Chapter 2: Modeling Distributions of Data
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Chapter 2: Modeling Distributions of Data
2.2A: Describing a Location in a Distribution
( Density Curve
A density curve is a curve that:
• is always on or above the _________________ axis, and
• has area exactly ___ underneath it.
A density curve describes the __________ pattern of a distribution. The ______ under the curve and above any interval of values on the horizontal axis is the ________________ of all observations that fall in that interval.
Mean vs. Median in a Density Curve
Mean: still the balancing point( Median: still the equal-areas point.
Sketch the mean and median for each of the following graphs…
Chapter 2: Modeling Distributions of Data
2.2: Normal Distributions
( Normal Distributions
One particularly important class of density curves are the Normal curves, which describe Normal distributions.
• All Normal curves are ___________________, single-peaked, and ________-shaped.
• A specific Normal curve is described by giving its mean ___ and standard deviation ____.
Definition:
A Normal distribution is described by a Normal density curve. Any particular normal distribution is completely specified by two numbers: its mean ( and standard deviation σ.
• The mean of a Normal distribution is the __________ of the symmetric normal curve.
• The standard deviation is the distance from the center to the _________-of-__________ points on either side.
• We abbreviate the normal distribution with mean ( and standard deviation σ as N((, σ).
( The 68-95-99.7 Rule (“The Empirical Rule”)
Although there are many Normal curves, they all have properties in common.
Definition:
In the Normal distribution with mean ( and standard deviation σ:
• Approximately 68% of the observations fall within σ of (.
• Approximately 95% of the observations fall within 2σ of (..
• Approximately 99.7% of the observations fall within 3σ of (.
Helmet Sizes:
The army reports that the distribution of head circumference among male soldiers is approximately normal with mean 22.8 inches and standard deviation 1.1 inches, (that is, N(22.8, 1.1)).
• What percent of soldiers have head circumference between 21.7 inches and 23.9 inches?
• What percent of soldiers have a head circumference greater than 23.9 inches?
• What percent of soldiers have a head circumference less than 20.6 inches?
• Between what head circumferences do the middle 95% of soldiers fall?
( The Standard Normal Distribution
All Normal distributions are the same if we measure in units of size σ from the mean ( as center.
Definition:
The standard Normal distribution is the Normal distribution with mean ___ and standard deviation ___. If a variable x has any Normal distribution N((,σ) with mean ( and standard deviation σ, then the standardized variable
z =
has the standard Normal distribution, N(0, 1).
( The Standard Normal Table
Because all Normal distributions are the same when we standardize, we can find areas under any Normal curve from a single table.
Definition:
Table A is a table of areas under the standard Normal curve. The table entry for each value z is the ________ under the curve to the _______ of z.
Use Table A to find:
• z < –2.2
• z > 1.34
• z > 5.2
• 0.58 < z < 1.79
• Find z that corresponds with the 20th percentile.
• Find z such that 45% of all observations are greater than z.
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