Normal Distribution: A function that represents the distribution of many random variables. This is represented by a Normal Model, usually as a bell-shaped curve.

- This curve is completely symmetric thus proving its normality. Thus, you can always assume that 50% of the data are greater than the mean and 50% are less than the mean.
- The graph is an idealized representation of the data's histogram
- Every normal set of data has a mean and a standard deviation
o They are used as the parameters of the set of data
- The figure above shows the relationship between standard deviation and the proportion of data in various parts of the bell curve. As you can see, roughly 68% of the data occur between 1 standard deviation less than the mean and 1 standard deviation greater than the mean. 95% are within 2 standard deviations of the mean, and 99.7% are within 3 standard deviations of the mean.
- Z-score: tells you how many standard deviations a value is from the mean. Equation:

The set of data must have only one peak and be symmetric to start, it cant just become normal.
The spread of the distribution (sometimes described by the range ) must be even and normal

Two things to find when given a bell curve
You can find the percent given the value How to On your calculator: 2nd distrubution
normal cdf (lower, upper bound of set , mean, standard deviation) ENTER

You can find the value given percent How to On your calculator:
2nd distributuion
inversnorm (percent to the left of the bell curve, mean , standard deviation)ENTER

Examples:
Mr. Killian's honors math analysis class took a quiz on logarithms. The scores were evenly distributed with a mean of 87 and a standard deviation of 2.7. If Rachel got a 93, then what percent of the class got her grade or higher?
Normalcdf(93,1E99,87,2.7) = 1.31%..GOOD JOB RACHEL!!

If Dylan scored in the top 25% of the class, then what was the lowest possible grade she could have gotten?
Invnorm(.75,87,2.7)= 88.82

You could also answer less quantitative questions.

Example:
Math SAT scores for students in a particular state are normally distributed with a mean of 610 and a standard deviation of 50 points. Are the following statements true or false?

1. If you choose a student at random, their most likely score is 610 (TRUE)
2. A student should be considered above average if they score a 600 (FALSE, more than half of the data are higher than 600, since exactly half of the data are above 610.) 3. More than half the students in the state scored between 560 and 660 (TRUE, this is the span from 1 standard deviation below the mean to 1 standard deviation above the mean, which includes roughly 68% of the data).
4. It should be considered pretty special to score above 760. (TRUE, since 0.3% of the data are more than 3 standard deviations from the mean, this means half of that, or 0.15%, is above 760 and 0.15% is below 460. Being in the top 0.15% is pretty special indeed!)
5. If you choose a student at random, they are just as likely to have scored a 600 as a 700. (FALSE, because the distribution is symmetric, points with equal probability are equidistant from the mean. 620 is as likely as 600, and 520 is as likely as 700.)

- This curve is completely symmetric thus proving its normality. Thus, you can always assume that 50% of the data are greater than the mean and 50% are less than the mean.

- The graph is an idealized representation of the data's histogram

- Every normal set of data has a

meanand astandard deviationo They are used as the parameters of the set of data

- The figure above shows the relationship between standard deviation and the proportion of data in various parts of the bell curve. As you can see, roughly 68% of the data occur between 1 standard deviation less than the mean and 1 standard deviation greater than the mean. 95% are within 2 standard deviations of the mean, and 99.7% are within 3 standard deviations of the mean.

- Z-score: tells you how many standard deviations a value is from the mean. Equation:

The set of data must have only one peak and be symmetric to start, it cant just become normal.

The spread of the distribution (sometimes described by the

range) must be even and normalTwo things to find when given a bell curve

You can find the percent given the value

How to On your calculator:2nd distrubution

normal cdf (lower, upper bound of set , mean, standard deviation)

ENTERYou can find the value given percent

How to On your calculator:2nd distributuion

inversnorm (percent to the left of the bell curve, mean , standard deviation)

ENTER

Examples:

Mr. Killian's honors math analysis class took a quiz on logarithms. The scores were evenly distributed with a mean of 87 and a standard deviation of 2.7. If Rachel got a 93, then what percent of the class got her grade or higher?

Normalcdf(93,1E99,87,2.7) = 1.31%..GOOD JOB RACHEL!!

If Dylan scored in the top 25% of the class, then what was the lowest possible grade she could have gotten?

Invnorm(.75,87,2.7)= 88.82

You could also answer less quantitative questions.

Example:

Math SAT scores for students in a particular state are normally distributed with a mean of 610 and a standard deviation of 50 points. Are the following statements true or false?

1. If you choose a student at random, their most likely score is 610 (TRUE)

2. A student should be considered above average if they score a 600 (FALSE, more than half of the data are higher than 600, since exactly half of the data are above 610.)

3. More than half the students in the state scored between 560 and 660 (TRUE, this is the span from 1 standard deviation below the mean to 1 standard deviation above the mean, which includes roughly 68% of the data).

4. It should be considered pretty special to score above 760. (TRUE, since 0.3% of the data are more than 3 standard deviations from the mean, this means half of that, or 0.15%, is above 760 and 0.15% is below 460. Being in the top 0.15% is pretty special indeed!)

5. If you choose a student at random, they are just as likely to have scored a 600 as a 700. (FALSE, because the distribution is symmetric, points with equal probability are equidistant from the mean. 620 is as likely as 600, and 520 is as likely as 700.)