Chapter 3 (Introducing density curves)
• When given a Histogram or list of data, we often are asked to
estimate the relative position of a particular data point.
What percent of Scores were under 40.
What percent of Scores were equal to or over 60?.What percent were equal
to 40 or larger but less than 50 (i.e. between 40 and 50) ?
Chapter 3 (Introducing density curves)
Is a score of 12 considered unusual? Explain.
Explain why a score of 44 is not considered unusual.
Chapter 3 (cont’d)
• Explain why a height of 70.5 inches (using the above chart) is
considered unusual.
.
Here is a histogram of vocabulary scores of 947 seventh graders
The smooth curve drawn over the histogram is a mathematical model
for the distribution (specifically , a Normal curve)
Is a score of 3 unusual?
Quantify that answer ! i.e. “What
is the “percent of scores” above
3?
How to answer the above , is the
subject of Ch 3 !
Ch 3 (cont’d)
• In Chapter 3, instead of using “sample” data
(histograms) we will use an entire
population to describe the relative position of
a data point.
• To do the above, we will need to define & use
“density curves”.
DENSITY CURVES:
• A density curve is an “idealized” mathematical
description of a population’s distribution.
• The area underneath the curve is exactly 1.
• In a density curve, the area under the curve
(to the left or to the right of a value)
represents the fraction of data values to the
left or right (Think of a histogram without
vertical lines between the intervals! )
Density Curve for IQ’s
Another “idealized” model of a populations distribution
IQ “normal curve”
m=100, s=15
The area under the curve (to the left or to the right of a value) represents the
fraction of data values to the left or right
(eg. P(x>130) ~2.5%, P(x<100) = 50%
How did I get the above percentiles????
1st ---Some Definitions/Notation
Population Mean : m, “Mu “ -- A measure of the center of
the entire population. It is the arithmetic mean of ALL
the data points
m
=
 xi
i.e. (sum the all the data values in the
population, divide by N). In practice a
VERY hard number to obtain.
N
Population Standard deviation: s , “sigma” A measure of
the spread of the entire population’s data values
s =  ( xi  x )
N
2
It is seldom calculated. Instead a large
sample is taken and the sample
standard deviate, “s” is used as the
“best estimate” for .
Chapter 3 - Some Definitions/Notation
x ~ N ( m,s )
x ~ N ( 7 ,1 . 5 )
“The variable x is Normally
distributed with a mean of m
and a standard deviation of s “
Implies tht there is a population
whose mean is 7 and standard
deviation is 1.5
IQ “normal curve”
x ~ N(100,15)
Let’s look at some “rule of thumbs” for normal curves.
68-95-99.7 Rule for
Any Normal Curve
68%
-s
95%
+s
µ
-2s
µ
+2s
99.7%
-3s
Essential Statistics
µ
Chapter 3
+3s
12
68-95-99.7 Rule for
Any Normal Curve
Essential Statistics
Chapter 3
13
Health and Nutrition Examination
Study of 1976-1980
• Heights of adult men, aged 18-24
– mean: 70.0 inches
– standard deviation: 2.8 inches
– heights follow a normal distribution, so we
have that heights of men are N(70, 2.8).
– DO ON BOARD-Essential Statistics
Chapter 3
14
Health and Nutrition Examination
Study of 1976-1980
• 68-95-99.7 Rule for men’s heights
 68%
are between 67.2 and 72.8 inches
[ µ  s = 70.0  2.8 ]
 95%
are between 64.4 and 75.6 inches
[ µ  2s = 70.0  2(2.8) = 70.0  5.6 ]
 99.7%
are between 61.6 and 78.4 inches
[ µ  3s = 70.0  3(2.8) = 70.0  8.4 ]
Essential Statistics
Chapter 3
15
Health and Nutrition Examination
Study of 1976-1980
• What proportion of men are less than 72.8
inches tall?
68%
16%
(by 68-95-99.7 Rule)
?
-1
+1
70
72.8
(height values)
? = 84%
Essential Statistics
Chapter 3
16
Health and Nutrition Examination
Study of 1976-1980
• What proportion of men are less than 68
inches tall?
?
68 70
(height values)
How many standard deviations is 68 from 70?
Essential Statistics
Chapter 3
17
Weds – Using Table A
• That’s it for the 68-95-99.7 rule.
• We will see how to find proportions (left,
right, in-between) any score (s)
• Note : HW is on StatsPortal
Z scores
ALL Normal Curves are the same –if we
measure in units of size s from the m.
Finding the distance a data point is away from
its mean – in standard deviations – is called
“standardizing”
How do we find the “proportion” of scores to
the left or right of a data point ?
• ANSWER: By finding the number of standard
deviations a score is AWAY from the mean –
i.e. find the
“z” score
then use Table A