Key Terms
Mathematical Model: an idealized description that gives a compact picture of the overall
pattern of the data but ignores minor irregularities as well as any outliers (p. 79).
Density Curve: a curve that (1) is always on or above the horizontal access, and (2) has
an area of exactly 1 underneath it. It describes the overall pattern of a distribution. The
area under the curve and above any range of values is the proportions of all observations
that fall in that range (p. 80).
Median of a Density Curve: the equal-areas point, the paint that divides the area under
the curve in half (p. 82).
Mean of a Density Curve: the balance point, at which the curve would balance if made of
solid material (p. 82).
σ: the Greek letter sigma stands for the standard deviation of a density curve (p. 83).
µ: the Greek letter mu stands for the mean of an idealized distribution (p. 83).
Outcomes: the observed results of a chance experiment (p. 84).
Simulation: the imitation of chance behavior, based on a model that accurately reflects
the experiment under consideration (p. 84).
Normal Curves: density curves that are symmetric, single-peaked, and bell-shaped. They
describe normal distributions. The mean is located at the center of the symmetric curve
and is
the same as the median (p. 85).
Inflection Points: the points at which a change of curvature in a density curve takes
place located at distance σ on either side of the mean µ (p. 86).
Standardized Value: often called a z-score; z = (x - µ)/σ (p. 94).
Standard Normal Distribution: the normal distribution N(0,1) with mean 0 and standard
deviation 1. If a variable x has a normal distribution N(µ,σ) with mean µ and standard
deviation σ, then the standardized variable has the standard normal distribution (p. 95).
Normal Probability Plot: provides a good assessment of the adequacy of the normal
model for a set of data (p. 105).
2.1: Density Curves and the Normal Distributions
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Density curves can sometimes describe the overall pattern of a distribution
A density curve always remains on or above the horizontal axis and has a total area of 1
The area gives the proportion of observations that fall in a range of values
Mean of a density curve is µ
Standard deviation of a density curve is σ
The mean, median, and quartiles can be located by eye
The 68-95-99.7 Rule (aka the empirical rule)
In the normal distribution with mean and standard deviation
68% of the observations fall within σ of the mean µ.
95% of the observations fall within 2σ of µ.
99.7% of the observations fall within 3σ of µ.
For symmetric density curves σ=µ
On a skewed density curve, the mean is located farther toward the tail
Normal distributions are described by normal curves
The mean is the center of the curve
The standard deviation is the distance from the mean to the inflection points on
either side
• 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.
• An observations percentile is the percent of the distribution that is at or to the left of the
observation.
• When the data is normally distributed, the curve is symmetrical and the mean = median
• When the distribution is skewed right, the mean > median
• When the distribution is skewed left, the mean < median
2.2: Standard Normal Calculations
• In order to standardize any observation x, use the following equation (which will give
you the z-score)
z=x-µ/σ
• The z-score tells you how many standard deviations x lies from the distribution mean.
• Table A gives the 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.
• The table entry for each value z is the area under the curve to the left of z.
Table entry is the area to the left of z.
• Finding normal proportions:
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Step 1: State the problem in terms of the observed variable x. Draw a picture of
the distribution and shade the area of interest under the curve.
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Step 2: Standardize x to restate the problem in terms of a standard normal
variable z. Draw a picture to show the area of interest under the standard
normal curve.
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Step 3: Find the required area under the standard normal curve, using Table A
and the fact that the total area under the curve is 1.
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Step 4: Write your conclusion in the context of the problem.
To check whether or not a set of data is normal, construct on normal probability plot on a
calculator.