The Normal Distribution Finite 9-3

What is Normal Distribution? The normal distribution is a descriptive model that describes real world situations. It is defined as a continuous frequency distribution of infinite range (can take any values not just integers as in the case of binomial and Poisson distribution). This is the most important probability distribution in statistics and important tool in analysis of epidemiological data and management science. The Normal distribution is also known as the Gaussian Distribution and the curve is also known as the Gaussian Curve, named after German Mathematician-Astronomer Carl Frederich Gauss. What is Normal Distribution?

Characteristics of Normal Distribution It links frequency distribution to probability distribution Has a Bell Shape Curve and is Symmetric It is Symmetric around the mean: Two halves of the curve are the same (mirror images) Characteristics of Normal Distribution

Characteristics of Normal Distribution Hence Mean = Median The total area under the curve is 1 (or 100%) Normal Distribution has the same shape as Standard Normal Distribution. Characteristics of Normal Distribution

Characteristics of Normal Distribution In a Standard Normal Distribution: The mean (μ ) = 0 and Standard deviation (σ) =1 Characteristics of Normal Distribution

Z Score (Standard Score) Z = X - μ Z indicates how many standard deviations away from the mean the point x lies. σ The relationship between the normal variable X and Z score is given by the Z score or standard score. Mu (μ) is the mean and sigma (σ) is the standard deviation of the population. Z Score (Standard Score)

Diagram of Normal Distribution Curve (z distribution) 0.135 Diagram of Normal Distribution Curve (z distribution) 0.3335 0.022 0.0015 This is the diagram of a normal distribution curve or z distribution. Note the bell shape of the curve and that its ends/tail don’t touch the horizontal axis below. As I mentioned earlier, the area under the curve equals 1 or 100%. Therefore, each half of the distribution from the center (that is from the mean is equal to 50%. Thus, the area from/above the mean up to 1 standard deviation is equal to 33.35%, area above +1 standard deviation is equal to 13.6%, the area above +2 standard deviation is equal to 2.2% and area above +3 standard deviations is equal to 0.1%. Since the other half is a mirror image, the percentage/proportion of area above -1 standard deviation is the same as the area above + 1 standard deviation i.e. it is 33.35%. And -2 standard deviation=+2 standard deviation and so forth….

Standard Normal Distribution

Probabilities in the Normal Distribution The distribution is symmetric, with a mean of zero and standard deviation of 1. The probability of a score between 0 and 1 is the same as the probability of a score between 0 and –1: both are .34. Thus, in the Normal Distribution, the probability of a score falling within one standard deviation of the mean is .68. Probabilities in the Normal Distribution

Normal Distribution How to utilize Google Sheets Find the area under the curve from negative infinity to x Normal Distribution

For general questions, without a specific mean or standard deviation, just use mean=0 and std dev=1. For normal distribution for the last term in the argument use “TRUE” Normal Distribution

Use normdist when you know the Z, or mean and standard deviation and a specific value and you want to know the percent of area under the curve Example: 0.75 standard deviations above the mean includes what percent of the population? Answer Normal Distribution

Note: you will often need to compute normdist twice, once for the upper bound, and once for the lower bound then subtract to find the total percent. Remember, the function provides the area from negative infinity up to the value x. Normal Distribution

Normal Distribution How to utilize Google Sheets Find the z value given a specific area under the curve Normal Distribution

Again, when working a generic problem, use a mean=0 and standard deviation=1. Normal Distribution

Use norminv when you know the area under the curve, and you want to find a specific Z value or ultimately an x value that results in this value. Example: Which z value results the lowest 25% of the population? Answer: Z=-0.6745 Normal Distribution

Assuming the normal heart rate (H Assuming the normal heart rate (H.R) in normal healthy individuals is normally distributed with Mean = 70 and Standard Deviation =10 beats/min 1) What area under the curve is above 80 beats/min? Exercises

Since M=70, then the area under the curve which is above 80 beats per minute corresponds to above + 1 standard deviation. The total shaded area corresponding to above 1+ standard deviation in percentage is 15.9% or Z= 15.9/100 =0.159. Or we can find the value of z by substituting the values in the formula Z= X-M/ standard deviation. Therefore, Z= 70-80/10 -10/10= -1.00 is the same as +1.00. The value of z from the table for 1.00 is 0.159. How do we interpret this? This means that 15.9% of normal healthy individuals have a heart rate above one standard deviation (greater than 80 beats per minute). Exercise

Exercise # 2 2) What area of the curve is above 90 beats/min? As in question, proceed by drawing the normal distribution curve and calculate the z value……. Exercise # 2

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3) What area of the curve is between 50-90 beats/min? Exercise # 3

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4) What area of the curve is above 100 beats/min? Exercise # 4

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5) What area of the curve is below 40 beats per min or above 100 beats per min? Exercise # 5

In this question, we need to calculate Z1 and Z2 In this question, we need to calculate Z1 and Z2. Therefore, Z1 =70-40/10, which is equal to 3. The z value of 3 is 0.015. Similarly, the value of Z2 is 70-100/10 which is equal to -3. Thus, the value of z is 0.015 And so, Z1+Z2 is equal to 0.3%. But how do we interpret this value? Please see the solution/answer #5 slide for its interpretation. Exercise # 5

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