The document discusses the normal distribution and some of its key properties. It also discusses the central limit theorem and how the distribution of sample means approaches a normal distribution as the sample size increases. Additionally, it covers how to transform a normally distributed variable into a standard normal variable using z-scores and how the normal distribution can be used to approximate the binomial distribution through a correction for continuity.

1. Statistics

2. 6-1 Normal Distribution
 Normal distribution is a continuous, symmetric,
bell-shaped distribution of a variable.
 The mathematic equation for a normal distribution is:
(X ) 2 /( 2 2
)
e
y
2
 e 2.718
 3.14
 = population mean
 =population standard deviation

3. Properties of the Theoretical
Distribution
1. A normal distribution is bell-shaped.
2. The mean, median, and mode are equal and are
located at the center of the distribution.
3. A normal distribution curve is unimodal.
4. The curve is symmetric about the mean, which is
equivalent to saying its shape is the same on both
sides of a vertical line passing through the center.

4. 5. The curve is continuous; that is, there are no gaps or
holes. For each value of X, there is a corresponding value
of Y.
6. The curve never touches the x axis. Theoretically, no
matter how far in either direction the curve extends, it
never meets the x axis-but it gets increasingly closer.
7. The total area under a normal distribution curve is equal
to 1.00, or 100%.
8. The area under the part of a normal curve that lies within
1 standard deviation of the mean is approx. 68%; within 2
standard deviations, about 95%; and within 3 standard
deviations, about 99.7%.

5. The Standard Normal Distribution
 The standard normal distribution is a normal
distribution with a mean of 0 and a standard deviation
of 1.
z2 / 2
e
y
2
 All normally distributed variables can be transformed
into the standard normally distributed variables by
using the standard score formula.
value m e an X
z z
s tandard de viation

6. 6-2 Application of the Normal
Distribution
 The z value is actually the number of standard
deviations that a particular X value is away from the
mean.
 When you must find the value of X, you can use the
following formula:
X z

7. Determining Normality
 Skewness can be checked by using Pearson’s index PI
of skewness. The formula is:
3( X me dian
)
PI
 If the index is greater than or equal to +1 or less than
or equal to -1, it can be concluded that the data is
significantly skewed.

8. 6-3 The Central Limit Theorem
 A sampling distribution of the sample means is a
distribution using the means computed from all
possible random samples of a specific size taken from a
population.
 Sampling error is the difference between the sample
measure and the corresponding population measure
due to the fact that the sample is not a perfect
representation of the population.

9. Properties of the Distribution Of
Sample Means
1. The mean of the sample means will be the same as
the population means
2. The standard deviation of the sample means will be
smaller than the standard deviation divided by the
square root of the sample size.
 The population mean is
X
 The standard deviation of sample means is
X
n

10.  The Central Limit Theorem
 As the sample size n increases without limit, the shape
of the distribution of the sample means taken without
replacement from a population with mean and the
standard deviation will approach a normal
distribution.
 If the sample size is sufficiently large the below formula
will be used.
X
z
n

11. 6-4 The Normal Approximation to
the Binomial Distribution
 A correction for continuity is a correction employed
when a continuous distribution is used to approximate
a discrete distribution.
n p
n p q