Sumukh Deshpande n Lecturer College of Applied Medical Sciences 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. You don’t know unless told! x f 5 8 9 12 You can NOT work out EMPIRICAL data

Empirical vs Theoretical Distributions 2 What are the frequencies? Now f = 2x +1 x f 5 11 8 17 9 19 12 25 You can calculate THEORETICAL DATA

The Normal Distribution: a theoretical function… (pdf) This is a bell shaped curve with different centers and spreads depending on  and  Note constants: =3.14159 e=2.71828

Bell-Shaped

The Normal Distribution f(X) Changing μ shifts 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

Why Study Normal Distribution? 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

Characteristics of a Normal Distribution 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.

Characteristics of a Normal Distribution 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.

68-95-99.7 Rule 68% of the data 95% of the data 99.7% of the data SAY: within 1 standard deviation either way of the mean within 2 standard deviations of the mean within 3 standard deviations either way of the mean WORKS FOR ALL NORMAL CURVES NO MATTER HOW SKINNY OR FAT

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!

Here is a table of z-scores! 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 - ???

Looking up probabilities in a SND table What is the area to the left of Z=2.46 in a standard normal curve? Z=2.46 Area is 99.31% Z=2.4?

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 %