6.1 Introduction to the Normal Curve
standard normal distribution:
N (0, 1) 0= µ 1= σ²
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.
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.
The larger the standard deviation, the
more area there will be in the tails of the distribution. Therefore, the curve will appear flatter.
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
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
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
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
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
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.
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
inflection point
is a point on the curve where the curvature changes.
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.
Changing the standard deviation parameter can have rather significant effects on the
shape of the distribution.
Normal distributions are all bell-shaped, but
the bells come in various shapes and sizes.