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Basic Statistics Refresher

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Outline

INTRODUCTION

NORMAL DISTRIBUTION

TOLERANCE INTERVALS

RESPONSE SURFACE MODELS

CE Setpoints – Statistics - p.3

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Learning Objectives

Introduction� Describe the difference between a bias and a random un certainty

Normal Distribution� Explain why “2-sigma” is commonly used to describe 95% coverage of a

normal distribution

Tolerance Intervals� Describe a 95/95 tolerance interval

� Explain the difference between a one-sided tolerance in terval and a two-sided tolerance interval

Response Surface Models� Define a Response Surface Model

CE Setpoints – Statistics - p.4

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INTRODUCTION

Outline

INTRODUCTION

NORMAL DISTRIBUTION

TOLERANCE INTERVALS

RESPONSE SURFACE MODELS

CE Setpoints – Statistics - p.5

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INTRODUCTION

Learning Objectives

Introduction� Describe the difference between a bias and a random u ncertainty

Normal Distribution� Explain why “2-sigma” is commonly used to describe 95% coverage of a

normal distribution

Tolerance Intervals� Describe a 95/95 tolerance interval

� Explain the difference between a one-sided tolerance interval and a two-sided tolerance interval

Response Surface Models� Define a Response Surface Model

CE Setpoints – Statistics - p.6

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INTRODUCTION

Deterministic vs. Statistical

Statistics allows for the treatment of random variability. � Deterministic – Single value used to represent a Param eter.

� Statistical – Distribution of value used to represent a parameter including random variability.

CE Setpoints – Statistics - p.7

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INTRODUCTION

Types of Uncertainty

Bias – Shifts the mean of the distribution to the left or to the right

Random – Impacts the shape of the distribution

CE Setpoints – Statistics - p.8

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INTRODUCTION

Types of Uncertainty

A base assumption throughout the setpoint methodology is that uncertainty parameters may be treated as symmetric and normally distributed.

Example� Tavg uncertainty = +4.0F/-4.8F

Option 1� Apply bounding random uncertainty

Option 2� Apply bias to nominal setting

Pay extra attention in these situations

Not always the case in reality

Bias = -0.4F

Random = +-4.4F

Bias = -0.0F

Random = +-4.8F

Bias = -0.4F

Random = +-4.4F

CE Setpoints – Statistics - p.9

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INTRODUCTION

Visualization

A Histogram is used to visualize how a collection of data points is distributed.Probability Density Functions (PDF) are often used to analytically estimate statistical distributions.

CE Setpoints – Statistics - p.10

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INTRODUCTION

Review Learning Objectives

Introduction� Describe the difference between a bias and a random u ncertainty

Normal Distribution� Explain why “2-sigma” is commonly used to describe 95% coverage of a

normal distribution

Tolerance Intervals� Describe a 95/95 tolerance interval

� Explain the difference between a one-sided tolerance interval and a two-sided tolerance interval

Response Surface Models� Define a Response Surface Model

CE Setpoints – Statistics - p.11

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NORMAL DIST.

Outline

INTRODUCTION

NORMAL DISTRIBUTION

TOLERANCE INTERVALS

RESPONSE SURFACE MODELS

CE Setpoints – Statistics - p.12

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NORMAL DIST.

Learning Objectives

Introduction� Describe the difference between a bias and a random u ncertainty

Normal Distribution� Explain why “2-sigma” is commonly used to describe 95% coverage of a

normal distribution

Tolerance Intervals� Describe a 95/95 tolerance interval

� Explain the difference between a one-sided tolerance interval and a two-sided tolerance interval

Response Surface Models� Define a Response Surface Model

CE Setpoints – Statistics - p.13

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NORMAL DIST.

Normal Distribution

Many of the setpoint calculations use probability distributions to model the real world variability in input parameters.

Simplified models are typically used.� Most common is a Normal distribution.

� � �1

� 2��� �

��

Where,

µ = mean

σ = standard deviation

CE Setpoints – Statistics - p.14

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NORMAL DIST.

Normal Distribution

A normal curve represents probability as the area under the curve.� Typically calculated using

tables because there is not a closed form solution for the integral.

� Student’s T distribution

� Note that the 2-sigma interval covers just over 95% of the data. µ-3σ µ-2σ µ-1σ µ µ+1σ µ+2σ µ+3σ

0.9545

0.9973

0.6827

CE Setpoints – Statistics - p.15

Two-sided Intervals

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NORMAL DIST.

Normal Distribution

A normal curve represents probability as the area under the curve.� Typically calculated using

tables because the is not closed form solution for the integral.

µ-3σ µ-2σ µ-1σ µ µ+1σ µ+2σ µ+3σ

0.9773

0.9987

0.8414

CE Setpoints – Statistics - p.16

One-sided Intervals

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NORMAL DIST.

Review of Learning Objectives

Introduction� Describe the difference between a bias and a random un certainty

Normal Distribution� Explain why “2-sigma” is commonly used to describe 95% coverage of a

normal distribution

Tolerance Intervals� Describe a 95/95 tolerance interval

� Explain the difference between a one-sided tolerance in terval and a two-sided tolerance interval

Response Surface Models� Define a Response Surface Model

CE Setpoints – Statistics - p.17

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TOLERANCE INTV.

Outline

INTRODUCTION

NORMAL DISTRIBUTION

TOLERANCE INTERVALS

RESPONSE SURFACE MODELS

CE Setpoints – Statistics - p.18

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TOLERANCE INTV.

Learning Objectives

Introduction� Describe the difference between a bias and a random u ncertainty

Normal Distribution� Explain why “2-sigma” is commonly used to describe 95% coverage of a

normal distribution

Tolerance Intervals� Describe a 95/95 tolerance interval

� Explain the difference between a one-sided tolerance interval and a two-sided tolerance interval

Response Surface Models� Define a Response Surface Model

CE Setpoints – Statistics - p.19

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TOLERANCE INTV.

Tolerance Interval

A finite data set is inadequate to characterize a parameter’s variability with 100% certainty.

A tolerance interval has two numbers associated with it, namely a confidence level and a coverage level.

The interval is made so that we can have specified confidence that at least the specified portion of the entire population is covered by the interval.

For example, a 99/95 tolerance interval means that there is 99% confidence that 95% of the population will be covered by the given interval.� The order of the terms is not always treated consistently (i.e. 99/95 vs.

95/99).

� Luckily, we usually use 95/95 so it does not matter.

CE Setpoints – Statistics - p.20

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TOLERANCE INTV.

Tolerance Interval

For example, a 99/95 tolerance interval means that there is 99% confidence that 95% of the population will be covered by the given interval.

The 99% refers to the confidence of the interval.

This is typically impacted by the number of samples available. � A large uncertainty factor must be applied if a small number of samples

is available or if a high confidence is desired.

� A small uncertainty factor can be applied is a large number of samples is available or if a low confidence is desired.

� The confidence of a best estimate calculation is 50% .

CE Setpoints – Statistics - p.21

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TOLERANCE INTV.

Tolerance Interval

The 95% is the portion of the population being covered.

Coverage can be one-sided or two-sided.

one-sidedtwo-sided

CE Setpoints – Statistics - p.22

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TOLERANCE INTV.

Tolerance Interval

Tolerance intervals for normal distributions are of the form:

The constants k are typically referred to as “k-factors” and are tabulated.

The K factor for a two-sided 95/95 tolerance interval with infinite samples is 1.96

The K factor for a one-sided 95/95 tolerance interval with infinite samples is 1.645

�� � � ∗ �(two−sided) �� � � ∗ �(lower) �� � � ∗ �(upper)

�� (is the sample population mean) �(is the sample population standard deviation)

CE Setpoints – Statistics - p.23

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TOLERANCE INTV.

Review of Learning Objectives

Introduction� Describe the difference between a bias and a random u ncertainty

Normal Distribution� Explain why “2-sigma” is commonly used to describe 95% coverage of a

normal distribution

Tolerance Intervals� Describe a 95/95 tolerance interval

� Explain the difference between a one-sided tolerance interval and a two-sided tolerance interval

Response Surface Models� Define a Response Surface Model

CE Setpoints – Statistics - p.24

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RSM

Outline

INTRODUCTION

NORMAL DISTRIBUTION

TOLERANCE INTERVALS

RESPONSE SURFACE MODELS

CE Setpoints – Statistics - p.25

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RSM

Learning Objectives

Introduction� Describe the difference between a bias and a random uncertain ty

Normal Distribution� Explain why “2-sigma” is commonly used to describe 95% cover age of a

normal distribution

Tolerance Intervals� Describe a 95/95 tolerance interval

� Explain the difference between a one-sided tolerance interv al and a two-sided tolerance interval

Response Surface Models� Define a Response Surface Model

CE Setpoints – Statistics - p.26

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RSM

Response Surface Model (RSM)

A response surface is a multi-dimensional fit of a particular response to a set of input parameters.

Typically used to estimate complex phenomenon in an efficient way.

CE Setpoints – Statistics - p.27

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RSM

Response Surface Model

To develop an RSM, a set of experimental results characterizing the design space is required. � For DNBR calculation, the experiments are explicit XCO BRA-IIIC runs

� The design space is defined by the min and max of each input parameter

� Typically inputs are varied at integer multiples of t heir standard deviation

One option is to evaluate all possible combinations of each parameter at a given set of levels.� For example, assume each parameter can be at -2σ, -1σ, 0σ, 1σ, or 2σ

� Running all combinations of 10 parameters at 5 level s results in 9,765,625 XCOBRA-IIIC runs

CE Setpoints – Statistics - p.28

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RSM

Response Surface Model

Design of Experiments (DOE)� A design of experiments can be used to limit the num ber of explicit runs

needed to build an RSM while minimizing the loss of information

� Many types of DOE exist• Box-Benhken design• Plackett-Burman• Cubic centered design

CE Setpoints – Statistics - p.29

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RSM

Review of Learning Objectives

Introduction� Describe the difference between a bias and a random uncertainty

Normal Distribution� Explain why “2-sigma” is commonly used to describe 9 5% coverage of a

normal distribution

Tolerance Intervals� Describe a 95/95 tolerance interval

� Explain the difference between a one-sided tolerance interval and a two-sided tolerance interval

Response Surface Models� Define a Response Surface Model

CE Setpoints – Statistics - p.30

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Exercise 2.1Companion Notebook > Statistics Tab > Exercise 2.1� Tolerance Intervals

CE Setpoints – Statistics - p.31

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Review of Learning Objectives

Introduction� Describe the difference between a bias and a random u ncertainty

Normal Distribution� Explain why “2-sigma” is commonly used to describe 95% coverage of a

normal distribution

Tolerance Intervals� Describe a 95/95 tolerance interval

� Explain the difference between a one-sided tolerance interval and a two-sided tolerance interval

Response Surface Models� Define a Response Surface Model

CE Setpoints – Statistics - p.32

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