6.1 Introduction to the Normal Curve

Another important property of a normal distribution is that the total area under the curve of a normal distribution is equal to

1 This value is derived from the interpretation of this area. -the total area under the curve is equivalent to the probability of randomly choosing a value from the distribution that is less than (or equal) to the largest value in the distribution. This probability certainly equals 1; therefore, the total area under the curve equals 1.

Properties of a Normal Distribution

1) symmetric and bell-shaped. 2) completely defined by its mean, μ, and standard deviation, σ. 3) The total area under a normal curve equals 1. 4) The x-axis is a horizontal asymptote

Properties of a Standard Normal Distribution

1) symmetric and bell-shaped. 2) completely defined by its mean, μ=0, and standard deviation, σ=1. 3) total area under equals 1. 4) x-axis is a horizontal asymptote

standard normal distribution:

N (0, 1) 0= µ 1= σ²

to convert a normal curve to a standard normal curve:

Remember: you can have a negative Z score

normal distributions with identical standard deviations

The only difference in the distributions is the central location, the mean.

The Standard Normal Distribution

The standard normal distribution is a special version of the normal distribution. The standard normal curve has all of the properties of a normal curve, and always has a mean of 0 and a standard deviation of 1.

To standardize a normal curve to the standard normal curve, we convert each X-value to

a standard score, z, using the formula:

A graphical representation of a normal curve is

a symmetric, bell-shaped curve centered above the mean of the distribution

In addition, the mean, mode, and median are

all equal. Note that the bell shape of the curve means that the majority of data will be in the middle of the distribution and the amount of data will taper off evenly in both directions from the center.

the value of X can be

any number on a real number line

to use the calculator for normal curve:

function: normal cdf default standard normal: µ=0 mean σ=1 standard deviation

The x-axis is a _____ for the normal distribution.

horizontal asymptote This says that the normal curve will approach the x-axis on both ends, but will never touch or cross it.

inflection point

is a point on the curve where the curvature changes.

The larger the standard deviation, the

more area there will be in the tails of the distribution. Therefore, the curve will appear flatter.

The most prevalent distribution is the

normal distribution, a continuous probability distribution for a given random variable X that is completely defined by its mean and standard deviation.

one standard deviation is the distance from the mean to

one of the inflection points.

Should we then need to convert back from a standard score to an X-value, we can

reverse the calculation and use the formula, X=σz+μ

Changing the standard deviation parameter can have rather significant effects on the

shape of the distribution.

Some other examples of data that are normally distributed over a large randomly selected sample would be

shoe size, weight, and pregnancy distribution.

when a problem says: Calculate the standard score of the given X value

that means you need to calculate the z score

Normal distributions are all bell-shaped, but

the bells come in various shapes and sizes.

the mean is always

the exact middle - the high point mean, median, mode are always the same spot on a continuous or real number line

An example of a data set that would produce an approximate normal distribution is

the height of 500 randomly selected men The random variable in this example is men's heights. The heights would be approximately normally distributed with a mean close to 69.2 inches. Heights of men produce a normal distribution because most men are fairly close to the same height, give or take a few inches. Very tall and very short men are rare.

Convert to z-scores when

the mean is not zero use the calculator! N (µ=25, σ=5) z=x-µ÷σ

While the mean defines the location, the standard deviation determines

the shape of the curve.

the normal curve's function can never equal

zero.

under the curve:

100% of values