In our earlier discussion of descriptive statistics, we introduced the average as a measure of main tendency and also variance and also standard deviation as procedures of variability. We deserve to now use these parameters to answer inquiries related come probability.

You are watching: Which of the following does not describe the standard normal distribution?

For a normally spread variable in a **population** the average is the ideal measure of central tendency, and the typical deviation(s) provides a measure up of variability.

The notation for a **sample native a population** is contempt different:

We have the right to use the mean and standard deviation to get a handle on probability. It transforms out that, as demonstrated in the figure below,

Approximately 68% of worths in the distribution are within 1 SD of the mean, i.e., above or below.P (µ - σ around 95% of worths in the distribution are within 2 SD of the mean.

P (µ - 2σ approximately 99% of values in the distribution are within 3 SD the the mean.

P (µ - 3σ

There are numerous variables that are generally distributed and can it is in modeled based upon the mean and standard deviation. For example,

BMI:µ=25.5, σ=4.0Systolic BP:µ=133, σ=22.5Birth Wgt. (gms) µ=3300, σ=500Birth Wgt. (lbs.) µ=7.3, σ=1.1The capability to address probability is facility by having plenty of distributions v different way and different standard deviations. The solution to this difficulty is to job these distributions onto a ** standard typical distribution** that will certainly make it simple to compute probabilities.

## The conventional Normal Distribution

The standard normal circulation is a one-of-a-kind normal circulation that has actually a mean=0 and also a typical deviation=1. This is an extremely useful for answering questions around probability, because, when we recognize how many standard deviations a particular result lies far from the mean, us can conveniently determine the probability of seeing a result greater or much less than that.

The figure below shows the percentage of observations that would lie within 1, 2, or 3 standard deviations from any kind of mean in a distribution that is an ext or less generally distributed. Because that a offered value in the distribution, the Z score is the variety of standard deviations over or below the mean. We deserve to think around probability native this.

What is the probability the a value*less than the mean*? The evident answer is 50%.What is the probability of a value less than ns SD

*below*the mean? P= 13.6+2.1+0.1=15.8%What is the probability that a value less than ns SD

*above*the mean? P= 34.1+34.1+13.6+2.1+0.1=84%

**Example:**What is the probability of a Z score much less than 0? Answer: P= 34.1+13.6+ 2.1+0.1=50%What is the probability the a Z score much less than +1? Answer: P= 34.1+34.1+13.6+2.1+0.1=84%

How many standard deviation devices a provided observation lies above or below the average is described as a **Z score**, and also there space tables and computer functions that deserve to tell us the probability of a value much less than a given Z score.

For example, in R:

> pnorm(0)<1> 0.5

The probability of one observation less than the typical is 50%.

> pnorm(1)<1> 0.8413447

The probability of an observation much less than 1 conventional deviation over the mean is 84.13%.

We can additionally look up the probability in a table of Z scores:

So, for any type of distribution the is much more or less generally distributed, if we determine how many standard deviation units a provided value is far from the mean (i.e., its equivalent Z score), then we can determine the probability the a value being less than or greater than that.

It is simple to determine how countless SD units a worth is from the median of a typical distribution:

In various other words, we identify how far a offered value is from the mean and then divide that through the traditional deviation to recognize the matching Z score.

For example, BMI among 60 year old males is normally distributed with µ=29 and also σ=6. What is the probability that a 60 year old masculine selected at random from this populace will have actually a BMI much less than 30? Stated another way, what ratio of the men have actually a BMI much less than 30?

BMI=30 is simply 0.17 SD units over the average of 29. So, every we have to do is look increase 0.17 in the table of Z scores to view what the probability that a value less than 30 is. Keep in mind that the table is set up in a very particular way.The entries in the middle of the table are locations under the standard normal curve **BELOW** the z score.The z score can be discovered by locating the units and also tenths place along the left margin and also the percentage percent place across the peak row.

From the table that Z scores we can see the Z=0.17 corresponds to a probability of 0.5676.

See more: Which Graph Is Used To Show Change In A Given Variable When A Second Variable Is Changed?

We can additionally look up the probability making use of R:

>pnorm(0.17)<1> 0.5674949

You can likewise have R immediately do the calculate of the Z score and also look up the probability by using the pnorm function with the parameters (the value, the mean, and the conventional deviation), e.g.:

# usage "pnorm(x,mean,SD)">pnorm(30,29,6)<1> 0.5661838

The table of probabilities for the conventional normal circulation gives the area (i.e., probability) ** below** a offered Z score, but the entire standard normal circulation has one area that 1, for this reason the

**a Z the 0.17 = 1-0.5675 = 0.4325.**

*area above*You deserve to compute the probability *above* the Z score directly in R:

>1-pnorm(0.17)<1> 0.4325051

**A Slightly various Example:**

Now consider what the probability that BMI

** Conclusion**: In this populace 69% of males who space 60 years old will have actually BMI