1.3 Density Curves and Normal Distributions
Density Curve
software can describe distribution by fitting a smooth curve to the data
less arbitrary than choosing classes for histogram
idealization that pictures the overall pattern of the data but ignores minor irregularities
and outliers
smooth approximation to the irregular bars of a histogram
Density Curve
•The curve is always on or above the horizontal axis.
•The curve has area exactly 1 underneath it.
• The area under the density curve and above any range of values is the proportion of all
observations that fall in that range.
Figure 1.23a
histogram areas of bars represent either counts or proportions of observations
red got 6.0 or lower (0.303 of all students)
red bars make up 0.303 of total area under all the bars
adjust the scale so total area of bars is 1 area of red bars will be 0.303
Figure 1.23b
red area under curve to the left of 6.0
scale is adjusted so total area under curve is exactly 1
area under curve= proportion
red area proportion= 0.293 (density curve provides good approximation)
Measuring center and spread for density curves
mode
 a peak point of a curve
median
 the point with half the total area on each side
mean
 the balance point (point at which curve would balance if made of solid material)
Why?
The small area in the long right tail tips the curve more than the same area near
the center.
(Figure 1.25)
•symmetric density curve mean and median are same
•skewed distribution pulls mean towards its tail (Figure 1.24)
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Figure 1.23
density curve exactly symmetric
histogram only approximately symmetric
A density curve is an idealized description of a distribution of data.
We need to
distinguish between the mean and standard deviation of the density curve and the
numbers
x
and
s
computed from the actual observations.
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 Spring '08
 ABDUS,S.
 Normal Distribution, Standard Deviation, Jen

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