3.3.2 - The Standard Normal Distribution

3.3.2 - The Standard Normal Distribution

A special case of the normal distribution has mean \(\mu = 0\) and a variance of \(\sigma^2 = 1\). The 'standard normal' is an important distribution.

Standard Normal Distribution

A standard normal distribution has a mean of 0 and variance of 1. This is also known as a z distribution. You may see the notation \(N(\mu, \sigma^2\)) where N signifies that the distribution is normal, \(\mu\) is the mean, and \(\sigma^2\) is the variance. A Z distribution may be described as \(N(0,1)\). Note that since the standard deviation is the square root of the variance then the standard deviation of the standard normal distribution is 1.

The Standard Normal Distribution
0.0 0.1 0.2 0.3 0.4 −2 −1 1 −3 3 0 2 X Standard Normal Distribution, N(0,1)

Finding Probabilities of a Standard Normal Random Variable

As we mentioned previously, calculus is required to find the probabilities for a Normal random variable.  Fortunately, we have tables and software to help us.

For any normal random variable, we can transform it to a standard normal random variable by finding the Z-score. Then we can find the probabilities using the standard normal tables.

Most statistics books provide tables to display the area under a standard normal curve. Look in the appendix of your textbook for the Standard Normal Table. We include a similar table, the Standard Normal Cumulative Probability Table so that you can print and refer to it easily when working on the homework.

Most standard normal tables provide the “less than probabilities”.  For example, if \(Z\) is a standard normal random variable, the tables provide \(P(Z\le a)=P(Z<a)\), for a constant, \(a\).

Example 3-9: Probability 'less than'

Find the area under the standard normal curve to the left of 0.87.

There are two main ways statisticians find these numbers that require no calculus! Click on the tabs below to see how to answer using a table and using technology.

A typical four-decimal-place number in the body of the Standard Normal Cumulative Probability Table gives the area under the standard normal curve that lies to the left of a specified z-value. The probability to the left of z = 0.87 is 0.8078 and it can be found by reading the table:

  1. Since z = 0.87 is positive, use the table for POSITIVE z-values.
  2. Go down the left-hand column, label z to "0.8."
  3. Then, go across that row until under the "0.07" in the top row.

You should find the value, 0.8078. Therefore,\(P(Z< 0.87)=P(Z\le 0.87)=0.8078\)

z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09
0.6 .7257 .7291 .7324 .7357 .7389 .7422 .7454 .7586 .7517 .7549
0.7 .7580 .7611 .7642 .7673 .7704 .7734 .7764 .7794 .7823 .7852
0.8 .7881 .7910 .7939 .7967 .7995 .8023 .8051 .8078 .8106 .8133
0.9 .8159 .8186 .8212 .8238 .8264 .8289 .8315 .8340 .8365 .8389

Using Minitab

To find the area to the left of z = 0.87 in Minitab...

  1. From the Minitab menu select Calc> Probability Distributions> Normal.
  2. Select Cumulative Probability.
  3. In the Input constant box, enter 0.87. Click OK

Cumulative Probability window in Minitab

You should see a value very close to 0.8078.

area left of .87

Example 3-10: Probability 'greater than'

Find the area under the standard normal curve to the right of 0.87.

Based on the definition of the probability density function, we know the area under the whole curve is one. Since we are given the “less than” probabilities in the table, we can use complements to find the “greater than” probabilities. Therefore,

\(P(Z>0.87)=1-P(Z\le 0.87)\).

Using the information from the last example, we have \(P(Z>0.87)=1-P(Z\le 0.87)=1-0.8078=0.1922\)

Using Minitab

Since we are given the “less than” probabilities when using the cumulative probability in Minitab, we can use complements to find the “greater than” probabilities. Therefore,

\(P(Z>0.87)=1-P(Z\le 0.87)\).

Using the information from the last example, we have \(P(Z>0.87)=1-P(Z\le 0.87)=1-0.8078=0.1922\)

You can also use the probability distribution plots in Minitab to find the "greater than."

  1. Select Graph> Probability Distribution Plot> View Probability and click OK.
  2. In the pop-up window select the Normal distribution with a mean of 0.0 and a standard deviation of 1.0.
  3. Select the Shaded Area tab at the top of the window.
  4. Select X Value.
  5. Enter 0.87 for X value.
  6. Select Right Tail.
  7. Click OK.

Probability distribution window in Minitab

Greater than Minitab graph

Example 3-11: Probability 'between'

Find the area under the standard normal curve between 2 and 3.

To find the probability between these two values, subtract the probability of less than 2 from the probability of less than 3. In other words,

\(P(2<Z<3)=P(Z<3)-P(Z<2)\)

\(P(Z<3)\) and \(P(Z<2)\) can be found in the table by looking up 2.0 and 3.0.

For 3.0...

z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09
2.8 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.9980
2.9 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.9986
3.0 0.9987 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9989 0.9990 0.9990
3.1 0.9990 0.9991 0.9991 0.9991 0.9992 0.9992 0.9992 0.9992 0.9993 0.9993

 

For 2.0...

z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09
1.8 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.9706
1.9 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.9767
2.0 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.9817
2.1 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.9857

 

\(P(2 < Z < 3)= P(Z < 3) - P(Z \le  2)= 0.9987 - 0.9772= 0.0215\).

Using Minitab

To find the area between 2.0 and 3.0 we can use the calculation method in the previous examples to find the cumulative probabilities for 2.0 and 3.0 and then subtract.

\(P(2 < Z < 3)= P(Z < 3) - P(Z \le  2)= 0.9987 - 0.9772= 0.0215\)

You can also use the probability distribution plots in Minitab to find the "between."

  1. Select Graph> Probability Distribution Plot> View Probability and click OK.
  2. In the pop-up window select the Normal distribution with a mean of 0.0 and a standard deviation of 1.0.
  3. Select the Shaded Area tab at the top of the window.
  4. Select X Value.
  5. Select Middle.
  6. Enter 2.0 for X value 1 and 3.0 for X value 2.
  7. Click OK.

Between 2 values Minitab graph

Percentiles of the Standard Normal Distribution

Recall from Lesson 1 that the \(p(100\%)^{th}\) percentile is the value that is greater than  \(p(100\%)\) of the values in a data set. We can use the standard normal table and software to find percentiles for the standard normal distribution.

The intersection of the columns and rows in the table gives the probability. If we look for a particular probability in the table, we could then find its corresponding Z value.

Example 3-12: Percentiles in the Standard Normal Distribution

Find the 10th percentile of the standard normal curve.

The question is asking for a value to the left of which has an area of 0.1 under the standard normal curve.

Since the entries in the Standard Normal Cumulative Probability Table represent the probabilities and they are four-decimal-place numbers, we shall write 0.1 as 0.1000 to remind ourselves that it corresponds to the inside entry of the table. We search the body of the tables and find that the closest value to 0.1000 is 0.1003. We look to the leftmost of the row and up to the top of the column to find the corresponding z-value.
The corresponding z-value is -1.28. Thus z = -1.28.  

z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09
-1.3 0.0968 0.0951 0.934 0.0918 0.0901 0.0885 0.0869 0.0853 0.0838 0.0823
-1.2 0.1150 0.1131 0.1112 0.1093 0.1075 0.1056 0.1038 0.1020 0.1003 00985
-1.1 0.1357 0.1335 0.1314 0.1292 0.1271 0.1251 0.1230 0.1210 0.1190 0.1170
-1.0 0.1587 0.1562 0.1539 0.1515 0.1492 0.1469 0.1446 0.1423 0.1401 0.1379

 
Therefore, the 10th percentile of the standard normal distribution is -1.28

Using Minitab

To find the 10th percentile of the standard normal distribution in Minitab...

  1. Select Calc> Probability Distributions> Normal.
  2. In the new window choose Inverse Cumulative Probability.
  3. Enter 0.1 in the Input constant box.
  4. Click OK.

You should see a value very close to -1.28.

10th percentile Minitab graph


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