►Finding the Area under a Normal Curve
► Area less than or equal to a value:
Example 1, Page 385
To find the proportion of three-year-old females who have a height less than 35 inches, we create a new variable with the value 35. Once the variable has been defined, go to Transform → Compute…
While in the compute variable window, create a name for the probabilities in the Target Variable box. In the Function Group box, select CDF and Noncentral CDF. CDF is the Cumulative Distribution Function.
In the Functions and Special Variables, select CDF.NORMAL.
Chapter 7. The Normal Probability Distribution 95
For this function to work, you need three values. The first is x, which is the variable we created earlier.
The second is the mean of the Normal distribution, and the third is the standard deviation of the Normal distribution. So, we will select CDF.NORMAL, and replace the question marks that appear with x, 38.72, and 3.17. When selecting the name of the variable we wish to use, we can either type the name (if it is short), or highlight the name, and push the ► button next to the Numeric Expression box.
Once the name and expression have been completed, push the OK button. This will create a new column in the data view window with the probabilities of observing an x value less than or equal to 35 inches. (You may need to change the number of decimal places shown in the variable view window to see some of the values).
Note: SPSS answers may vary slightly from book answers due to the book rounding to 2 decimal places for the z-score. SPSS does not round in its calculations.
So, 12.03% of three-year-old females have a height less than 35 inches.
► Area greater than or equal to a value:
To find the area greater than or equal to a value, use 1-CDF.NORMAL as the equation. The CDF is always the area to the left, so we can use the complement rule to find the area to the right.
► Area between two values:
Example 3, Page 387
To find the area between two values, you can start with typing in one value in the first column. This value is irrelevant, as it serves only to define the sample size. . Once the variable has been defined, go to Transform → Compute…
While in the compute variable window, create a name for the probabilities in the Target Variable box. In the Function Group box, select CDF and Noncentral CDF. CDF is the Cumulative Distribution Function.
In the Functions and Special Variables, select CDF.NORMAL. Instead of using a variable, x, for the first value, we type in the larger of the two x values. We then subtract the CDF.NORMAL of the smaller of the two x values from the first.
Once the name and expression have been completed, push the OK button. This will create a new column in the data view window with the probabilities of observing an x value between 35 and 40. (You may need to change the number of decimal places shown in the variable view window to see some of the values).
Chapter 7. The Normal Probability Distribution 97
If you don’t create the dummy variable first, you will get an error, as SPSS will not know how many answers to create. If you have more than one value, the answer will be repeated as many times as you have values in the data set.
The probability of finding a three-year-old female between 35 inches and 40 inches is .53652.
►Finding Values of Normal Random Variables
► Area less than or equal to a value:
Example 4, Page 388
When looking for a specific value of x under the Normal Distribution, SPSS must look for values using a cumulative distribution. So, probabilities must be areas to the left of x. If the information does not come in this way, it must be changed to fit how the computer can use it.
In this example, we are looking for the 20th percentile, which is the point at which 20% of the information is less than or equal to x, or the cumulative probability is .20. To find the height of three-year-old females which marks the 20th percentile, we create a new variable with the value .20. Once the variable has been defined, go to Transform → Compute…
While in the compute variable window, create a name for the probabilities in the Target Variable box. In the Function Group box, select Inverse DF. Inverse DF is the Inverse Distribution Function. In the Functions and Special Variables, select IDF.NORMAL.
For this function to work, you need three values. The first is the probability to the left, which is the variable we created earlier. This only works with the cumulative probability. The second is the mean of the Normal distribution, and the third is the standard deviation of the Normal distribution. So, we will select IDF.NORMAL, and replace the question marks that appear with probability, 38.72, and 3.17. When selecting the name of the variable we wish to use, we can either type the name (if it is short), or highlight the name, and push the ► button next to the Numeric Expression box.
Once the name and expression have been completed, push the OK button. This will create a new column in the data view window with the x value. (You may need to change the number of decimal places shown in the variable view window to see some of the values).
Chapter 7. The Normal Probability Distribution 99
The height of a three-year-old female that separates the top 80% from the bottom 20% is 36.05.
► Area more than or equal to a value:
Example 6, Page 389
To get the x value which marks where 1% of the information is to the right of that value, we have to use the complement rule. We know that if 1% is to the right, then 99% is to the left, and that is the information we give to SPSS. We can then follow the steps for area to the left.
There are two advantages of using SPSS in these situations. The first is that we don’t need to find the closest value in the table, as SPSS has the complete table to work from. In other words, we don’t have to worry about rounding for the z-scores. The second is we can look up the values of more than one probability at a time, without adding work.