Transcript Chapter 2

Describing Location
in a Distribution
Measures of Relative Standing and Density
Curves
Measures of Relative Standing
O Percentiles- the percent of data that lies at
or below a particular value
O Standardized test score reports
O Baby weight/height/head size
O Ogive- cumulative relative frequency graph is
a percentile graph
Example
O Suppose Jenny earns an 86 (out of 100) on
her next Statistics exam
O Is that a good grade? Should she be happy
or disappointed with this grade?
Example Continued
O Here are the scores of all 25 students in
Jenny’s class. How did she perform on this
test relative to her classmates?
O 79, 81, 80, 77, 73, 83, 74, 93, 78, 80, 75,
67, 73, 77, 83, 86, 90, 79, 85, 83, 89, 84,
82, 77, 72
O Find the mean, median, and standard
deviation because the data would produce a
symmetric histogram.
Example Cont.
O Mean = 80
O Median = 80
O Standard Deviation = 6.07
O Jenny did better than average!
How many standard deviations above the
mean is Jenny?
Measures of Relative Standing
O Standardized value (z – score)- how many
standard deviations away from the mean a
given observation is
O Useful with symmetric distributions
O Positive z score means above the mean
O Negative z score means below the mean
Example
O Jenny’s z score
Z=
86 −80
6.07
= 0.99
Converting Jenny’s original score to standard
deviation units is called standardizing!
Z- Scores
O Used to compare the relative standing of
individuals in different distributions
O The next day Jenny got her Chemistry test
back and saw she earned an 82. The test
scores were symmetric with a mean of 72
and a standard deviation of 4.
O Did Jenny do better on the Chem test or the
Stats test?
Percentiles
O Different Stats books will calculate the
percentile differently.
O There is no exact science
O The median is always the 50th percentile
O One formula is…
Percentile =
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑣𝑎𝑙𝑢𝑒𝑠 𝑏𝑒𝑙𝑜𝑤 𝑋+0.5
𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑣𝑎𝑙𝑢𝑒𝑠
* 100
Chebyshev’s InequalityNot on AP Exam
O Proportion of values from a data set that will fall
within k standard deviations of the mean will be
1
at least 1 - 2 , where k is a number greater than
𝑘
1 (k is not necessarily an integer). Works for any
shape!
O At least ¾ or 75% of the data values fall within 2
standard deviations of the mean
O 1 – 1/22 = 1 – ¼ = ¾ = 75%
8
O
9
or 88.89% of the data values will fall within 3
standard deviations of the mean
O 1 – 1/ 32 = 1 – 1/9 = 8/9 = 88.89%
Density Curve
O Always on or above the horizontal axis and
has an area of exactly 1 underneath it
O The area under the curve for any given
interval is equal to the proportion of all
observations
Notation used for density
curves
O Observed data
Idealized Data
O (Sample)
(population)
Parameters
O Statistics
MEAN
O
O
s
standard deviation
μ (mu)
σ (sigma)
Median of a Density Curve
O Median is the equal areas point (the point
with half the area under the curve to the left
and half to the right)
O The quartiles divide the area under the
curve into quarters
Mean of a Density Curve
O Balance point of the curve- the curve would
be balanced here if it were made of solid
material
The median and mean are the same for a
symmetric density curve. Both lie at the
center of the curve.
The mean of a skewed curve is pulled away
from the median in the direction of the long
tail.
Density Curves
O Rough estimate by eye for mean, median,
and quartiles, but not standard deviation
O This is an idealized description of the
distribution of data
O Distinguish between the mean and standard
deviations of samples and parameters by
using different notations