Density Curves and Normal Distributions

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Transcript Density Curves and Normal Distributions

Density Curves and Normal
Distributions
Density Curves
• So far we have worked only with jagged
histograms and stem plots to analyze data
• As we begin to explore more fully the many
statistical calculations and analyses one can
perform on data it will become clear that
working with smooth curves is much easier
than jagged histograms
Density Curves Continued…
• A density curve is a smooth curve that describes
the overall pattern of a distribution by showing
what proportions of observations (not counts) fall
into a range of values.
Density Curves Continued…
• Areas under a density curve represent
proportions of observations
• The scale of a density curve is adjusted in
such a way that the total area under the
curve is always equal to 1
Mean and Median of Density
Curves
• Median: The point which divides the area
under the curve in half
• Mean: The point at which the curve would
balance if made out of solid material
For a symmetric Density curve…
For a skewed density curve…
Constructing a simple density
curve for dice simulation…
1)
2)
3)
We will simulate rolling a 6 sided die 120 times using the command
randInt(1,6,120)
L2

Though our histogram looks jagged, we can
approximate this histogram using a density curve:
20
1
2
3
4
5
1
2
3
4
5
6
1
6
6
1
6
6
Based on the density curve above what proportion of outcomes
fall within the following intervals:
1) x  5
2) 0  x  2
3) 3  x  6
4) x  6
Normal Curves
• A Particularly important class of density curves
• Symmetric, single peak, bell shape
• The mean of a density curve (including the normal
curve) is denoted by  and the standard deviation
is denoted by 
• All Normal distributions have the same overall
shape. Any differences can be explained by  and 
In a normal distribution with mean  and standard
deviation  :
- 68% of the observations fall within  of the mean 
- 95% of the observations fall within 2 of the mean 
- 99.7% of the observations fall within 3 of the
mean 
Percentiles
The percentile of a score, x, is the percentage of scores which
fall at or below the score.
percentile( x)  Number of scores  x *100
total number of scores
Pk = Score at the kth percentile rank.
Example: If you scored an 89 on a recent statistics test, and the
other scores in the class are listed below:
a) what is your percentile ranking?
b) What is P40
Scores on test:
29, 45, 50, 69, 70, 70, 71, 80, 83, 84, 88, 89, 89, 90, 91, 93, 95, 98,
99, 100
Standard Normal Calculations
Often times we are asked to compare the scores of two
pieces of data that do not come from the same
distribution. In order to decide which score is in fact
higher, we must first standardize the scores
If x is an observation from a distribution that has
A mean  and a standard deviation  then
the standardize value of x (often called the z-score) is:
z
x

The z-score tells us how many standard deviations a
particular piece of data is away from the mean of the
distribution. It therefore allows us to make comparisons
across distributions:
Example: Let’s say I gave two tests. On test 1 the mean was
68 and the standard deviation was 10. On test two the mean
was 85, and the standard deviation was 4. A student who in
the first test got a score of 83 claims that, relatively speaking,
his score is better than a score of 87 received by his friend on
test 2. Is he right? (assume both tests had approximately bell
shaped distributions)