Construction Engineering 221
Normal distribution part II
Normal distribution
• Many measurements are normally
distributed and can be converted to a
standardized distribution in order to use ztables
• To calculate probabilities, it is typical to
pick an acceptable level (say, .05) and
calculate probability that z is greater than
or less than that value (this will be
important in hypothesis testing)
Normal distribution
• Example:
– Concrete cylinder test results (in psi) are
1000, 1200, 1150, 1100, and 1230. You need
to have three of six cylinders exceed 1200 psi
before you can strip forms. What is the
probability that the sixth cylinder will exceed
1200 psi?
– First calculate the mean and sd of the sample
• Mean = 1136, sd = 90
Normal distribution
• Z-statistic for the sixth cylinder is
– 1200-1136/90 = .711
– Z= .711, P = .26115
– Probability is (1 - .76115), or .2389, 24%
chance
– must add 50% area from left of the mean and
subtract remaining area to find probability
Normal distribution
• What if the specifications read that the
average of six cylinders must be 1200?
– 1200 X 6 = 7200
– Σ (1-5) = 5680
– Cylinder 6 must test at 7200-5680 = 1520 for
average to be 1200
– Z = 1520-1200/176.5 = 1.81
– A(x) = .46485
– P= 1-(.5 + .46485) = .0352, or 3% chance
Normal distribution
• What if you asked to change the
specification to read the average of the 5
highest of 6 tests (eliminate the outlier of
1000) must equal 1200?
– 1200 + 1150 + 1100 + 1230 +X/5 = 1200
– X= 1320 (sixth cylinder must test at 1320
– Z= 1320-1200/83.37 = 1.44
– A(x) = .42507
– P= 1-.92507, or 7.5%
Normal Distribution
• Is this important?
– Assume an 18 million dollar job with a 12 month
completion schedule
– 1.5 million per month/ 22 workdays per = 68,200 per
day
– .03 X 68200 = $2046
– .075 X 68200 = $5115
– risk savings of over $3000 for a simple change
– Consider you make (50,000 X1.4)/264 or $265/day
Normal distribution
• Normal approximation of the Binomial
distribution is used when the number of
trials is large
– Math gets cumbersome
– Large sample sizes approach continuous
distributions
– Use normal approximations when values
exceed those included in binomial tables
– Use when n*π >5 (or n* (1-π)), or when n>30
Normal distribution
• Do 6 and 10 on page 82