AP Stats
Chapter 2: Describing Location in a Distribution
2.1 Measures of Relative Standing & Density Curves
density curve – describes the overall pattern of a distribution & has an area
of 1 underneath it (Theoretical !!)
density curve
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The median (M) of a density curve is the point that divides the area under the curve in half. (ie: equal areas on each
side)
The mean (x̅) of a density curve is the “balance point”. Point that the curve would balance at if made of solid
material.
A density curve is an idealized description of the distribution of data.
Values calculated from a density curve are theoretical and use different symbols (used for global things).
Mean
Greek letter mu μ (x̅ for data)
Standard deviation Greek letter sigma σ
(s for data)
Uniform Density Curves looks like this:
Find the proportion of observations within the given interval.
P (x ≤ 2) =
P (x < 2) =
P (2 < x ≤ 7) =
P (x > 6) =
P (x = 2) =
What would be the Median (M)? M =
Example:
Find the proportion of observations within the given interval
1.0
.75
.5
.25
0
0
.25
.5
.75
1.0
1.25
P(0 < X < 2) =
P(.25 < X < .5) =
P(.25 < X < .75) =
P(1.25 < X < 1.75) =
P(.5 < X < 1.5) =
P(1.75 < X < 2) =
What would be the Median (M)? M =
1.5
1.75
2.0
Example: A density curve fits the model y = .25x
Graph the line.
Use the area under this density curve to find the proportion of observations within the given interval
P(1 < X < 2) =
P(.5 < X < 2.5) =
If the curve starts at x = 0, what value of x does it end at?
What value of x is the median?
What value of x is the 62.5th percentile?
1.0
0.5
1
2
Measuring Relative Standing Percentiles
Percentile - percent of observations less than or equal to a particular observation
EXAMPLE #1 (PART B): scores: 92, 91, 85, 77, 79, 88, 99, 69, 73, 84
A score of _____ is the ______ percentile?
a) 79
b) 88
c) 99
Homework
p 122
6, 7
p128
9 -13 (not 13a)
3