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Using the Normal Curve
The normal curve is used to find the probability that a value falls within a certain standard deviation away from the mean.
Learning Objectives

Calculate the probability that a variable is within a certain range by finding its zvalue and using the Normal curve

Apply the 689599.7 rule for quick calculations
Key Points
 In order to use the normal curve to find probabilities, the observed value must first be standardized using the following formula:
$z=\frac{x\mu }{\sigma }$ .  To calculate the probability that a variable is within a range, we have to find the area under the curve. Luckily, we have tables to make this process fairly easy.
 When reading the table, we must note that the leftmost column tells you how many sigmas above the the mean the value is to one decimal place (the tenths place), the top row gives the second decimal place (the hundredths), and the intersection of a row and column gives the probability.
 It is important to remember that the table only gives the probabilities to the left of the zvalue and that the normal curve is symmetrical.
 In a normal distribution, approximately 68% of values fall within one standard deviation of the mean, approximately 95% of values fall with two standard deviations of the mean, and approximately 99.7% of values fall within three standard of the mean.
Terms

zvalue
the standardized value of an observation found by subtracting the mean from the observed value, and then dividing that value by the standard deviation; also called zscore

standard deviation
a measure of how spread out data values are around the mean, defined as the square root of the variance
Full Text
ZValue
The functional form for a normal distribution is a bit complicated. It can also be difficult to compare two variables if their mean and or standard deviations are differentfor example, heights in centimeters and weights in kilograms, even if both variables can be described by a normal distribution. To get around both of these conflicts, we can define a new variable:
This variable gives a measure of how far the variable is from the mean (x−), then "normalizes" it by dividing by the standard deviation (σ). This new variable gives us a way of comparing different variables. The zvalue tells us how many standard deviations, or "how many sigmas," the variable is from its respective mean.
Areas Under the Curve
To calculate the probability that a variable is within a range, we have to find the area under the curve. Normally, this would mean we'd need to use calculus. However, statisticians have figured out an easier method, using tables, that can typically be found in your textbook or even on your calculator.
Standard Normal Table
This table can be used to find the cumulative probability up to the standardized normal value z.
These tables can be a bit intimidating, but you simply need to know how to read them. The leftmost column tells you how many sigmas above the the mean to one decimal place (the tenths place).The top row gives the second decimal place (the hundredths).The intersection of a row and column gives the probability.
For example, if we want to know the probability that a variable is no more than 0.51 sigmas above the mean, P(z<0.51), we look at the 6^{th} row down (corresponding to 0.5) and the 2^{nd} column (corresponding to 0.01). The intersection of the 6^{th} row and 2^{nd} column is 0.6950, which tells us that there is a 69.50% percent chance that a variable is less than 0.51 sigmas (or standard deviations) above the mean.
A common mistake is to look up a zvalue in the table and simply report the corresponding entry, regardless of whether the problem asks for the area to the left or to the right of the zvalue. The table only gives the probabilities to the left of the zvalue. Since the total area under the curve is 1, all we need to do is subtract the value found in the table from 1. For example, if we wanted to find out the probability that a variable is more than 0.51 sigmas above the mean, P(z>0.51), we just need to calculate 1P(z<0.51)=10.6950=0.3050, or 30.5%.
There is another note of caution to take into consideration when using the table: The table provided only gives values for positive zvalues, which correspond to values above the mean. What if we wished instead to find out the probability that a value falls below a zvalue of 0.51, or 0.51 standard deviations below the mean? We must remember that the standard normal curve is symmetrical , meaning that P(z<0.51)=P(z>0.51), which we calculate above to be 30.5%.
Symmetrical Normal Curve
This images shows the symmetry of the normal curve. In this case, P(z2.01).
We may even wish to find the probability that a variable is between two zvalues, such as between 0.50 and 1.50, or P(0.50).
689599.7 Rule
Although we can always use the zscore table to find probabilities, the 689599.7 rule helps for quick calculations. In a normal distribution, approximately 68% of values fall within one standard deviation of the mean, approximately 95% of values fall with two standard deviations of the mean, and approximately 99.7% of values fall within three standard deviations of the mean .
689599.7 Rule
Dark blue is less than one standard deviation away from the mean. For the normal distribution, this accounts for about 68% of the set, while two standard deviations from the mean (medium and dark blue) account for about 95%, and three standard deviations (light, medium, and dark blue) account for about 99.7%.
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Key Term Reference
 deviation
 Appears in these related concepts: Applications of Statistics, Outliers, and Variance
 distribution
 Appears in these related concepts: Application of Knowledge, Monte Carlo Simulation, and Selling to Consumers
 mean
 Appears in these related concepts: Mean, Variance, and Standard Deviation of the Binomial Distribution, The Mean Value Theorem, Rolle's Theorem, and Monotonicity, and Understanding Statistics
 normal distribution
 Appears in these related concepts: Shapes of Sampling Distributions, The Average and the Histogram, and Standard Deviation: Definition and Calculation
 probability
 Appears in these related concepts: Particle in a Box, The Addition Rule, and Rules of Probability for Mendelian Inheritance
 range
 Appears in these related concepts: Range, The Derivative as a Function, and Visualizing Domain and Range
 variable
 Appears in these related concepts: Related Rates, Math Review, and Psychology and the Scientific Method: From Theory to Conclusion
 zscore
 Appears in these related concepts: The Standard Normal Curve, Calculating a Normal Approximation, and ZScores and Location in a Distribution
Sources
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Cite This Source
Source: Boundless. “Using the Normal Curve.” Boundless Statistics. Boundless, 21 Jul. 2015. Retrieved 28 Aug. 2015 from https://www.boundless.com/statistics/textbooks/boundlessstatisticstextbook/sampling7/errorsinsampling31/usingthenormalcurve161315/