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Section 2.1Density Curves & the Normal DistributionsAP StatisticsBerkley High School/CASA
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AP Statistics, Section 2.1, Part 1 2
Density Curve
“Mathematical Models” The area under the
curve is related to the distribution of values
“Idealized description”
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AP Statistics, Section 2.1, Part 1 3
Density Curves
Characteristics Always above the x-
axis Area always equal to 1
The area under the curve and above any range of values is the proportion of all observations that fall in that range.
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AP Statistics, Section 2.1, Part 1 4
Median and Mean Since the area
represents portions of the population, the Median is the spot where the area to the left is the same as the area to the right.
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AP Statistics, Section 2.1, Part 1 5
Median and Mean
The Mean represents the “balance point”.
Imagine a “see-saw”
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AP Statistics, Section 2.1, Part 1 6
Median and Mean
The Mean is always pulled towards the tail in a skewed distribution
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AP Statistics, Section 2.1, Part 1 7
“Idealized Distributions” We use different
notation for density curve (which represent entire populations) as compared to data sets (which represent samples)
population a ofdeviation standardpopulation a ofmean
sample a ofdeviation standard sample a ofmean
sx
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AP Statistics, Section 2.1, Part 1 8
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AP Statistics, Section 2.1, Part 1 9
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AP Statistics, Section 2.1, Part 1 10
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AP Statistics, Section 2.1, Part 1 11
Normal Distribution
A special, bell shaped, symmetric, single-peaked distribution
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AP Statistics, Section 2.1, Part 1 12
Normal Distribution
Because of the symmetry, the mean and median are the same and at the line symmetry
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AP Statistics, Section 2.1, Part 1 13
Normal Distribution
The inflection points (where the curve starts to flatten out) represent the width of the standard deviation
μ μ+σμ-σ
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AP Statistics, Section 2.1, Part 1 14
68-95-99.7 Rule
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AP Statistics, Section 2.1, Part 1 15
68-95-99.7 Rule with N(64.5,2.5)
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AP Statistics, Section 2.1, Part 1 16
Assignment
Exercises 2.1 – 2.15 odd, The Practice of Statistics