Inverse Problems: Percentiles and Quantiles

1
Inverse Problems Percentiles and Quantiles

• If you fall into the 98th percentile in a
  population, this means that 98 of the population
  falls below you.
• E.g. Karen scores in the 99th percentile for the
  Bursary English exam in 1999. This means that 99
  of all the students who sat the 1999 Bursary
  English exam had lower marks than Karen.
• The lower quartile is the 25th percentile
• The median is the 50th percentile
• The upper quartile is the 75th percentile

2
Quantiles

• A quantile is the same as a percentile but is
  expressed as a proportion. E.g. the 75th
  percentile would be the 0.75 quantile.
• More formally, if 0 ? p ? 1, then the p-quantile
  qp is defined to be the value such that Pr(X ?
  qp) p.
• This is just a complicated way of saying that the
  lower tail probability for some value qp is p
• Check the following with Minitab or Excel. For
  the Normal distribution the
• 0.95 quantile is approximately 1.6449
• 0.975 quantile is approximately 1.96
• 0.99 quantile is approximately 2.326

3
Using a computer program to find to find
percentiles/quantiles for the Normal distribution

• Last time we gave the computer an x value, a mean
  and a standard deviation and it gave us back a
  lower tail probability
• This time, we provide the computer with a
  probability, a mean and a standard deviation, and
  we want an x value
• Most software packages return x values for lower
  tail probabilities. That is, given p, the
  computer returns an x such that Pr(X ? x) p.
• This means that we must rephrase our questions in
  terms of lower tail probabilities before we can
  answer them
• This of course is where the most skill is involved

4

• Assume that we have phrased our question as a
  lower tail probability.
• In Minitab
• Enter the probabilities for which you require an
  x into a column (say C1)
• Select Calc gt Probability Distributions gt Normal
• Click on the Inverse Cumulative Probability radio
  button
• Enter the mean into the mean box and the standard
  deviation into the standard deviation box
• Enter the column in which your probabilities are
  stored (C1) into the input box and click on OK.
• In Excel
• Enter the probability into a cell, say A1
• Click on an adjacent cell (say B1) and then click
  on the paste function button fx
• Select the Statistical from the function
  category and NORMINV from the function name and
  click on OK
• Enter the address of the probability (A1) into
  the probability box, the mean into the mean box,
  and the standard deviation into the std_deviation
  box and click on okay
• Use Fill Down if you have more than one
  probability

5
Example

• Suppose that IQ scores are Normally distributed
  with a mean of 100 and a standard deviation of
  15. Use the Minitab output to answer the
  following questions
• Find the IQ score of the bottom 20 of the people
  in the population (i.e. find the 20th percentile)
• What IQ score is exceed by only the top 10?
• Find the interquartile range for IQ scores.

Normal with mean 100.000 and standard deviation
15.0000 P( X lt x) x P( X lt x)
x 0.1000 80.7767 0.7500
110.1173 0.2000 87.3757 0.8000
112.6243 0.2500 89.8827 0.9000
119.2233
6
(a)
(b)
(c)
7
Exercise

• Assume that the natural gestation period for
  human births is approximately Normally
  distributed with a mean 266 days and a standard
  deviation of about 16 days. Use the Minitab
  output to answer the following questions
• What is the maximum gestation for the bottom 15
  of births?
• What is the range of gestation period for the
  central 60 of births?
• What is the maximum gestation period exceed by
  90 of births?

Normal with mean 100.000 and standard deviation
15.0000 P( X lt x) x P( X lt x)
x 0.1000 80.7767 0.6000
103.8002 0.1500 84.4535 0.8000
112.6243 0.2000 87.3757 0.9000
119.2233
8
(a)
(b)
(c)
9
Working in standard units

• Sometimes it is useful to measure distance in
  terms on the number of standard deviations from
  the mean.
• The z-score measures the distance in terms on the
  number of standard deviations from the mean
• To find the z-score for a particular value x, we
  subtract the mean and divide by the standard
  deviation, i.e.
• If x is smaller than the mean then the z-score is
  negative
• If x is larger than the mean then the z-score is
  positive
• Example If
  find the z-score for
• (a) x 28 (b) x 16 (c) x 30 (d) x 13

10
Example

• A student sat 3 exams. Her results are Maths 72,
  English 68, Art History 64. Assume that the
  results of each of these exams are approximately
  Normally distributed as follows
• Maths m 63 s 11
• English m 58 s 13
• Art History m 54 s 10
• Assume that large numbers of students sat each of
  the three exams. In which subject is this student
  the strongest in terms of ranking with those who
  sat the same exam as her?

11
The Standard Normal distribution

• If X is Normally distributed with a mean of m
  and a standard deviation of s, i.e.
  then the random variable corresponding to
  the z-score
• is Normal with mean 0 and standard deviation 1
• The Normal distribution with m 0 and s 1 is
  called the standard Normal distribution and we
  usually write