# Set up packages for lecture. Don't worry about understanding this code, but
# make sure to run it if you're following along.
import numpy as np
import babypandas as bpd
import pandas as pd
from matplotlib_inline.backend_inline import set_matplotlib_formats
import matplotlib.pyplot as plt
from scipy import stats
import otter
set_matplotlib_formats("svg")
plt.style.use('ggplot')

np.set_printoptions(threshold=20, precision=2, suppress=True)
pd.set_option("display.max_rows", 7)
pd.set_option("display.max_columns", 8)
pd.set_option("display.precision", 2)

# Setup to start where we left off last time
keep_cols = ['business_name', 'inspection_date', 'inspection_score', 'risk_category', 'Neighborhoods', 'Zip Codes']
restaurants_full = bpd.read_csv('data/restaurants_full.csv').get(keep_cols)
bakeries = restaurants_full[restaurants_full.get('business_name').str.lower().str.contains('bake')]
bakeries = bakeries[bakeries.get('inspection_score') >= 0] # Keeping only the rows where we know the inspection score

# Animation
from IPython.display import IFrame, display

def show_clt_slides():
    src = "https://docs.google.com/presentation/d/e/2PACX-1vTcJd3U1H1KoXqBFcWGKFUPjZbeW4oiNZZLCFY8jqvSDsl4L1rRTg7980nPs1TGCAecYKUZxH5MZIBh/embed?start=false&loop=false&delayms=3000"
    width = 960
    height = 509
    display(IFrame(src, width, height))

C:\Users\janin\Anaconda3\lib\site-packages\scipy\__init__.py:146: UserWarning: A NumPy version >=1.16.5 and <1.23.0 is required for this version of SciPy (detected version 1.23.1
  warnings.warn(f"A NumPy version >={np_minversion} and <{np_maxversion}"

Lecture 27 – Review, Conclusion

DSC 10, Winter 2023

Announcements

• The Final Exam is tomorrow at 3PM in Galbraith Hall 242.
  • See this EdStem announcement for details.
  • Assigned seats will be emailed to you by Friday.
  • We will check IDs.
  • You'll have 2 hours, 50 minutes to work on the exam.
  • No questions during the exam.
  • The DSC 10 Reference Sheet will be provided. No calculators or other notes.
  • Practice with old exam problems at practice.dsc10.com.
• If at least 80% of the class fills out both CAPEs and the End of Quarter Survey, then the entire class gets 0.5% of extra credit on their overall grade.
  • We value your feedback!

Agenda

• More review.
• Working on personal projects.
• Demo: Gapminder 🌎.
• Some parting thoughts.

Bakeries 🧁

Consider this population of bakeries in San Francisco.

bakeries

	business_name	inspection_date	inspection_score	risk_category	Neighborhoods	Zip Codes
327	Le Marais Bakery Castro	2018-08-06T00:00:00.000	90.0	Moderate Risk	NaN	NaN
365	Pho Luen Fat Bakery & Restaurant	2019-04-08T00:00:00.000	76.0	Low Risk	NaN	NaN
372	Brioche Bakery & Cafe	2019-01-31T00:00:00.000	88.0	Low Risk	NaN	NaN
...	...	...	...	...	...	...
53954	Fancy Wheatfield Bakery	2019-03-04T00:00:00.000	83.0	Moderate Risk	NaN	NaN
54102	New Hollywood Bakery & Restaurant	2016-08-30T00:00:00.000	74.0	High Risk	NaN	NaN
54171	Speciality's Cafe and Bakery	2019-04-29T00:00:00.000	89.0	Moderate Risk	NaN	NaN

1216 rows × 6 columns

score_bins = np.arange(50, 102, 2)
bakeries.plot(kind='hist', y='inspection_score', density=True, bins=score_bins, ec='w', figsize=(10, 5),
              title='Population Distribution');

For reference, the mean and standard deviation of the population distribution are calculated below.

bakeries.get('inspection_score').describe()

count    1216.00
mean       84.20
std         8.35
          ...   
50%        86.00
75%        90.00
max       100.00
Name: inspection_score, Length: 8, dtype: float64

In this case we happen to have the inspection scores for all members of the population, but in reality we won't. So let's instead take a random sample of 200 bakeries from the population.

np.random.seed(23) # Ignore this

sample_of_bakeries = bakeries.sample(200) # SOLUTION
sample_of_bakeries

	business_name	inspection_date	inspection_score	risk_category	Neighborhoods	Zip Codes
33359	Universal Bakery Inc.	2019-01-28T00:00:00.000	83.0	Low Risk	2.0	28859.0
19980	Cherry Blossom Bakery 2	2016-06-28T00:00:00.000	90.0	Moderate Risk	NaN	NaN
29825	Waterfront Bakery	2018-06-07T00:00:00.000	94.0	Low Risk	32.0	308.0
...	...	...	...	...	...	...
4835	Marla Bakery	2018-09-10T00:00:00.000	91.0	High Risk	NaN	NaN
26932	PRINCESS BAKERY	2016-08-16T00:00:00.000	79.0	Low Risk	5.0	28861.0
34201	Castro Tarts Cafe and Bakery Inc.	2017-08-23T00:00:00.000	82.0	Low Risk	NaN	NaN

200 rows × 6 columns

sample_of_bakeries.plot(kind='hist', y='inspection_score', density=True, bins=score_bins, ec='w', figsize=(10, 5),
                        title='Sample Distribution');

Note that since we took a large, random sample of the population, we expect that our sample looks similiar to the population and has a similar mean and SD.

sample_of_bakeries.get('inspection_score').describe() 

count    200.00
mean      84.67
std        8.38
          ...  
50%       87.00
75%       91.25
max       98.00
Name: inspection_score, Length: 8, dtype: float64

Indeed, the sample mean is quite close to the population mean, and the sample standard deviation is quite close to the population standard deviation.

Let's suppose we want to estimate the population mean (that is, the mean inspection score of all bakeries in SF).

One estimate of the population mean is the mean of our sample.

sample_of_bakeries.get('inspection_score').mean()

84.665

However, our sample was random and could have been different, meaning our sample mean could also have been different.

Question: What's a reasonable range of possible values for the sample mean? What is the distribution of the sample mean?

The Central Limit Theorem

The Central Limit Theorem (CLT) says that the probability distribution of the sum or mean of a large random sample drawn with replacement will be roughly normal, regardless of the distribution of the population from which the sample is drawn.

show_clt_slides()

To see an empirical distribution of the sample mean, let's take a large number of samples directly from the population and compute the mean of each one.

Remember, in real life we wouldn't be able to do this, since we wouldn't have access to the population.

sample_means = np.array([])

# BEGIN SOLUTION
for i in np.arange(5000):
    sample_mean = bakeries.sample(200).get('inspection_score').mean()
    sample_means = np.append(sample_means, sample_mean)
# END SOLUTION

sample_means

array([84.34, 85.02, 83.79, ..., 84.64, 84.49, 84.17])

bpd.DataFrame().assign(sample_means=sample_means).plot(kind='hist', density=True, ec='w', bins=25, figsize=(10, 5));

Unsurprisingly, the distribution of the sample mean is bell-shaped. The CLT told us that!

The CLT also tells us that

$$\text{SD of Distribution of Possible Sample Means} = \frac{\text{Population SD}}{\sqrt{\text{sample size}}}$$

Let's try this out.

np.std(bakeries.get('inspection_score')) / np.sqrt(200)

0.5904894545352809

np.std(sample_means)

0.5469232018985846

Pretty close! Remember that sample_means is an array of simulated sample means; the more samples we simulate, the closer that np.std(sample_means) will get to the SD described by the CLT.

Note that in practice, we won't have the SD of the population, since we'll usually just have a single sample. In such cases, we can use the SD of the sample as an estimate of the SD of the population:

np.std(sample_of_bakeries.get('inspection_score')) / np.sqrt(200)

0.5909855116667413

Using the CLT, we have that the distribution of the sample mean:

• is roughly normal,
• is centered at the population mean (for which the sample mean is an estimate), and
• has a standard deviation of $\frac{\text{Population SD}}{\sqrt{\text{sample size}}}$ (which can be estimated using $\frac{\text{Sample SD}}{\sqrt{\text{sample size}}}$).

Using this information, we can build a confidence interval for where we think the population mean might be. A 95% confidence interval for the population mean is given by

$$ \left[ \text{sample mean} - 2\cdot \frac{\text{sample SD}}{\sqrt{\text{sample size}}}, \ \text{sample mean} + 2\cdot \frac{\text{sample SD}}{\sqrt{\text{sample size}}} \right] $$

sample_mean = sample_of_bakeries.get('inspection_score').mean()
sample_std = np.std(sample_of_bakeries.get('inspection_score'))

[sample_mean - 2 * sample_std / np.sqrt(200), sample_mean + 2 * sample_std / np.sqrt(200)] # SOLUTION

[83.48302897666652, 85.8469710233335]

Concept Check ✅ – Answer at cc.dsc10.com

Using a single sample of 200 bakeries, how can we estimate the median inspection score of all bakeries in San Francisco with an inspection score? What technique should we use?

A. Standard hypothesis testing

B. Permutation testing

C. Bootstrapping

D. The Central Limit Theorem

Click for the answer after you've entered your guess above. Don't scroll any further.

Bootstrapping. The CLT only applies to sample means (and sums), not to any other statistics.

There is no CLT for sample medians, so instead we'll have to resort to bootstrapping to estimate the distribution of the sample median.

Recall, bootstrapping is the act of sampling from the original sample, with replacement. This is also called resampling.

# The median of our original sample – this is just one number
sample_of_bakeries.get('inspection_score').median() # SOLUTION

87.0

# The median of a single bootstrap resample – this is just one number
sample_of_bakeries.sample(200, replace=True).get('inspection_score').median() # SOLUTION

86.0

Let's resample repeatedly.

np.random.seed(23) # Ignore this

boot_medians = np.array([])

# BEGIN SOLUTION
for i in np.arange(5000):
    boot_median = sample_of_bakeries.sample(200, replace=True).get('inspection_score').median()
    boot_medians = np.append(boot_medians, boot_median)
# END SOLUTION

boot_medians

array([87. , 85. , 86.5, ..., 87.5, 88. , 86. ])

bpd.DataFrame().assign(boot_medians=boot_medians).plot(kind='hist', density=True, ec='w', bins=10, figsize=(10, 5));

Note that this distribution is not at all normal.

To compute a 95% confidence interval, we take the middle 95% of the bootstrapped medians.

# BEGIN SOLUTION
left = np.percentile(boot_medians, 2.5)
right = np.percentile(boot_medians, 97.5)

[left, right]
# END SOLUTION

[85.0, 88.0]

Discussion Question

Which of the following interpretations of this confidence interval are valid?

1. 95% of SF bakeries have an inspection score between 85 and 88.
2. 95% of the resamples have a median inspection score between 85 and 88.
3. There is a 95% chance that our sample has a median inspection score between 85 and 88.
4. There is a 95% chance that the median inspection score of all SF bakeries is between 85 and 88.
5. If we had taken 100 samples from the same population, about 95 of these samples would have a median inspection score between 85 and 88.
6. If we had taken 100 samples from the same population, about 95 of the confidence intervals created would contain the median inspection score of all SF bakeries.

Click for the answer after you've entered your guess above. Don't scroll any further.

The correct answers are Option 2 and Option 6.

Physicians 🩺

The setup

You work as a family physician. You collect data and you find that in 6354 patients, 3115 were children and 3239 were adults.

You want to test the following hypotheses:

• Null Hypothesis: Family physicians see an equal number of children and adults.
• Alternative Hypothesis: Family physicians see more adults than they see children.

Concept Check ✅ – Answer at cc.dsc10.com

Which test statistic(s) could be used for this hypothesis test? Which values of the test statistic point towards the alternative?

A. Proportion of children seen
B. Number of children seen
C. Number of children minus number of adults seen
D. Absolute value of number of children minus number of adults seen

There may be multiple correct answers; choose one.

Click for the answer after you've entered your guess above. Don't scroll any further.

All of these but the last one would work for this alternative. Small values of these statistics would favor the alternative. If the alternative was instead "Family physicians see a different number of children and adults", the last option would work while the first three wouldn't.

Let's use option B, the number of children seen, as a test statistic. Small values of this statistic favor the alternative hypothesis.

How do we generate a single value of the test statistic?

np.random.multinomial(6354, [0.5, 0.5])[0] # SOLUTION

3172

As usual, let's simulate the test statistic many, many times (10,000).

test_stats = np.array([])

# BEGIN SOLUTION
for i in np.arange(10000):
    stat = np.random.multinomial(6354, [0.5, 0.5])[0]
    test_stats = np.append(test_stats, stat)
# END SOLUTION

test_stats

array([3204., 3213., 3172., ..., 3150., 3198., 3213.])

bpd.DataFrame().assign(test_stats=test_stats) \
               .plot(kind='hist', density=True, ec='w', figsize=(10, 5), bins=20);
plt.axvline(3115, lw=3, color='black', label='observed statistic')
plt.legend();

Recall that you collected data and found that in 6354 patients, 3115 were children and 3239 were adults.

Concept Check ✅ – Answer at cc.dsc10.com

What goes in blank (a)?

p_value = np.count_nonzero(test_stats __(a)__ 3115) / 10000

A. >=

B. >

C. <=

D. <

Click for the answer after you've entered your guess above. Don't scroll any further.

<=

# Calculate the p-value

Concept Check ✅ – Answer at cc.dsc10.com

What do we do, assuming that we're using a 5% p-value cutoff?

A. Reject the null

B. Fail to reject the null

C. It depends

Click for the answer after you've entered your guess above. Don't scroll any further.

Fail to reject the null, since the p-value is above 0.05.

Note that while we used np.random.multinomial to simulate the test statistic, we could have used np.random.choice, too:

choices = np.random.choice(['adult', 'child'], p=[0.5, 0.5], size=6354, replace=True) # SOLUTION
choices

array(['adult', 'adult', 'adult', ..., 'child', 'child', 'adult'],
      dtype='<U5')

np.count_nonzero(choices == 'child') # SOLUTION

3142

Concept Check ✅ – Answer at cc.dsc10.com

Is this an example of bootstrapping?

A. Yes, because we are sampling with replacement.

B. No, this is not bootstrapping.

Click for the answer after you've entered your guess above. Don't scroll any further.

No, this is not bootstrapping. Bootstrapping is when we resample from a single sample; here we're simulating data under the assumptions of a model.

Personal projects

Using Jupyter Notebooks after DSC 10

• You may be interested in working on data science projects of your own.
• In this video, we show you how to make blank notebooks and upload datasets of your own to DataHub.
• Depending on the classes you're in, you may not have access to DataHub. Eventually, you'll want to install Jupyter Notebooks on your computer.
  • Anaconda is a great way to do that, as it also installs many commonly used packages.
  • You may want to download your work from DataHub so you can refer to it after the course ends.
  • Remember, all babypandas code is regular pandas code, too!

Finding data

These sites allow you to search for datasets (in CSV format) from a variety of different domains. Some may require you to sign up for an account; these are generally reputable sources.

Note that all of these links are also available at rampure.org/find-datasets.

• Data is Plural
• FiveThirtyEight.
• CORGIS.
• Kaggle Datasets.
• Google’s dataset search.
• DataHub.io.
• Data.world.
• R datasets.
• Wikipedia. (Use this site to extract and download tables as CSVs.)
• Awesome Public Datasets GitHub repo.
• Links to even more sources.

Domain-specific sources of data

• Sports: Basketball Reference, Baseball Reference, etc.
• US Government Sources: census.gov, data.gov, data.ca.gov, data.sfgov.org, FBI’s Crime Data Explorer, Centers for Disease Control and Prevention.
• Global Development: data.worldbank.org, databank.worldbank.org, WHO.
• Transportation: New York Taxi trips, Bureau of Transportation Statistics, SFO Air Traffic Statistics.
• Music: Spotify Charts.
• COVID: Johns Hopkins.
• Any Google Forms survey you’ve administered! (Go to the results spreadsheet, then go to “File > Download > Comma-separated values”.)

Tip: if a site only allows you to download a file as an Excel file, not a CSV file, you can download it, open it in a spreadsheet viewer (Excel, Numbers, Google Sheets), and export it to a CSV.

Join a DS3 Project Group 🤝

The Data Science Student Society organizes project groups, which are a great way to get experience and build your resume. Keep your eye out for applications!

• Students have the opportunity to join a team to pursue a unique data science project that will last two quarters.
• At the end of the project, teams will have developed a polished, complete personal project which they will showcase to their peers, faculty, and companies in the data science industry.
• Great for if you don't yet have a lot of experience and are looking for that first data science project to get you started.

Demo: Gapminder 🌎

plotly

• All of the visualizations (scatter plots, histograms, etc.) in this course were created using a library called matplotlib.
  • This library was called under-the-hood everytime we wrote df.plot.
• plotly is a different visualization library that allows us to create interactive visualizations.
• You may learn about it in a future course, but we'll briefly show you some cool visualizations you can make with it.

import plotly.express as px

Gapminder dataset

Gapminder Foundation is a non-profit venture registered in Stockholm, Sweden, that promotes sustainable global development and achievement of the United Nations Millennium Development Goals by increased use and understanding of statistics and other information about social, economic and environmental development at local, national and global levels. - Gapminder Wikipedia

gapminder = px.data.gapminder()
gapminder

	country	continent	year	lifeExp	pop	gdpPercap	iso_alpha	iso_num
0	Afghanistan	Asia	1952	28.80	8425333	779.45	AFG	4
1	Afghanistan	Asia	1957	30.33	9240934	820.85	AFG	4
2	Afghanistan	Asia	1962	32.00	10267083	853.10	AFG	4
...	...	...	...	...	...	...	...	...
1701	Zimbabwe	Africa	1997	46.81	11404948	792.45	ZWE	716
1702	Zimbabwe	Africa	2002	39.99	11926563	672.04	ZWE	716
1703	Zimbabwe	Africa	2007	43.49	12311143	469.71	ZWE	716

1704 rows × 8 columns

The dataset contains information for each country for several different years.

gapminder.get('year').unique()

array([1952, 1957, 1962, 1967, 1972, 1977, 1982, 1987, 1992, 1997, 2002,
       2007], dtype=int64)

Let's start by just looking at 2007 data (the most recent year in the dataset).

gapminder_2007 = gapminder[gapminder.get('year') == 2007]
gapminder_2007

	country	continent	year	lifeExp	pop	gdpPercap	iso_alpha	iso_num
11	Afghanistan	Asia	2007	43.83	31889923	974.58	AFG	4
23	Albania	Europe	2007	76.42	3600523	5937.03	ALB	8
35	Algeria	Africa	2007	72.30	33333216	6223.37	DZA	12
...	...	...	...	...	...	...	...	...
1679	Yemen, Rep.	Asia	2007	62.70	22211743	2280.77	YEM	887
1691	Zambia	Africa	2007	42.38	11746035	1271.21	ZMB	894
1703	Zimbabwe	Africa	2007	43.49	12311143	469.71	ZWE	716

142 rows × 8 columns

Scatter plot

We can plot life expectancy vs. GDP per capita. If you hover over a point, you will see the name of the country.

px.scatter(gapminder_2007, x='gdpPercap', y='lifeExp', hover_name='country')

In future courses, you'll learn about transformations. Here, we'll apply a log transformation to the x-axis to make the plot look a little more linear.

px.scatter(gapminder_2007, x='gdpPercap', y='lifeExp', log_x=True, hover_name='country')

Animated scatter plot

We can take things one step further.

px.scatter(gapminder,
           x = 'gdpPercap',
           y = 'lifeExp', 
           hover_name = 'country',
           color = 'continent',
           size = 'pop',
           size_max = 60,
           log_x = True,
           range_y = [30, 90],
           animation_frame = 'year',
           title = 'Life Expectancy, GDP Per Capita, and Population over Time'
          )

Watch this video if you want to see an even-more-animated version of this plot.

Animated histogram

px.histogram(gapminder,
            x = 'lifeExp',
            animation_frame = 'year',
            range_x = [20, 90],
            range_y = [0, 50],
            title = 'Distribution of Life Expectancy over Time')

Choropleth

px.choropleth(gapminder,
              locations = 'iso_alpha',
              color = 'lifeExp',
              hover_name = 'country',
              hover_data = {'iso_alpha': False},
              title = 'Life Expectancy Per Country',
              color_continuous_scale = px.colors.sequential.tempo
)

Parting thoughts

From Lecture 1: What is "data science"?

Data science is about drawing useful conclusions from data using computation. Throughout the quarter, we touched on several aspects of data science:

• In the first 4 weeks, we used Python to explore data.
  • Lots of visualization 📈📊 and "data manipulation", using industry-standard tools.

• In the next 4 weeks, we used data to infer about a population, given just a sample.
  • Rely heavily on simulation, rather than formulas.

• In the last 2 weeks, we used data from the past to predict what may happen in the future.
  • A taste of machine learning 🤖.

• In future courses – including DSC 20 and 40A, which you may be taking next quarter – you'll revisit all three of these aspects of data science.

Thank you!

This course would not have been possible without...

• Our 2 graduate TAs: Dylan Stockard and Dasha Veraksa.
• Our 17 undergraduate tutors: Gabriel Cha, Eric Chen, Charlie Gillet, Vanessa Hu, Dylan Lee, Anthony Li, Jasmine Lo, Linda Long, Aishani Mohapatra, Harshita Saha, Abel Seyoum, Selim Shaalan, Yutian (Skylar) Shi, Tony Ta, Zairan Xiang, Diego Zavalza, and Lauren (Luran) Zhang.

• Learn more about tutoring – it's fun, and you can be a tutor as early as your 3rd quarter at UCSD!
• Keep in touch! dsc10.com/staff
  • After grades are released, we'll make a post on EdStem where you can ask course staff for advice on courses and UCSD more generally.

Good luck on your finals! 🎉

And see you tomorrow at 3PM in Galbraith Hall 242. ⏰