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Math 243: Lecture File 2
N. Christopher Phillips
2 April 2009
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From histograms to density curves
Data: 12 17 21 23 24 26 29 31 31 39 Histogram with class width 10:
10 20 30 40
1
2
3
4
5
6
Area of red bar: 10 · 2 = 20. Total area: 10 · 2 + 10 · 5 + 10 · 3 = 100. Fraction of the data covered by the red bar: 20/100 = 0. 2.
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Rescale vertical axis so that total area is 1
10 20 30 40
1
2
3
4
5
6
10 20 30 40
Rescale vertical axis so that total area is 1 (continued)
10 20 30 40
Area of red bar: 10 · 0 .02 = 0. 2. Total area: 10 · 0 .02 + 10 · 0 .05 + 10 · 0 .03 = 1.
The fraction of the data covered by the red bar is still 0. 2 , but this is now just the area of the red bar.
Data: 12 17 21 23 24 26 29 31 31 39
10 20 30 40
1
2
3
4
5
6
10 20 30 40
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Data: 12 17 21 23 24 26 29 31 31 39 Histogram with class width 5:
10 20 30 40
Area of red section: 5 · 1 + 5 · 1 = 10. Total area: 5 · 1 + 5 · 1 + 5 · 3 + · · · = 50. Fraction of the data covered by the red section: 10/50 = 0. 2.
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Rescale vertical axis so that total area is 1
10 20 30 40
10 20 30 40
Rescale vertical axis so that total area is 1 (continued)
10 20 30 40
Area of red section: 5 · 0 .02 + 5 · 0 .02 = 0. 2. Total area: 5 · 0 .02 + 5 · 0 .02 + 5 · 0 .06 + · · · = 1.
The fraction of the data covered by the red section is still 0. 2 , but this is now just the area of the red section.
100 classes.
100 000
200 000
300 000
400 000
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300 classes.
20 000
40 000
60 000
80 000
100 000
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1000 classes.
10 000
20 000
30 000
40 000
One way to think about density curves
When you see a density curve, imagine that it is a histogram in which the classes are so narrow that each individual bar in the histogram is too small to see.
An example of a density curve
Here is a density curve:
1 2 3 4 5
Conditions: The curve is always on or above the horizontal axis. The total area under the curve is 1. (If you have seen Math 242: The integral of the function is 1.) N. Christopher Phillips () Math 243: Lecture File 2 2 April 2009 17 / 48
Meaning of the density curve
A density curve represents the distribution of a very large data set. (In principle, the data set should be infinite; the curve is only an approximation for a large finite data set.)
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Meaning of the density curve (continued)
1 2 3 4 5
The shaded area is the fraction of the data that lies between 2 and 3.
Meaning of the density curve (continued)
1 2 3 4 5
The shaded area is the fraction of the data that lies between 1 and 2 or between 3 and 3. 5.
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From histograms to density curves, for the normal
distribution.
The following histograms show larger and larger numbers of data points chosen randomly from the standard Normal distribution, sometimes with the Normal curve superimposed. Observe that small numbers of normally distributed data points are somewhat irregular, but very large numbers are very regular.
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100 observations.
1000 observations.
104 observations.
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106 observations, curve not shown.
-4 -2 0 2 4
10000
20000
30000
40000
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106 observations, curve shown.
-4 -2 2 4
10000
20000
30000
40000
107 observations, curve not shown.
-4 -2 0 2 4
20000
40000
60000
80000
Comparing Normal distributions (continued)
Jane Wang’s z-score was about 1. 24561.
Math ACT scores are roughly N(20. 7 , 5 .0). Quincy Michaels got 27 on the math ACT. Assuming the tests measure the same thing, did he do better or worse than Jane Wang?
Compare z-scores: his z-score is
z =
x − μ σ
So he did slightly better.
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Comparing Normal distributions (continued)
John Doe’s z-score on the Math SAT (roughly N(518, 114)) was − 2. What was his actual score?
In z =
x − μ σ
solve for x to get x = μ + zσ. So John Doe’s score was
x = μ + zσ = 518 − 2 · 114 = 290.
(Compare: 2 standard deviations below the mean.)
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The rule of thumb (“68–95–99.7 rule”)
In the distribution N(μ, σ) (Normal with mean μ and standard deviation σ), About 68% of the observations are within one standard deviation of the mean. About 95% of the observations are within two standard deviations of the mean. About 99.7% of the observations are within three standard deviations of the mean.
Example: Math SAT scores are roughly N(518, 114). What fraction of Math SAT scores are in the range (404, 632)?
The interval is (μ − σ, μ + σ), so the answer is about 68%.
Note that this idealized model assumes scores of exactly 404 and 632 do not occur.
The rule of thumb (“68–95–99.7 rule”) (continued)
Example: Math SAT scores are roughly N(518, 114). What fraction of Math SAT scores are larger than 746?
z = x − μ σ
so we want to know what fraction of scores are more than two standard deviations above the mean.
The rule of thumb says that about 95% are within two standard deviations of the mean. So about 100% − 95% = 5% are more than two standard deviations away from the mean. By symmetry, half of these are more than two standard deviations above the mean (and the other half are more than two standard deviations below from the mean). So the answer is about 2 .5%.
The rule of thumb (“68–95–99.7 rule”) (continued)
400 600 800
By the rule of thumb, the unshaded region has area about 0. 95. So the two shaded regions together have area about 0. 05. We are interested in the one on the right, which has half the area, or area about 0. 025.
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Using Table A
Table A (pages 684 and 685) gives the area under the standard Normal curve to the left of (below) the specified value of z. Example:
The shaded region is at − 1 .32 and below. Look at the row in Table A labelled “− 1 .3” and the column labelled “0.02”, and read off the number 0 .0934 for the shaded area. N. Christopher Phillips () Math 243: Lecture File 2 2 April 2009 42 / 48
Using Table A (continued)
Note: One can do problems like this directly with most calculators. However, the standardization idea is important anyway. Table A will be provided on exams (without the pictures).
Example: Math SAT scores are roughly N(518, 114). What fraction of Math SAT scores are larger than 746?
z = x − μ σ
as before. Look at the row in Table A labelled “2.0” and the column labelled “0.00”, and read off the number 0. 9772. This tells you that the fraction about 0.9772 of the data has z-scores less than 2. Therefore the fraction about 1 − 0 .9772 = 0. 0228 , or about 2.28%, has z-scores above 2. Thus, about 2.28% of math SAT scores are above 746.
The rule of thumb gave about 2.5%.
Using Table A (continued)
Example: Math SAT scores are roughly N(518, 114). What fraction of Math SAT scores are less than 600?
z =
x − μ σ
Look at the row in Table A labelled “0.7” and the column labelled “0.02”, and read off the number 0. 7642. This tells you that the fraction about 0 .7642 of the data has z-scores less than 0. 72. Thus, about 76.42% of math SAT scores are below 600.
