The Standard Normal Distribution

The standard normal distribution is a normal circulation through a mean of zero and standard deviation of 1. The traditional normal circulation is centered at zero and also the degree to which a offered measurement deviates from the expect is provided by the traditional deviation. For the typical normal circulation, 68% of the observations lie within 1 traditional deviation of the mean; 95% lie within two typical deviation of the mean; and 99.9% lie within 3 typical deviations of the suppose. To this allude, we have actually been making use of "X" to signify the variable of interest (e.g., X=BMI, X=elevation, X=weight). However, when utilizing a traditional normal distribution, we will certainly usage "Z" to describe a variable in the conmessage of a conventional normal distribution. After standarization, the BMI=30 discussed on the previous web page is shown below lying 0.16667 devices above the expect of 0 on the traditional normal circulation on the appropriate.

You are watching: For a normal distribution, a positive value of z indicates that

====

Due to the fact that the area under the standard curve = 1, we have the right to start to even more specifically define the probabilities of particular observation. For any type of offered Z-score we deserve to compute the location under the curve to the left of that Z-score. The table in the structure below mirrors the probabilities for the conventional normal distribution.Examine the table and also note that a "Z" score of 0.0 lists a probcapability of 0.50 or 50%, and a "Z" score of 1, interpretation one typical deviation over the intend, lists a probcapability of 0.8413 or 84%. That is bereason one traditional deviation over and below the expect encompasses around 68% of the area, so one standard deviation above the expect represents half of that of 34%. So, the 50% listed below the suppose plus the 34% above the suppose offers us 84%.

Probabilities of the Standard Typical Distribution Z


This table is arranged to carry out the location under the curve to the left of or less of a specified worth or "Z value". In this case, bereason the mean is zero and also the typical deviation is 1, the Z worth is the variety of typical deviation devices ameans from the mean, and also the area is the probcapability of observing a value much less than that particular Z worth. Keep in mind also that the table mirrors probabilities to two decimal locations of Z. The devices area and also the first decimal place are displayed in the left hand column, and also the second decimal place is presented across the optimal row.

But let"s acquire earlier to the question around the probcapability that the BMI is much less than 30, i.e., P(XDistribution of BMI and Standard Typical Distribution

====

The location under each curve is one but the scaling of the X axis is different. Note, but, that the locations to the left of the daburned line are the same. The BMI circulation arrays from 11 to 47, while the standardized normal circulation, Z, arrays from -3 to 3. We desire to compute P(X Z score, additionally referred to as a standardized score:

*

where μ is the expect and σ is the conventional deviation of the variable X.

In order to compute P(X standardizing):

*

Hence, P(X Another Example

Using the exact same distribution for BMI, what is the probcapability that a male aged 60 has BMI exceeding 35? In other words, what is P(X > 35)? Aacquire we standardize:

*

We now go to the conventional normal circulation table to look up P(Z>1) and also for Z=1.00 we find that P(Z1)=1-0.8413=0.1587. Interpretation: Almost 16% of men aged 60 have BMI over 35.

Normal Probability Calculator

*

Z-Scores through R

As an different to looking up normal probabilities in the table or making use of Excel, we have the right to use R to compute probabilities. For example,

> pnorm(0)

<1> 0.5

A Z-score of 0 (the intend of any distribution) has 50% of the area to the left. What is the probcapability that a 60 year old male in the population over has actually a BMI much less than 29 (the mean)? The Z-score would certainly be 0, and also pnorm(0)=0.5 or 50%.

What is the probcapacity that a 60 year old guy will certainly have actually a BMI much less than 30? The Z-score was 0.16667.

> pnorm(0.16667)

<1> 0.5661851

So, the probabilty is 56.6%.

What is the probability that a 60 year old male will certainly have a BMI greater than 35?

35-29=6, which is one typical deviation over the suppose. So we have the right to compute the area to the left

> pnorm(1)

<1> 0.8413447

and also then subtract the outcome from 1.0.

1-0.8413447= 0.1586553

So the probability of a 60 year ld guy having a BMI greater than 35 is 15.8%.

Or, we can usage R to compute the whole thing in a single action as follows:

> 1-pnorm(1)

<1> 0.1586553

Probability for a Range of Values

What is the probability that a male aged 60 has actually BMI between 30 and also 35? Note that this is the exact same as asking what propercentage of males aged 60 have actually BMI in between 30 and also 35. Specifically, we want P(30 Answer

Now take into consideration BMI in woguys. What is the probability that a female aged 60 has actually BMI less than 30? We usage the exact same approach, however for women aged 60 the mean is 28 and also the traditional deviation is 7.

Answer

What is the probability that a female aged 60 has BMI exceeding 40? Specifically, what is P(X > 40)?

40) = P(Z > (40-28/7 = 12/7 = 1.71.

See more: Why Are Models Based On Assumptions? ? A Model Assumptions — Explained

Now we must compute P(Z>1.71). If we look up Z=1.71 in the standard normal circulation table, we discover that P(Z 40?", CAPTIONSIZE, 2, CGCOLOR, "#c00000", PADX, 5, 5, PADY, 5, 5,BUBBLECLOSE, STICKY, CLOSECLICK, CLOSETEXT, "", BELOW, RIGHT, BORDER, 1, BGCOLOR, "#c00000", FGCOLOR, "#ffffff", WIDTH, 600, TEXTSIZE, 2, TEXTCOLOR, "#000000", CAPCOLOR, "#ffffff");" onfocus="rerotate overlib("Aacquire we standardize P(X > 40) = P(Z > (40-28/7 = 12/7 = 1.71.

Now we have to compute P(Z>1.71). If we look up Z=1.71 in the standard normal distribution table, we discover that P(Z 40?", CAPTIONSIZE, 2, CGCOLOR, "#c00000", PADX, 5, 5, PADY, 5, 5,BUBBLECLOSE, STICKY, CLOSECLICK, CLOSETEXT, "", BELOW, RIGHT, BORDER, 1, BGCOLOR, "#c00000", FGCOLOR, "#ffffff", WIDTH, 600, TEXTSIZE, 2, TEXTCOLOR, "#000000", CAPCOLOR, "#ffffff");">Answer

return to peak | previous page | next page


Content ©2016. All Rights Reserved.Date last modified: July 24, 2016.Wayne W. LaMorte, MD, PhD, MPH