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Inverse normal calculations

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Inverse normal calculations

Convert the statement to a less-than area, then use the signed standard score in the inverse standardisation formula. Use the empirical rule only when an approximation is explicitly requested

\[x=\mu+z\sigma\]
  1. 1Convert to a less-than area Complement an upper-tail probability before finding the quantile \(\displaystyle \Pr(Z>z)=0.025\;\Rightarrow\;\Pr(Z<z)=0.975\)
  2. 2Distinguish the empirical approximation The empirical rule uses rounded areas and integer standard scores, not exact inverse-CDF quantiles \(\displaystyle \Pr(Z<-1)\approx0.16,\quad\Pr(Z<2)\approx0.975\)
  3. 3Use inverse standardisation Add the signed standard-score displacement to the mean \(\displaystyle x=\mu+z\sigma\)
  4. 4Halve a symmetric complement Split the area outside a symmetric central interval equally between the two tails \(\displaystyle \Pr(|Z|<2)\approx0.95\;\Rightarrow\;\Pr(Z>2)\approx\frac{1-0.95}{2}=0.025\)
  5. 5Use numerical inverse quantiles when requested For a specified probability, use the inverse normal function at the requested precision rather than replacing it by an integer empirical score \(\displaystyle \Phi^{-1}(0.975)\approx1.959964\ne2\)
Year 12, VCE Units 3 & 4, Chapter 13, 4 difficulty levels, Calculator allowedBuild a worksheet

Inverse normal calculations

Convert to a less-than area before inverse standardisation, and state whether the empirical approximation or a numerical inverse quantile is being used

\[x=\mu+z\sigma\]
  1. Convert to a less-than area Complement an upper-tail probability before finding the quantile \(\displaystyle \Pr(Z>z)=0.025\;\Rightarrow\;\Pr(Z<z)=0.975\)
  2. Distinguish the empirical approximation The empirical rule uses rounded areas and integer standard scores, not exact inverse-CDF quantiles \(\displaystyle \Pr(Z<-1)\approx0.16,\quad\Pr(Z<2)\approx0.975\)
  3. Use inverse standardisation Add the signed standard-score displacement to the mean \(\displaystyle x=\mu+z\sigma\)
  4. Halve a symmetric complement Split the area outside a symmetric central interval equally between the two tails \(\displaystyle \Pr(|Z|<2)\approx0.95\;\Rightarrow\;\Pr(Z>2)\approx\frac{1-0.95}{2}=0.025\)
  5. Use numerical inverse quantiles when requested For a specified probability, use the inverse normal function at the requested precision rather than replacing it by an integer empirical score \(\displaystyle \Phi^{-1}(0.975)\approx1.959964\ne2\)

Worked example

A measured length in millimetres has the displayed normal model. Use the empirical-rule approximation to estimate the boundary with the displayed upper-tail area

\[X\sim N(70,25),\quad\Pr(X>x)\approx0.025\]
  1. \(\displaystyle \Pr(X<x)\approx1-0.025=0.975\)Complement the approximate upper-tail area
  2. \(\displaystyle z\approx2\)The stated empirical rule places the boundary about two standard deviations above the mean
  3. \(\displaystyle \sigma=\sqrt{25}=5,\quad x=\mu+z\sigma\)Use the standard deviation in inverse standardisation
  4. \(\displaystyle x\approx70+2\times5=80\text{ mm}\)Report an empirical estimate, not an exact normal percentile

Common mistakes

Upper-tail area used as a lower-tail area
✕\(\displaystyle \Pr(Z>z)=0.16\;\Rightarrow\;\Pr(Z<z)=0.16\)✓\(\displaystyle \Pr(Z>z)=0.16\;\Rightarrow\;\Pr(Z<z)=1-0.16=0.84\)
An upper-tail probability must be complemented before it is used as a less-than area
Unsigned standard score
✕\(\displaystyle x=50+1\times4=54\)✓\(\displaystyle x=50+(-1)\times4=46\)
A boundary below the mean has a negative z-value, so the signed displacement must be kept
Variance used in inverse standardisation
✕\(\displaystyle x=50+1\times16=66\)✓\(\displaystyle x=50+1\times4=54\)
The inverse formula uses the standard deviation, while 16 is the variance

Difficulty levels

introFind a standard score from a lower-tail empirical probability
standardFind a boundary value or central half-width from an empirical probability statement
challengeRecover a mean, standard deviation or pair of parameters from inverse normal conditions
examChoose the correct area conversion or inverse setup before finding one unknown value

Questions about this topic

Why complement an upper-tail probability?+

An inverse normal function uses the area to the left of the boundary

\[\Pr(Z<z)=1-\Pr(Z>z)\]
Why can a standard score be negative?+

Its sign records the direction from the mean, so a boundary below the mean has a negative score

\[x<\mu\;\Rightarrow\;z<0\]
How are quantiles and percentiles related?+

A quantile is identified by the probability to its left. Empirical-rule percentages are rounded approximations rather than exact numerical inverse quantiles

\[\Pr(X<x_p)=p\quad\Longleftrightarrow\quad x_p\text{ is the }100p\text{th percentile}\]
Can I use a calculator?+

A calculator is allowed unless a question is marked no calculator

Which year level and chapter is this?+

Year 12, VCE Units 3 & 4, Mathematical Methods, chapter 13: Normal distribution

In this chapter

Read normal parameters and describe their graph effectCalculate probabilities for normal distributionsCalculate a standard score or actual valueUse z-score boundaries to estimate normal-model percentages and countsCompare observations from different distributions using z-scoresInverse normal calculationsInfer a normal-model mean or standard deviationConditional probabilities with normal intervalsMixed probability applications with the normal distribution