The standard normal distribution is a normal distribution with a average of zero and also standard deviation that 1. The standard normal distribution is focused at zero and also the degree to which a given measurement deviates indigenous the average is provided by the standard deviation. For the standard normal distribution, 68% of the monitorings lie in ~ 1 typical deviation the the mean; 95% lie within two standard deviation of the mean; and 99.9% lie within 3 standard deviations of the mean. Come this point, we have actually been making use of "X" to signify the variable of interest (e.g., X=BMI, X=height, X=weight). However, once using a traditional normal distribution, us will usage "Z" to describe a variable in the paper definition of a standard normal distribution. ~ standarization, the BMI=30 discussed on the previous page is shown below lying 0.16667 units over the typical of 0 top top the traditional normal circulation on the right.

You are watching: In a standard normal distribution, the range of values of z is from

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Since the area under the conventional curve = 1, we can start to much more precisely specify the probabilities of specific observation. For any given Z-score we can compute the area under the curve come the left of the Z-score. The table in the frame listed below shows the probabilities because that the conventional normal distribution.Examine the table and note the a "Z" score that 0.0 list a probability that 0.50 or 50%, and a "Z" score of 1, an interpretation one standard deviation above the mean, lists a probability that 0.8413 or 84%. The is due to the fact that one standard deviation **above and below** the typical encompasses about 68% the the area, so one standard deviation above the mean represents fifty percent of the of 34%. So, the 50% listed below the mean plus the 34% above the mean gives us 84%.

## Probabilities the the typical Normal circulation Z

This table is organized to administer the area under the curve come the left of or less of a mentioned value or "Z value". In this case, due to the fact that the median is zero and the typical deviation is 1, the Z value is the number of standard deviation units away from the mean, and the area is the probability that observing a value less than that details Z value. Note additionally that the table reflects probabilities to two decimal areas of Z. The units place and also the an initial decimal location are shown in the left hand column, and the 2nd decimal location is displayed across the peak row.

But let"s get ago to the question around the probability that the BMI is much less than 30, i.e., P(XDistribution the BMI and also Standard normal Distribution

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The area under every curve is one but the scaling that the X axis is different. Note, however, that the locations to the left of the dashed line room the same. The BMI circulation ranges native 11 to 47, if the standardized normal distribution, Z, varieties from -3 to 3. We desire to compute P(X Z score, additionally called a **standardized score**:

where μ is the mean and σ is the conventional deviation the the change X.

In order come compute P(X standardizing):

Thus, P(X an additional Example

Using the same distribution for BMI, what is the probability that a masculine aged 60 has actually BMI **exceeding** 35? In other words, what is P(X > 35)? Again we standardize:

We currently go to the standard normal circulation table to look increase P(Z>1) and also for Z=1.00 we find that P(Z1)=1-0.8413=0.1587. Interpretation: practically 16% of guys aged 60 have actually BMI end 35.

## Normal Probability Calculator

## Z-Scores through R

As an alternative to spring up typical probabilities in the table or making use of Excel, we have the right to use R come compute probabilities. Because that example,

> pnorm(0)

<1> 0.5

A Z-score the 0 (the mean of any kind of distribution) has 50% the the area to the left. What is the probability that a 60 year old man in the populace above has a BMI much less than 29 (the mean)? The Z-score would be 0, and also pnorm(0)=0.5 or 50%.

What is the probability the a 60 year old man will have actually a BMI less than 30? The Z-score to be 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 have actually a BMI * greater* 보다 35?

35-29=6, i beg your pardon is one typical deviation above the mean. For this reason we deserve to compute the area come the left

> pnorm(1)

<1> 0.8413447

and climate subtract the an outcome from 1.0.

1-0.8413447= 0.1586553

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

Or, we have the right to use R to compute the whole thing in a single step as follows:

> 1-pnorm(1)

<1> 0.1586553

## Probability for a selection of Values

What is the probability the a male aged 60 has actually BMI between 30 and 35? note that this is the exact same as questioning what proportion of guys aged 60 have BMI between 30 and also 35. Special, we want P(30 Answer

Now think about BMI in women. What is the probability that a female age 60 has BMI less than 30? We use the very same approach, yet for ladies aged 60 the average is 28 and the traditional deviation is 7.

Answer

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

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

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Now we have to compute P(Z>1.71). If we look up Z=1.71 in the traditional normal distribution table, we uncover 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="return overlib("Again 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 increase Z=1.71 in the traditional normal circulation table, we uncover 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

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