The normal distribution is a widely used probability distribution that is bell-shaped and symmetric around the mean. It is characterized by two parameters: the mean (μ) and the standard deviation (σ). The standard normal distribution has a mean of 0 and standard deviation of 1. Any normal distribution can be converted to a standard normal distribution using standardization, allowing probabilities to be easily calculated using standard normal probability tables. These tables provide the probability that a standard normal random variable Z will fall below various z-values.
Computer Oriented Numerical Analysis
What is interpolation?
Many times, data is given only at discrete points such as .
So, how then does one find the value of y at any other value of x ?
Well, a continuous function f(x) may be used to represent the data values with f(x) passing through the points (Figure 1). Then one can find the value of y at any other value of x .
This is called interpolation
Newton’s Divided Difference Formula:
To illustrate this method, linear and quadratic interpolation is presented first.
Then, the general form of Newton’s divided difference polynomial method is presented.
Numerical method (curve fitting)
***TOPICS ARE****
Linear Regression
Multiple Linear Regression
Polynomial Regression
Example of Newton’s Interpolation Polynomial And example
Example of Newton’s Interpolation Polynomial And example
Computer Oriented Numerical Analysis
What is interpolation?
Many times, data is given only at discrete points such as .
So, how then does one find the value of y at any other value of x ?
Well, a continuous function f(x) may be used to represent the data values with f(x) passing through the points (Figure 1). Then one can find the value of y at any other value of x .
This is called interpolation
Newton’s Divided Difference Formula:
To illustrate this method, linear and quadratic interpolation is presented first.
Then, the general form of Newton’s divided difference polynomial method is presented.
Numerical method (curve fitting)
***TOPICS ARE****
Linear Regression
Multiple Linear Regression
Polynomial Regression
Example of Newton’s Interpolation Polynomial And example
Example of Newton’s Interpolation Polynomial And example
The Normal Distribution:
There are different distributions namely Normal, Skewed, and Binomial etc.
Objectives:
Normal distribution its properties its use in biostatistics
Transformation to standard normal distribution
Calculation of probabilities from standard normal distribution using Z table.
Normal distribution:
- Certain data, when graphed as a histogram (data on the horizontal axis, frequency on the vertical axis), creates a bell-shaped curve known as a normal curve, or normal distribution.
- Two parameters define the normal distribution, the mean (µ) and the standard deviation (σ).
Properties of the Normal Distribution:
Normal distributions are symmetrical with a single central peak at the mean (average) of the data.
The shape of the curve is described as bell-shaped with the graph falling off evenly on either side of the mean.
Fifty percent of the distribution lies to the left of the mean and fifty percent lies to the right of the mean.
-The mean, the median, and the mode fall in the same place. In a normal distribution the mean = the median = the mode.
- The spread of a normal distribution is controlled by the
standard deviation.
In all normal distributions the range ±3σ includes nearly
all cases (99%).
Uni modal:
One mode
Symmetrical:
Left and right halves are mirror images
Bell-shaped:
With maximum height at the mean, median, mode
Continuous:
There is a value of Y for every value of X
Asymptotic:
The farther the curve goes from the mean, the closer it gets to the X axis but it never touches it (or goes to 0).
The total area under a normal distribution curve is equal to 1.00, or 100%.
Using Normal distribution for finding probability:
While finding out the probability of any particular observation we find out the area under the curve which is covered by that particular observation. Which is always 0-1.
Transforming normal distribution to standard normal distribution:
Given the mean and standard deviation of a normal distribution the probability of occurrence can be worked out for any value.
But these would differ from one distribution to another because of differences in the numerical value of the means and standard deviations.
To get out of this problem it is necessary to find a common unit of measurement into which any score could be converted so that one table will do for all normal distributions.
This common unit is the standard normal distribution or Z
score and the table used for this is called Z table.
- A z score always reflects the number of standard deviations above or below the mean a particular score or value is.
where
X is a score from the original normal distribution,
μ is the mean of the original normal distribution, and
σ is the standard deviation of original normal distribution.
Steps for calculating probability using the Z-
score:
-Sketch a bell-shaped curve,
- Shade the area (which represents the probability)
-Use the Z-score formula to calculate Z-value(s)
-Look up Z-values in table
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2. The Normal Distribution
• Properties
– Bell shaped
1
– Area under curve equals 1
– Symmetric around the mean μ
– Mean = median = Mode
– Two tails approach the horizontal axis – never
touch axis
– Empirical rule applies
– Two parameters – μ and σ 2
3. How does the standard deviation affect the shape of the distribution
s=2
s =3
s =4
m = 11
How does the mean affect the location of the distribution
s=2
m = 10 m = 11 m = 12
3
4. The Normal Distribution
Mathematical model expressed as:
1 xm
2
1
2 s
f ( x) e , x
2s
where 3.14159 and e 2.71828
- notation N(μ ; σ2)
4
5. STANDARDISING THE
RANDOM VARIABLE
• Seen how different means and std dev’s
generate different normal distributions
• This means a very large number of
probability tables would be needed to
provide all possible probabilities
• We therefore standardise the random
variable x so that only one set of tables is
needed
5
6. STANDARDISING THE
RANDOM VARIABLE
• A normal random variable x can be
converted to a standard normal variable
(denoted Z) by using the following
standardisation formula:-
xm
z , x any value of the random variable X
s
6
7. The Standard Normal Distribution
• Different values of μ and σ generate
different normal distributions
• The random variable X can be
standardised
– mean = μ = 0
– standard deviation = σ = 1
xm
z , x any value of the random variable X
s 7
8. The Standard Normal Distribution
μ=0
s=1
z
-3 -2 -1 0 1 2 3
z-values on the horizontal axis
• distance between the mean and the point
represented by z in terms of standard deviation 8
9. Finding Normal Probabilities
• Example
– Marks for a semester test is normally
distributed, with a mean of 60
and a standard deviation of 8
– X ~ N(60 , 82)
– If we need to determine the
probability that the mark will x
50 60 65
be between 50 and 65,
we need to determine the size of the shaded area
– Before calculating the probabilities the x-values
need to be transformed to z-values 9
10. P(-∞ < Z < z)
Standard normal probabilities have been
calculated and are provided in a table
The tabulated probabilities
correspond to the area between
0 z
Z = -∞ and some z > 0
z 0.00 0.01 → 0.05 0.06 → 0.09
0.0 0.5000 0.5040 0.5199 0.5239 0.5359
0.1 0.5398 0.5438 0.5596 0.5636 0.5753
↓
1.0 0.8413 0.8438 0.8531 0.8554 0.8621
1.1 0.8643 0.8665 0.8749 0.8770 0.8830
1.2 0.8849 0.8869 0.8944 0.8962 0.9015
10
↓
11. • Example continue
– If X denotes the test mark, we seek the
probability
– P(50 < X < 65)
– Transform the X to the standard normal
variable Z
Every normal variable X m Therefore, once
with some m and s, Z probabilities for Z are
can be transformed s calculated, probabilities
into this Z E(Z) V(Z) of any normal variable
11
μ=0 σ2 = 1 can be found
12. • Example continue Mean = μ = 60 minutes
Standard deviation = σ = 8 minutes
X m
50 -- 60 < Z < 65 -- 60 )
X m
P(50 < X < 65) = P(
8s 8s
= P(-1.25 < Z < 0.63) X-m
Z=
s
To complete the calculation we need to compute
the probability under the standard normal distribution12
13. • How to use the z-table to calculate
probabilities Example
Determine the following probability: P(Z > 1.05) = ?
1 - P(Z < 1.05)
= P(Z > 1.05)
0 1.05
P(Z > 1.05) = 1 – 0.8531 = 0.1469
13
14. • How to use the z-table to calculate probabilities
Example
Determine the following probability: P(-2.12 < Z < 1.32) = ?
0.9066
P(-∞ < Z < 1.32) = 0.9066
0,5
1.32
P(-2.12 < Z < +∞) = 0.9830
0.9830
0,5
-2.12
+ -2.12 0 1.32
P(-2.12 < Z < 1.32) = (0.9066 + 0.9830) - 1 = 0.8896 14
15. • Example
– Determine the following probabilities:
P(0.73 < Z < 1.40) = ?
P(-∞ < Z < 1.40) = 0.9192
0.9192
1.40
P(-∞ < Z < 0.73) = 0.7673
0.7673 0 0.73 1.40
0.73
P(0.73 < Z < 1.40) = 0.9192 – 0.7673 = 0.1519 15
16. The symmetry of the normal distribution
makes it possible to calculate probabilities
for negative values of Z using the table as
follows:
P(-z < Z < +∞) = P(-∞ < Z < z) 16
17. • Example student marks - continued
P(50 < X < 65) = P(-1.25 < Z < 0.63) 0.7357
0.8944
= 0.8944 + 0.7357 – 1 0.8944
0.8944
0.8944
0.8944 0.8944
= 0.6301 z = 0.63
In this example z = -1.25 , because of symmetry read 1.25
z 0.00 0.01 → 0.05 0.06 → 0.09
0.0 0.5000 0.5040 0.5199 0.5239 0.5359
0.1 0.5398 0.5438 0.5596 0.5636 0.5753
↓
1.0 0.8413 0.8438 0.8531 0.8554 0.8621
1.1 0.8643 0.8665 0.8749 0.8770 0.8830
1.2 0.8849 0.8869 0.8944 0.8962 0.9015 17