How to Calculate the Area Under a Normal Curve

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You scored a 12 on the math test and you want to know how you did compared to everyone else who took the test. If you plot everyone's score, you will see that the shape resembles a bell curve -- called the normal distribution in statistics. If your data fit a normal distribution, you can convert the raw score to a z-score and use the z-score to compare your standing to everyone else's in the group. This is called estimating the area under the curve.

    Make sure your data are normally distributed. A normal distribution or curve is shaped like a bell with most of the scores in the center, and less the farther the score falls from the center. A standardized normal distribution has a mean of zero and a standard deviation of one. The mean is in the middle of the distribution with half of the scores on the left and half of the scores on the right. The area under the curve is 1.00 or 100 percent. The easiest way to determine that your data are normally distributed is to use a statistical software program such as SAS or Minitab and conduct the Anderson Darling Test of Normality. Given that your data is normal, you can calculate z-score.

    Calculate the mean of your data. To calculate the mean, add up each individual score and divide by the total number of scores. For example, if the sum of all the math scores is 257 and 20 students took the test, the mean would be 257/20 = 12.85.

    Calculate the standard deviation. Subtract each individual score from the mean. If you have a score of 12, subtract this from the mean 12.85 and you get (-0.85). Once you have subtracted each of the individual scores from the mean, square each by multiplying it by itself: (-0.85) * (-0.85) is 0.72. Once you have done this for each of the 20 scores, add all of these together and divide by the total number of scores minus one. If the total is 254.55, divide by 19, which will be 13.4. Finally, take the square root of 13.4 to get 3.66. This is the standard deviation of your population of scores.

    Calculate z-score by using the following formula: score - mean/standard deviation. Your score of 12 -12.85 (the mean) is -(0.85). Dividing the standard deviation of 12.85 results in a z-score of (-0.23). This z-score is negative, meaning that the raw score of 12 was below the mean for the population, which was 12.85. This z-score is exactly 0.23 standard deviation units below the mean.

    Look up the z-value to find the area under the curve up to your z-score. Resource two provides this table. Usually, this kind of table will show the bell-shaped curve and a line indicating your z-score. All of the area below that z-score will be shaded, indicating this table is for looking up scores up to a particular z-score. Ignore the negative sign. For z-score 0.23, look up the first part, 0.2, in the column to the left, and intersect this value with the 0.03 along the top row of the table. The z-value is 0.5910. Multiply this value by 100, showing that 59 percent of the test scores were lower than 12.

    Calculate the percentage of scores either above or below your z-score by looking up the z-value in the one-tailed z-table, such as Table One in Resource 3. Tables of this type will show two bell-shaped curves, with the number below a z-score shaded on one curve and the number above a z-score shaded in the second bell curve. Ignore the (-) sign. Look up the z-value in the same way as before, noting a z-value of 0.4090. Multiply this value by 100 to get the percentage of scores falling either above or below the score of 12, which is 41 percent, meaning 41% of the scores were either below 12 or above 12.

    Calculate the percentage of scores both above and below your z-score by using a table with a picture of one bell shaped curve with both the lower tail (left side) and the upper tail (right side) shaded (Table Two in Resource 3). Again, ignore the negative sign and look up the value 0.02 in the column and 0.03 in the row headings to get the z-value of 0.8180. Multiply this number by 100, showing 82 percent of scores on the math test fall both above and below your score of 12.


About the Author

Brenda Scottsdale is a licensed psychologist, a six sigma master black belt and a certified aerobics instructor. She has been writing professionally for more than 15 years in scientific journals, including the "Journal of Criminal Justice and Behavior" and various websites.