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CVE 472 Statistical Techniques in Hydrology. 1/47
Assist. Prof. Dr. Bertuğ Akıntuğ
Civil Engineering ProgramMiddle East Technical University
Northern Cyprus Campus
CVE 472CVE 472
3. NORMAL DISTRIBUTION3. NORMAL DISTRIBUTION
CVE 472 Statistical Techniques in Hydrology. 2/47
3. NORMAL DISTRIBUTION3. NORMAL DISTRIBUTION
Outline
General Normal Distribution
Per-productive Properties
Standard Normal Distribution
Central Limit Theorem
Constructing Normal Curves for Data
CVE 472 Statistical Techniques in Hydrology. 3/47
3. NORMAL DISTRIBUTION3. NORMAL DISTRIBUTION
The Normal Distribution
‘Bell Shaped’SymmetricalMean, Median and Mode
are EqualLocation is determined by the mean, µSpread is determined by the standard deviation, σ
The random variable has an infinite theoretical range: + ∞ to − ∞ Mean = Median= Mode
X
px(x)
µ
σ
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3. NORMAL DISTRIBUTION3. NORMAL DISTRIBUTION
By varying the parameters µ and σ, we obtain different normal distributions
The Normal Distribution
Many Normal Distributions
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3. NORMAL DISTRIBUTION3. NORMAL DISTRIBUTION
The Normal Distribution
X
f(X)
µ
σ
Changing µ shifts the distribution left or right.
Changing σ increases or decreases the spread.
(continued)
The Normal Distribution Shape
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3. NORMAL DISTRIBUTION3. NORMAL DISTRIBUTION
The Normal Distribution
Small standard deviation
Large standard deviation
The Normal Distribution Shape
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3. NORMAL DISTRIBUTION3. NORMAL DISTRIBUTION
The Normal Distribution
The formula for the normal probability density function is
Where e = the mathematical constant approximated by 2.71828π = the mathematical constant approximated by 3.14159µ = the population meanσ = the population standard deviationX = any value of the continuous variable
2µ)/σ](1/2)[(Xx e
2π1(x)p −−=
σ
The Normal Probability Density Function
CVE 472 Statistical Techniques in Hydrology. 8/47
3. NORMAL DISTRIBUTION3. NORMAL DISTRIBUTION
Outline
General Normal Distribution
Per-productive Properties
Standard Normal Distribution
Central Limit Theorem
Constructing Normal Curves for Data
CVE 472 Statistical Techniques in Hydrology. 9/47
3. NORMAL DISTRIBUTION3. NORMAL DISTRIBUTION
Perproductive Properties
If Xi for i=1, 2, … n are independently and normally distributed with mean, µ and variance, σ2
X ~ N(µ,σ2)
Then Yi = a + bXi is normally distributed with
Y ~ N(a+bµ, b2σ2)
Solve Example 5.1 p.86
CVE 472 Statistical Techniques in Hydrology. 10/47
3. NORMAL DISTRIBUTION3. NORMAL DISTRIBUTION
The Sum of Two Independent Normal Random Variables
The sum of two independent Normal random variables is also normally distributed.
If X1 ~ N(µ1, σ12) and X2 ~ N(µ2, σ2
2) are independent random variables, then
( )22
212121 σσ,µµN~XXY +++=
CVE 472 Statistical Techniques in Hydrology. 11/47
3. NORMAL DISTRIBUTION3. NORMAL DISTRIBUTION
Linear Combinations of Independent Normal Random Variables
The two results presented so far can be synthesized into the following general result.If Xi ~ N(µi, σi
2), 1≤ i ≤ n, are independent random variables and if ai, 1≤ i ≤ n, and b are constants, then
whereµ = a + b1 µ 1 + … + bn µ n
and σ2 = b1
2 σ12 + … + bn
2 σn2
( )2nn11 σµ,N~XbXbaY +++= K
CVE 472 Statistical Techniques in Hydrology. 12/47
3. NORMAL DISTRIBUTION3. NORMAL DISTRIBUTION
Outline
General Normal Distribution
Per-productive Properties
Standard Normal Distribution
Central Limit Theorem
Constructing Normal Curves for Data
CVE 472 Statistical Techniques in Hydrology. 13/47
3. NORMAL DISTRIBUTION3. NORMAL DISTRIBUTION
The Standardized Normal
Any normal distribution (with any mean and standard deviation combination) can be transformed into the standardized normal distribution (Z)
Need to transform X units into Z units
CVE 472 Statistical Techniques in Hydrology. 14/47
3. NORMAL DISTRIBUTION3. NORMAL DISTRIBUTION
Translation to the Standardized Normal Distribution
Translate from X to the standardized normal (the “Z”distribution) by subtracting the mean of X and dividing by its standard deviation:
σµXZ −
=
The Z distribution always has mean = 0 and standard deviation = 1
CVE 472 Statistical Techniques in Hydrology. 15/47
3. NORMAL DISTRIBUTION3. NORMAL DISTRIBUTION
The Standardized Normal Probability Density Function
The formula for the standardized normal probability density function is
Where e = the mathematical constant approximated by 2.71828π = the mathematical constant approximated by 3.14159z = any value of the standardized normal distribution
2(1/2)zz e
2π1(z)p −=
CVE 472 Statistical Techniques in Hydrology. 16/47
3. NORMAL DISTRIBUTION3. NORMAL DISTRIBUTION
The StandardizedNormal Distribution
Also known as the “Z” distributionMean is 0Standard Deviation is 1
z
gz (z)
0
1
Values above the mean have positive Z-values, values below the mean have negative Z-values
CVE 472 Statistical Techniques in Hydrology. 17/47
3. NORMAL DISTRIBUTION3. NORMAL DISTRIBUTION
Example
If X is distributed normally with mean of 100 and standard deviation of 50, the Z value for X = 200 is
This says that X = 200 is two standard deviations (2 increments of 50 units) above the mean of 100.
2.050
100200σ
µXZ =−
=−
=
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3. NORMAL DISTRIBUTION3. NORMAL DISTRIBUTION
Comparing X and Z units
Z100
2.00200 X
Note that the distribution is the same, only the scale has changed. We can express the problem in original units (X) or in standardized units (Z)
(µ = 100, σ = 50)
(µ = 0, σ = 1)
CVE 472 Statistical Techniques in Hydrology. 19/47
3. NORMAL DISTRIBUTION3. NORMAL DISTRIBUTION
Finding Normal Probabilities
Probability is the area under thecurve!
a b X
f(X) P a X b( )≤
Probability is measured by the area under the curve
≤P a X b( )<<=
(Note that the probability of any individual value is zero)
CVE 472 Statistical Techniques in Hydrology. 20/47
3. NORMAL DISTRIBUTION3. NORMAL DISTRIBUTION
f(X)
Xµ
Probability as Area Under the Curve
0.50.5
The total area under the curve is 1.0, and the curve is symmetric, so half is above the mean, half is below
1.0)XP( =∞<<−∞
0.5)XP(µ =∞<<0.5µ)XP( =<<−∞
CVE 472 Statistical Techniques in Hydrology. 21/47
3. NORMAL DISTRIBUTION3. NORMAL DISTRIBUTION
Empirical Rules
µ ± 1σ encloses about 68% of X’s
f(X)
Xµ µ+1σµ-1σ
What can we say about the distribution of values around the mean? There are some general rules:
σσ
68.26%
CVE 472 Statistical Techniques in Hydrology. 22/47
3. NORMAL DISTRIBUTION3. NORMAL DISTRIBUTION
The Empirical Rule
µ ± 2σ covers about 95% of X’s
µ ± 3σ covers about 99.7% of X’s
xµ2σ 2σ
xµ3σ 3σ
95.44% 99.73%
(continued)
CVE 472 Statistical Techniques in Hydrology. 23/47
3. NORMAL DISTRIBUTION3. NORMAL DISTRIBUTION
The Standardized Normal Table
The Cumulative Standardized Normal table in the textbook (Appendix table E.2) gives the probability less than a desired value for Z (i.e., from negative infinity to Z)
Z0 2.00
0.9772Example:P(Z < 2.00) = 0.9772
CVE 472 Statistical Techniques in Hydrology. 24/47
3. NORMAL DISTRIBUTION3. NORMAL DISTRIBUTION
The Standardized Normal Table
The value within the table gives the probability from Z = − ∞ up to the desired Z value.9772
2.0P(Z < 2.00) = 0.9772
The row shows the value of Z to the first decimal point
The column gives the value of Z to the second decimal point
2.0
...
(continued)
Z 0.00 0.01 0.02 …
0.0
0.1
CVE 472 Statistical Techniques in Hydrology. 25/47
3. NORMAL DISTRIBUTION3. NORMAL DISTRIBUTION
General Procedure for Finding Probabilities
Draw the normal curve for the problem in terms of X
Translate X-values to Z-values
Use the Standardized Normal Table
To find P(a < X < b) when X is distributed normally:
CVE 472 Statistical Techniques in Hydrology. 26/47
3. NORMAL DISTRIBUTION3. NORMAL DISTRIBUTION
Finding Normal Probabilities
Suppose X is normal with mean 8.0 and standard deviation 5.0Find P(X < 8.6)
X
8.68.0
CVE 472 Statistical Techniques in Hydrology. 27/47
3. NORMAL DISTRIBUTION3. NORMAL DISTRIBUTION
Suppose X is normal with mean 8.0 and standard deviation 5.0. Find P(X < 8.6)
Z0.120X8.68
µ = 8σ = 10
µ = 0σ = 1
(continued)
Finding Normal Probabilities
0.125.0
8.08.6σ
µXZ =−
=−
=
P(X < 8.6) P(Z < 0.12)
CVE 472 Statistical Techniques in Hydrology. 28/47
3. NORMAL DISTRIBUTION3. NORMAL DISTRIBUTION
Z
0.12
Z .00 .01
0.0 .5000 .5040 .5080
.5398 .5438
0.2 .5793 .5832 .5871
0.3 .6179 .6217 .6255
Solution: Finding P(Z < 0.12)
.5478.02
0.1 .5478
Standardized Normal Probability Table (Portion)
0.00
= P(Z < 0.12)P(X < 8.6)
CVE 472 Statistical Techniques in Hydrology. 29/47
3. NORMAL DISTRIBUTION3. NORMAL DISTRIBUTION
Upper Tail Probabilities
Suppose X is normal with mean 8.0 and standard deviation 5.0. Now Find P(X > 8.6)
X
8.68.0
CVE 472 Statistical Techniques in Hydrology. 30/47
3. NORMAL DISTRIBUTION3. NORMAL DISTRIBUTION
Now Find P(X > 8.6)…(continued)
Z
0.120
Z
0.12
0.5478
0
1.000 1.0 - 0.5478 = 0.4522
P(X > 8.6) = P(Z > 0.12) = 1.0 - P(Z ≤ 0.12)
= 1.0 - 0.5478 = 0.4522
Upper Tail Probabilities
CVE 472 Statistical Techniques in Hydrology. 31/47
3. NORMAL DISTRIBUTION3. NORMAL DISTRIBUTION
Probability Between Two Values
Suppose X is normal with mean 8.0 and standard deviation 5.0. Find P(8 < X < 8.6)
P(8 < X < 8.6)
= P(0 < Z < 0.12)
Z0.120X8.68
05
88σ
µXZ =−
=−
=
0.125
88.6σ
µXZ =−
=−
=
Calculate Z-values:
CVE 472 Statistical Techniques in Hydrology. 32/47
3. NORMAL DISTRIBUTION3. NORMAL DISTRIBUTION
Z
0.12
Solution: Finding P(0 < Z < 0.12)
0.0478
0.00
= P(0 < Z < 0.12)P(8 < X < 8.6)
= P(Z < 0.12) – P(Z ≤ 0)= 0.5478 - .5000 = 0.0478
0.5000
Z .00 .01
0.0 .5000 .5040 .5080
.5398 .5438
0.2 .5793 .5832 .5871
0.3 .6179 .6217 .6255
.02
0.1 .5478
Standardized Normal Probability Table (Portion)
CVE 472 Statistical Techniques in Hydrology. 33/47
3. NORMAL DISTRIBUTION3. NORMAL DISTRIBUTION
Suppose X is normal with mean 8.0 and standard deviation 5.0. Now Find P(7.4 < X < 8)
X
7.48.0
Probabilities in the Lower Tail
CVE 472 Statistical Techniques in Hydrology. 34/47
3. NORMAL DISTRIBUTION3. NORMAL DISTRIBUTION
Probabilities in the Lower Tail
Now Find P(7.4 < X < 8)…
X7.4 8.0
P(7.4 < X < 8)
= P(-0.12 < Z < 0)
= P(Z < 0) – P(Z ≤ -0.12)
= 0.5000 - 0.4522 = 0.0478
(continued)
0.0478
0.4522
Z-0.12 0
The Normal distribution is symmetric, so this probability is the same as P(0 < Z < 0.12)
CVE 472 Statistical Techniques in Hydrology. 35/47
3. NORMAL DISTRIBUTION3. NORMAL DISTRIBUTION
Steps to find the X value for a known probability:1. Find the Z value for the known probability2. Convert to X units using the formula:
Finding the X value for a Known Probability
ZσµX +=
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3. NORMAL DISTRIBUTION3. NORMAL DISTRIBUTION
Finding the X value for a Known Probability
Example:Suppose X is normal with mean 8.0 and standard deviation 5.0. Now find the X value so that only 20% of all values are below this X
X? 8.0
0.2000
Z? 0
(continued)
CVE 472 Statistical Techniques in Hydrology. 37/47
3. NORMAL DISTRIBUTION3. NORMAL DISTRIBUTION
Find the Z value for 20% in the Lower Tail
20% area in the lower tail is consistent with a Z value of -0.84
Z .03
-0.9 .1762 .1736
.2033
-0.7 .2327 .2296
.04
-0.8 .2005
Standardized Normal Probability Table (Portion)
.05
.1711
.1977
.2266
…
…
…
…X? 8.0
0.2000
Z-0.84 0
1. Find the Z value for the known probability
CVE 472 Statistical Techniques in Hydrology. 38/47
3. NORMAL DISTRIBUTION3. NORMAL DISTRIBUTION
2. Convert to X units using the formula:
Finding the X value
80.3
0.5)84.0(0.8
ZσµX
=
−+=
+=
So 20% of the values from a distribution with mean 8.0 and standard deviation 5.0 are less than 3.80
CVE 472 Statistical Techniques in Hydrology. 39/47
3. NORMAL DISTRIBUTION3. NORMAL DISTRIBUTION
Standard Normal Distribution
Solve Example 5.3 p.87
Example 5.4 p.88
Example 5.5 p.88
CVE 472 Statistical Techniques in Hydrology. 40/47
3. NORMAL DISTRIBUTION3. NORMAL DISTRIBUTION
Outline
General Normal Distribution
Per-productive Properties
Standard Normal Distribution
Central Limit Theorem
Constructing Normal Curves for Data
CVE 472 Statistical Techniques in Hydrology. 41/47
3. NORMAL DISTRIBUTION3. NORMAL DISTRIBUTION
Averaging Independent Normal Random Variables
If Xi ~ N(µ, σ2), 1≤ i ≤ n, are independent random variables, then their average is distributed X
nσµ,N~X
2
Central Limit Theorem
CVE 472 Statistical Techniques in Hydrology. 42/47
3. NORMAL DISTRIBUTION3. NORMAL DISTRIBUTION
Outline
General Normal Distribution
Per-productive Properties
Standard Normal Distribution
Central Limit Theorem
Constructing Normal Curves for Data
CVE 472 Statistical Techniques in Hydrology. 43/47
3. NORMAL DISTRIBUTION3. NORMAL DISTRIBUTION
Constructing Normal Curves For Data
Frequently the histogram of a set of observed data suggests that the data may be approximated by a normal distribution.
One way to investigate the goodness of this approximation is by superimposing a normal curve on the frequency histogram and visually compare the two distributions.
CVE 472 Statistical Techniques in Hydrology. 44/47
3. NORMAL DISTRIBUTION3. NORMAL DISTRIBUTION
Constructing Normal Curves For Data
This is possible by plotting both observed relative frequency and expected (according to normal distribution) relative frequency and visually compare two.
In order to find expected relative frequency for any class interval, standard normal table or software such as EXCELL or MATLAB can be used.
EXCEL normdist()MATLAB normcdf()
CVE 472 Statistical Techniques in Hydrology. 45/47
3. NORMAL DISTRIBUTION3. NORMAL DISTRIBUTION
Constructing Normal Curves For Data
ExampleData: Kentucky River peak flows 1895-1960Xmin= 20600 cfsXmax= 115000 cfsµ = sample mean = 67509 cfsσ = sample standard deviation = 20952 cfsTo find expected relative frequency for class mark 25000 with 10000 class interval (class limits 20000 – 30000) using EXCEL:f25000 =normdist(30000, µ, σ,TRUE)-normdist(20000, µ, σ,TRUE)
= 0.0250MATLAB:f25000 =normcdf(30000, µ, σ)-normcdf(20000, µ, σ)
= 0.0250
CVE 472 Statistical Techniques in Hydrology. 46/47
3. NORMAL DISTRIBUTION3. NORMAL DISTRIBUTION
Distribution Comparison
Comparison of normal distribution with the observed distribution of peak flows in the Kentucky River.
2 3 4 5 6 7 8 9 10 11 12
x 104
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
PEAK FLOW (cfs)
RE
LATI
VE
FR
EQ
UE
NC
Y
Normal DistributionObserved Distribution
CVE 472 Statistical Techniques in Hydrology. 47/47
3. NORMAL DISTRIBUTION3. NORMAL DISTRIBUTION
Excersises
Page 94 and 95:5.1 (a) and (b) only5.25.35.65.75.85.145.18