Entrepôt du Hockey and participating Sports Experts stores. ^ CT Money is collected on the number of whole litres of fuel purchased rounded down to the nearest whole litre when you show your Triangle card, mobile App or key fob and pay with cash or debit. Rate subject to change and could vary by location, see local gas bar for details. Not all Gas+ locations have premium fuel. Must go inside kiosk to collect rewards with App or key fob. Conditions apply. Visit triangle.com for full program rules and Partner location information.

In this example maximum and minimum disk stackup height is calculated by adding nominal dimension with upper and lower tolerance respectively considering required process capability value is 3σ.