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BIOSTATISTICS (BST 211). Sumukh Deshpande n Lecturer College of Applied Medical Sciences. Lecture 5. Statistics = Skills for life. Normal Distribution 1. Empirical vs Theoretical Distributions 1. What are the frequencies?. Freq Tables in previous lectures are empirical.
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BIOSTATISTICS (BST 211) Sumukh DeshpandenLecturerCollege of Applied Medical Sciences Lecture 5 Statistics = Skills for life.
Empirical vs Theoretical Distributions 1 What are the frequencies? Freq Tables in previous lectures are empirical. You don’t know unless told! You can NOT work out EMPIRICAL data
Empirical vs Theoretical Distributions 2 What are the frequencies? Now f = 2x +1 You can calculate THEORETICAL DATA
This is a bell shaped curve with different centers and spreads depending on and The Normal Distribution:a theoretical function… (pdf) Note constants: =3.14159 e=2.71828
The Normal Distribution f(X) Changingμshifts the distribution left or right. Changing σ increases or decreases the spread. s m X The good news is: I am NOT expecting much maths
7-3 Why Study Normal Distribution? • Natural phenomena follow Normal distribution. • Growth, size, height, weight, IQ, weather, …. All more or less follow Normal Distribution • If you learn Normal Distribution you might be able to predict behaviour and have better control
7-3 Characteristics of a Normal Distribution • The normal curve is bell-shaped and has a single peak at the exact center of the distribution. • The arithmetic mean, median, and mode of the distribution are equal and located at the peak. • Half the area under the curve is above the peak, and the other half is below it.
7-4 Characteristics of a Normal Distribution • The normal probability distribution is symmetrical about its mean. • The normal probability distribution is asymptotic - the curve gets closer and closer to the x-axis but never actually touches it.
The beauty of the Normal Curve: No matter what and are, the area between - and + is about 68%; the area between -2 and +2 is about 95%; and the area between -3 and +3 is about 99.7%. Nearly all values fall within 3 standard deviations.
68% of the data 95% of the data 99.7% of the data 68-95-99.7 Rule
Another beauty of the Normal Curve: If a dataset (x1, x2,…..xn) follows the Normal distribution with mean m and SD s, then the derived data set [(x1 – m)/s, (x2 – m)/s, ... (xn – m)/s] also follows Normal distribution BUT with mean m = 0 and s = 1. N(m , s) N(0 , 1) Normal distribution with mean m = 0 and s = 1 is called STANDARD NORMAL DISTRIBUTION (SND)
STANDARD NORMAL DISTRIBUTION (SND) SND values, also known as z-scores, are tabulated and you are NOT expected to learn them by heart.. You just need to know how to use these tables!
N(0 ,1) Here is a table of z-scores! What do I do with it?
What do you need to do? • Learn how to convert X to Z • Learn how to use SND Tables
Practice problem If birth weights in a population are normally distributed with a mean of 2 kg and a standard deviation of 0.15 kg, • What is the chance of obtaining a birth weight of 2.369 kg or heavier? • What is the chance of obtaining a birth weight of 1.8 kg or lighter?
Answer a / 1 • What is the chance of obtaining a birth weight of 2.369 kg or heavier? Remember! Z-scores refer to the area on the left of z value (≤) In this case, we want 2.369 kg or HEAVIER…. Z ≥ 2.46
Answer a / 2 Remember! Z-scores refer to the are on the left of z value (≤) In this case, we want 2.369 kg or HEAVIER…. Z ≥ 2.46 P(Z ≥ 2.46) = 1 – P(Z ≤ 2.46) = 1 - ???
Z=2.4? Looking up probabilities in a SND table What is the area to the left of Z=2.46 in a standard normal curve? Area is 99.31% Z=2.46
Answer a /3 Remember! Z-scores refer to the are on the left of z value (≤) In this case, we want 2.369 kg or HEAVIER…. Z ≥ 2.369 P(Z ≥ 2.46) = 1 – P(Z ≤ 2.46) = 1 -??? Therefore P(Z ≥ 2.46) = 1 – P(Z ≤ 2.46) = 1 – 0.9931 = 0.0069 = 0.69%
Answer b b. What is the chance of obtaining a birth weight of 1.8 kg or lighter? • P(-Z) = 1 - P(Z) • P(Z ≤ -1.33) = 1 - P(Z≤1.33) =1 – 0.9082 = 9.18 %