More notebooks for Think Stats More notebooks for Think Stats
More notebooks for Think Stats As I mentioned in the previous post, I am getting ready to teach Data Science in the spring, so I... More notebooks for Think Stats

More notebooks for Think Stats

As I mentioned in the previous post, I am getting ready to teach Data Science in the spring, so I am going back through Think Stats and updating the Jupyter notebooks.  I am done with Chapters 1 through 6 now.

If you are reading the book, you can get the notebooks by cloning this repository on GitHub, and running the notebooks on your computer.  Or you can read (but not run) the notebooks on GitHub:

Chapter 4 Notebook (Chapter 4 Solutions)
Chapter 5 Notebook (Chapter 5 Solutions)
Chapter 6 Notebook (Chapter 6 Solutions)

I’ll post the next batch soon; in the meantime, here are some of the examples from Chapter 5, including one of my favorites: How tall is the tallest person in Pareto World?

Exercise: In the BRFSS (see Section 5.4), the distribution of heights is roughly normal with parameters µ = 178 cm and σ = 7.7 cm for men, and µ = 163 cm and σ = 7.3 cm for women.
In order to join Blue Man Group, you have to be male between 5’10” and 6’1” (see http://bluemancasting.com). What percentage of the U.S. male population is in this range? Hint: use scipy.stats.norm.cdf.
scipy.stats contains objects that represent analytic distributions
In [27]:
import scipy.stats
For example scipy.stats.norm represents a normal distribution.
In [28]:
mu = 178
sigma = 7.7
dist = scipy.stats.norm(loc=mu, scale=sigma)
type(dist)
Out[28]:
scipy.stats._distn_infrastructure.rv_frozen
A “frozen random variable” can compute its mean and standard deviation.
In [29]:
dist.mean(), dist.std()
Out[29]:
(178.0, 7.7000000000000002)
It can also evaluate its CDF. How many people are more than one standard deviation below the mean? About 16%
In [30]:
dist.cdf(mu-sigma)
Out[30]:
0.15865525393145741
How many people are between 5’10” and 6’1″?
In [31]:
# Solution

low = dist.cdf(177.8)    # 5'10"
high = dist.cdf(185.4)   # 6'1"
low, high, high-low
Out[31]:
(0.48963902786483265, 0.83173371081078573, 0.34209468294595308)
Exercise: To get a feel for the Pareto distribution, let’s see how different the world would be if the distribution of human height were Pareto. With the parameters xm = 1 m and α = 1.7, we get a distribution with a reasonable minimum, 1 m, and median, 1.5 m.
Plot this distribution. What is the mean human height in Pareto world? What fraction of the population is shorter than the mean? If there are 7 billion people in Pareto world, how many do we expect to be taller than 1 km? How tall do we expect the tallest person to be?
scipy.stats.pareto represents a pareto distribution. In Pareto world, the distribution of human heights has parameters alpha=1.7 and xmin=1 meter. So the shortest person is 100 cm and the median is 150.
In [32]:
alpha = 1.7
xmin = 1       # meter
dist = scipy.stats.pareto(b=alpha, scale=xmin)
dist.median()
Out[32]:
1.5034066538560549
What is the mean height in Pareto world?
In [33]:
# Solution

dist.mean()
Out[33]:
2.4285714285714288
What fraction of people are shorter than the mean?
In [34]:
# Solution

dist.cdf(dist.mean())
Out[34]:
0.77873969756528805
Out of 7 billion people, how many do we expect to be taller than 1 km? You could use dist.cdf or dist.sf.
In [35]:
# Solution

(1 - dist.cdf(1000)) * 7e9, dist.sf(1000) * 7e9
Out[35]:
(55602.976430479954, 55602.976430479954)
How tall do we expect the tallest person to be?
In [36]:
# Solution

# One way to solve this is to search for a height that we
# expect one person out of 7 billion to exceed.

# It comes in at roughly 600 kilometers.

dist.sf(600000) * 7e9            
Out[36]:
1.0525452731613427
In [37]:
# Solution

# Another way is to use `ppf`, which evaluates the "percent point function", which
# is the inverse CDF.  So we can compute the height in meters that corresponds to
# the probability (1 - 1/7e9).

dist.ppf(1 - 1/7e9)
Out[37]:
618349.61067595053
Exercise: The Weibull distribution is a generalization of the exponential distribution that comes up in failure analysis (see http://wikipedia.org/wiki/Weibull_distribution). Its CDF is
$mathrm{CDF}(x) = 1 − exp[−(x / λ)^k]$
Can you find a transformation that makes a Weibull distribution look like a straight line? What do the slope and intercept of the line indicate?
Use random.weibullvariate to generate a sample from a Weibull distribution and use it to test your transformation.
Generate a sample from a Weibull distribution and plot it using a transform that makes a Weibull distribution look like a straight line.
thinkplot.Cdf provides a transform that makes the CDF of a Weibull distribution look like a straight line.
In [38]:
sample = [random.weibullvariate(2, 1) for _ in range(1000)]
cdf = thinkstats2.Cdf(sample)
thinkplot.Cdf(cdf, transform='weibull')
thinkplot.Config(xlabel='Weibull variate', ylabel='CCDF')
Exercise: For small values of n, we don’t expect an empirical distribution to fit an analytic distribution exactly. One way to evaluate the quality of fit is to generate a sample from an analytic distribution and see how well it matches the data.
For example, in Section 5.1 we plotted the distribution of time between births and saw that it is approximately exponential. But the distribution is based on only 44 data points. To see whether the data might have come from an exponential distribution, generate 44 values from an exponential distribution with the same mean as the data, about 33 minutes between births.
Plot the distribution of the random values and compare it to the actual distribution. You can use random.expovariate to generate the values.
In [39]:
import analytic

df = analytic.ReadBabyBoom()
diffs = df.minutes.diff()
cdf = thinkstats2.Cdf(diffs, label='actual')

n = len(diffs)
lam = 44.0 / 24 / 60
sample = [rand

Allen Downey

Allen Downey

I am a Professor of Computer Science at Olin College in Needham MA, and the author of Think Python, Think Bayes, Think Stats and several other books related to computer science and data science. Previously I taught at Wellesley College and Colby College, and in 2009 I was a Visiting Scientist at Google, Inc. I have a Ph.D. from U.C. Berkeley and B.S. and M.S. degrees from MIT. Here is my CV. I write a blog about Bayesian statistics and related topics called Probably Overthinking It. Several of my books are published by O’Reilly Media and all are available under free licenses from Green Tea Press.