CHAPTER 3 – THE NORMAL DISTRIBUTION
TOPICS COVERED - Sections shown with numbers as in e-book
Any topic listed on this document and not covered in class must be studied “On Your Own” (OYO)
Section 3.1 – DENSITY CURVES (pg. 67)
 From histograms to density curves
 Characteristics of a density curve
 Area, probability, proportion
 Uniform distribution
 Sketch a density curve for a given specified shape
Section 3.2 – DESCRIBING DENSITY CURVES (pg. 71)
 Mean and median and mode of a density curve
o Right skewed
o Left skewed
o symmetric
 Notations for mean and standard deviation of a density curve
Section 3.3 – NORMAL DISTRIBUTIONS (pg. 73)
 Shape of a normal density curves
 Normal distributions
 Location of mean, median and mode
 Complete description of a normal distribution with the mean and the standard deviation
o Parameters of a normal distribution – notation used
 How does the change of the mean (mu) and the standard deviation (sigma) affect a normal distribution
Section 3.4 – THE 68 – 95 – 99.7 RULE (pg. 74)
 The 68-95-99.7 rule for exactly normal distributions
 For a given normal distribution: N(mu, sigma), show a sketch and label the mean and the scores that are 1,
2, and 3 standard deviations away from the mean. Use the graph to answer questions of the type:
o 68% of the scores are between ______ and ______
o 95% of the scores are between ______ and ______
o 99.7% of the scores are between ______ and ______
o Give the % (area/probability/proportion) between two given scores
o Give the % (area/probability/proportion) below a certain score
o Give the % (area/probability/proportion) above a certain score
Section 3.5 – THE STANDARD NORMAL DISTRIBUTION (pg. 77)
 Standardized values of a given score in a distribution: the z-score
o Finding z-scores for any given observation in a distribution
 The standard normal distribution
 Parameters of the standard normal distribution – notation used
Section 3.6 – FINDING NORMAL PROPORTIONS (pg. 79)
 Read example – method is explained on the next section
Section 3.7 – USING THE STANDARD NORMAL TABLE (pg. 81)
 Using the table to find areas under the standard normal curve:
o Areas to the left of a certain z score
o Areas to the right of a certain z score
o Areas between any two z scores
 Finding probabilities/areas/proportions for any normal distribution
Section 3.8 – FINDING A VALUE GIVEN A PROPORTION (pg. 83)
 From probabilities/areas/proportions to z scores
 From probabilities/areas/proportions to a score in a normal distribution
Summary
We can sometimes describe the overall pattern of a distribution by a density curve. A density curve has total
area 1 underneath it. An area under a density curve gives the proportion of observations that fall in a range of
values.

A density curve is an idealized description of the overall pattern of a distribution that smooths out the
irregularities in the actual data. We write the mean of a density curve as μ and the standard deviation of a
density curve as σ to distinguish them from the mean and standard deviation s of the actual data.

The mean, the median, and the quartiles of a density curve can be located by eye. The mean μ is the
balance point of the curve. The median divides the area under the curve in half. The quartiles and the median
divide the area under the curve into quarters. The standard deviation σ cannot be located by eye on most
density curves.

The mean and median are equal for symmetric density curves. The mean of a skewed curve is located
farther toward the long tail than is the median.

The Normal distributions are described by a special family of bell-shaped, symmetric density curves,
called Normal curves. The mean μ and standard deviation σ completely specify a Normal distribution N(μ,
σ). The mean is the center of the curve, and σ is the distance from μ to the change-of-curvature points on either
side.

To standardize any observation x, subtract the mean of the distribution and then divide by the standard
deviation. The resulting z-score
says how many standard deviations x lies from the distribution mean.

All Normal distributions are the same when measurements are transformed to the standardized scale. In
particular, all Normal distributions satisfy the 68 − 95 − 99.7 rule, which describes what percent of
observations lie within one, two, and three standard deviations of the mean.

If x has the N(μ, σ) distribution, then the standardized variable z = (x − μ)/σ has the standard Normal
distribution N(0, 1) with mean 0 and standard deviation 1. Table A gives the cumulative proportions of
standard Normal observations that are less than z for many values of z. By standardizing, we can use Table A for
any Normal distribution.