import pandas as pd
import numpy as np
import os
pd.options.plotting.backend = 'plotly'

Lecture 8 – Unfaithful Data, Hypothesis Testing

DSC 80, Spring 2023

Agenda

• Messy data.
• Unfaithful data.
• Hypothesis testing.

Messy data

More data type ambiguities

• 1649043031 looks like a number, but is probably a date.

  • As we saw earlier, Unix timestamps count the number of seconds since January 1st, 1970.

• "USD 1,000,000" looks like a string, but is actually a number and a unit.

• 92093 looks like a number, but is really a zip code (and isn't equal to 92,093).

• Sometimes, False appears in a column of country codes. Why might this be? 🤔

Example: The Norway problem 🇳🇴

import yaml

player = '''
name: Magnus Carlsen
age: 32
country: NO
'''

yaml.safe_load(player)

{'name': 'Magnus Carlsen', 'age': 32, 'country': False}

Unfaithful data

Is the data "faithful" to the DGP?

• In other words, how well does the data represent reality?

• Does the data contain unrealistic or "incorrect" values?

  • Dates in the future for events in the past.
  • Locations that don't exist.
  • Negative counts.
  • Misspellings of names.
  • Large outliers.

Is the data "faithful" to the DGP?

• Does the data violate obvious dependencies?
  • Age and birthday don't match.
• Was the data entered by hand?
  • Spelling errors.
  • Fields shifted.
  • Did the form require fields or provide default values?
• Are there obvious signs of data falsification (also known as "curbstoning")?
  • Repeated names.
  • Fake looking email addresses.
  • Repeated use of uncommon names or fields.

Example: Police vehicle stops 🚔

The dataset we're working with contains all of the vehicle stops that the San Diego Police Department made in 2016.

stops = pd.read_csv(os.path.join('data', 'vehicle_stops_2016_datasd.csv'))
stops.head()

	stop_id	stop_cause	service_area	subject_race	subject_sex	subject_age	timestamp	stop_date	stop_time	sd_resident	arrested	searched	obtained_consent	contraband_found	property_seized
0	1308198	Equipment Violation	530	W	M	28	2016-01-01 00:06:00	2016-01-01	0:06	Y	N	N	N	N	N
1	1308172	Moving Violation	520	B	M	25	2016-01-01 00:10:00	2016-01-01	0:10	N	N	N	NaN	NaN	NaN
2	1308171	Moving Violation	110	H	F	31	2016-01-01 00:14:00	2016-01-01	0:14	NaN	NaN	NaN	NaN	NaN	NaN
3	1308170	Moving Violation	Unknown	W	F	29	2016-01-01 00:16:00	2016-01-01	0:16	N	N	N	NaN	NaN	NaN
4	1308197	Moving Violation	230	W	M	52	2016-01-01 00:30:00	2016-01-01	0:30	N	N	N	NaN	NaN	NaN

stops.shape

(103051, 15)

Data types

Are the data types correct? If not, are they easily fixable?

stops.head(1)

	stop_id	stop_cause	service_area	subject_race	subject_sex	subject_age	timestamp	stop_date	stop_time	sd_resident	arrested	searched	obtained_consent	contraband_found	property_seized
0	1308198	Equipment Violation	530	W	M	28	2016-01-01 00:06:00	2016-01-01	0:06	Y	N	N	N	N	N

stops.info()

<class 'pandas.core.frame.DataFrame'>
RangeIndex: 103051 entries, 0 to 103050
Data columns (total 15 columns):
 #   Column            Non-Null Count   Dtype 
---  ------            --------------   ----- 
 0   stop_id           103051 non-null  int64 
 1   stop_cause        103044 non-null  object
 2   service_area      103051 non-null  object
 3   subject_race      102920 non-null  object
 4   subject_sex       102865 non-null  object
 5   subject_age       100284 non-null  object
 6   timestamp         102879 non-null  object
 7   stop_date         103051 non-null  object
 8   stop_time         103051 non-null  object
 9   sd_resident       83689 non-null   object
 10  arrested          84400 non-null   object
 11  searched          83330 non-null   object
 12  obtained_consent  4791 non-null    object
 13  contraband_found  4969 non-null    object
 14  property_seized   4924 non-null    object
dtypes: int64(1), object(14)
memory usage: 11.8+ MB

Unfaithfulness

• Are there suspicious values?
• If a value is suspicious, can we trust the observation?
• For example, consider 'subject_age' – some are too high to be true, some are too low to be true.

stops['subject_age'].unique()

array(['28', '25', '31', '29', '52', '24', '20', '50', '23', '54', '53',
       '35', '38', '18', '48', '68', '45', '63', '49', '42', '27', '19',
       '55', '32', '47', '33', '41', '59', '60', '58', '26', '36', '40',
       '39', '21', '64', '30', '43', '17', '51', '34', '56', '44', '22',
       '69', '46', '16', '57', '37', '65', '72', '67', '66', '70', '62',
       '73', '74', '0', '77', nan, '89', '79', '61', '78', '99', '75',
       '85', '82', '71', '15', '80', '81', '93', '84', '76', '2', '4',
       '86', '91', '83', '88', '98', '87', 'No Age', '9', '100', '14',
       '95', '96', '92', '119', '1', '90', '163', '5', '114', '94', '10',
       '212', '220', '6', '145', '97', '120'], dtype=object)

ages = pd.to_numeric(stops['subject_age'], errors='coerce')
ages.describe()

count    99648.000000
mean        37.277697
std         14.456934
min          0.000000
25%         25.000000
50%         34.000000
75%         47.000000
max        220.000000
Name: subject_age, dtype: float64

Ages range all over the place, from 0 to 220. Was a 220 year old really pulled over?

stops[ages > 100]

	stop_id	stop_cause	service_area	subject_race	subject_sex	subject_age	timestamp	stop_date	stop_time	sd_resident	arrested	searched	obtained_consent	contraband_found	property_seized
25029	1333254	Equipment Violation	820	W	F	119	2016-03-19 22:20:00	2016-03-19	22:20	N	N	N	NaN	NaN	NaN
34179	1375483	Moving Violation	510	W	M	163	2016-04-18 16:35:00	2016-04-18	16:35	N	N	N	NaN	NaN	NaN
36968	1378369	Moving Violation	240	A	F	114	2016-04-28 11:44:00	2016-04-28	11:44	NaN	NaN	NaN	NaN	NaN	NaN
63570	1405478	Moving Violation	110	V	M	212	2016-08-04 18:10:00	2016-08-04	18:10	Y	N	N	NaN	NaN	NaN
70267	1411694	Moving Violation	310	H	F	220	2016-08-30 18:28:00	2016-08-30	18:28	Y	N	N	NaN	NaN	NaN
77038	1418885	Moving Violation	830	B	M	145	2016-09-22 20:35:00	2016-09-22	20:35	Y	N	Y	N	N	NaN
99449	1440889	Equipment Violation	120	W	F	120	2016-12-15 19:28:00	2016-12-15	19:28	Y	N	N	NaN	NaN	NaN

What about all of the stops that involved people under the legal driving age?

ages[ages < 16].value_counts()

0.0     218
15.0     27
2.0       6
14.0      5
4.0       4
1.0       2
10.0      2
9.0       1
5.0       1
6.0       1
Name: subject_age, dtype: int64

stops[ages < 16]

	stop_id	stop_cause	service_area	subject_race	subject_sex	subject_age	timestamp	stop_date	stop_time	sd_resident	arrested	searched	obtained_consent	contraband_found	property_seized
547	1308628	Moving Violation	120	W	M	0	2016-01-03 16:20:00	2016-01-03	16:20	NaN	NaN	NaN	NaN	NaN	NaN
747	1308820	Moving Violation	Unknown	H	F	0	2016-01-04 22:13:00	2016-01-04	22:13	N	N	N	NaN	NaN	NaN
1686	1309896	Moving Violation	120	W	M	15	2016-01-08 19:30:00	2016-01-08	19:30	N	N	N	NaN	NaN	NaN
1752	1309975	Moving Violation	Unknown	O	M	0	2016-01-08 23:20:00	2016-01-08	23:20	N	N	N	NaN	NaN	NaN
2054	1310377	Equipment Violation	430	H	F	0	2016-01-10 16:42:00	2016-01-10	16:42	Y	N	N	NaN	NaN	NaN
...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...
100465	1442806	Moving Violation	810	O	M	15	2016-12-20 10:45:00	2016-12-20	10:45	Y	N	N	NaN	NaN	NaN
101495	1443500	Moving Violation	320	W	M	0	2016-12-23 22:30:00	2016-12-23	22:30	NaN	NaN	NaN	NaN	NaN	NaN
101784	1443745	Moving Violation	Unknown	O	M	0	2016-12-26 19:30:00	2016-12-26	19:30	NaN	NaN	NaN	NaN	NaN	NaN
101790	1443747	Moving Violation	320	W	F	0	2016-12-26 20:00:00	2016-12-26	20:00	NaN	NaN	NaN	NaN	NaN	NaN
102848	1444647	Moving Violation	320	W	M	0	2016-12-31 17:45:00	2016-12-31	17:45	NaN	NaN	NaN	NaN	NaN	NaN

267 rows × 15 columns

Unfaithful 'subject_age'

• Ages of 'No Age' and 0 are likely explicit null values.
• What do we do about the exceptionally small and large ages?
  • Do we throw the entire row away, even if the rest of row is well-formed?
• What about the 14 and 15 year olds?
  • Each has more than one occurrence – these could be real entries!

Human-entered data

• Which fields were likely entered by a human?
• Which fields were likely generated by code?
  • What was the original source?

Let's look at all unique stop causes. Notice that there are three different causes related to bicycles, which should probably all fall under the same cause.

stops['stop_cause'].value_counts()

Moving Violation                      75200
Equipment Violation                   26234
Radio Call/Citizen Contact              571
Muni, County, H&S Code                  319
Personal Knowledge/Informant            289
No Cause Specified on a Card            184
Suspect Info (I.S., Bulletin, Log)      181
Personal Observ/Knowledge                45
MUNI, County, H&S Code                   14
Bicycle                                   2
Suspect Info                              2
BICYCLE                                   1
B & P                                     1
Bicycle Bicycle                           1
Name: stop_cause, dtype: int64

Let's plot the distribution of ages, within a reasonable range (15 to 85). What do you notice? How could we address this?

ages[(ages > 15) & (ages <= 85)].plot(kind='hist')

Now let's look at the first few and last few rows of stops.

stops[['timestamp', 'stop_date', 'stop_time']].head()

	timestamp	stop_date	stop_time
0	2016-01-01 00:06:00	2016-01-01	0:06
1	2016-01-01 00:10:00	2016-01-01	0:10
2	2016-01-01 00:14:00	2016-01-01	0:14
3	2016-01-01 00:16:00	2016-01-01	0:16
4	2016-01-01 00:30:00	2016-01-01	0:30

stops[['timestamp', 'stop_date', 'stop_time']].tail(10)

	timestamp	stop_date	stop_time
103041	NaN	2016-12-13	0.398611111
103042	NaN	2016-12-15	0.383333333
103043	NaN	2016-12-15	0.397916667
103044	NaN	2016-12-19	0:91
103045	NaN	2016-12-20	0:82
103046	NaN	2016-12-20	0.397916667
103047	NaN	2016-12-21	0:73
103048	NaN	2016-12-21	0:94
103049	NaN	2016-12-29	0:81
103050	NaN	2016-12-29	-0:81

Do you think '-0:81' is a time that a computer would record?

Unfaithful data vs. outliers

• Unfaithful data are data that don't accurately represent the data generating process.
• Outliers are "unusual" observations, unlike the rest of the data. They may be real, or they may be unfaithful.
  • For instance, it's possible that a 102-year old was pulled over for speeding.
• The two are hard to tell apart; doing so often requires research and domain knowledge.

Watch out for...

• Consistently "incorrect" values.

  • Example: Recorded ages of -1 or 99.
  • These are often "default" values, often used when a value is missing.
  • Solution: Change the value to the correct one if it is known!

• Abnormal artifacts from the data collection process.

  • Example: Spikes in recorded ages at round numbers (25, 30, 35, 40), or spikes in recorded COVID cases on Mondays.
  • Solution: Try "smoothing", e.g. binning the ages.

• Unreasonable outliers.

  • Example: Age of 200.
  • Solution: Not sure. Could remove the row. Could be indicative of a bug in the data collection process. Could be real!

Missing values

Where'd you go?

• Missing, or null, values in a dataset can occur from:

  • Intentional logic, where a value doesn't make sense.
  • A non-response in the measurement process.
  • Mistakes in the data recording process.

• Missing values are most often encoded with NULL, None, NaN, '', 0, etc.

• Important: While you may have "dealt" with null values in Project 1, the extent to which you handled them was calling .fillna(0). As we'll see over the next few weeks, we must take more care in treating null values!

Common representations of "null"

• All forms of 0 (e.g. 0, '0', 'zero') are common substitutes for null, but as we'll see, simply filling nulls with 0 is not always useful statistically.
• -1 is common if a column must be non-negative.
• 1900 and 1970 are common if a non-null date is required.
  • Remember, Unix time starts counting from January 1, 1970.

Common representations of "null"

• Some common representations for "null" are also real values themselves!
• For instance, the point 0°00'00.0"N+0°00'00.0"E in the South Atlantic Ocean is called "Null Island."

• This person's name is Mr. Null!
• Watch: Null Island: The Busiest Place That Doesn't Exist.

Missing values in the stops dataset

What are the non-NaN null values in the stops dataset?

• 'service_area': 'Unknown'.
• 'subject_age': 0, 'No Age'.
• Others?

stops

	stop_id	stop_cause	service_area	subject_race	subject_sex	subject_age	timestamp	stop_date	stop_time	sd_resident	arrested	searched	obtained_consent	contraband_found	property_seized
0	1308198	Equipment Violation	530	W	M	28	2016-01-01 00:06:00	2016-01-01	0:06	Y	N	N	N	N	N
1	1308172	Moving Violation	520	B	M	25	2016-01-01 00:10:00	2016-01-01	0:10	N	N	N	NaN	NaN	NaN
2	1308171	Moving Violation	110	H	F	31	2016-01-01 00:14:00	2016-01-01	0:14	NaN	NaN	NaN	NaN	NaN	NaN
3	1308170	Moving Violation	Unknown	W	F	29	2016-01-01 00:16:00	2016-01-01	0:16	N	N	N	NaN	NaN	NaN
4	1308197	Moving Violation	230	W	M	52	2016-01-01 00:30:00	2016-01-01	0:30	N	N	N	NaN	NaN	NaN
...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...
103046	1443046	Moving Violation	110	H	M	40	NaN	2016-12-20	0.397916667	Y	N	N	NaN	NaN	NaN
103047	1443132	Moving Violation	310	W	M	No Age	NaN	2016-12-21	0:73	N	N	N	NaN	NaN	NaN
103048	1443169	Moving Violation	Unknown	H	M	26	NaN	2016-12-21	0:94	Y	N	N	NaN	NaN	NaN
103049	1447090	Moving Violation	430	F	F	63	NaN	2016-12-29	0:81	N	N	N	NaN	NaN	NaN
103050	1445548	Moving Violation	320	O	F	34	NaN	2016-12-29	-0:81	Y	N	N	NaN	NaN	NaN

103051 rows × 15 columns

Finding null values in pandas

• Null values are encoded using NumPy's NaN value, which is of type float.
• The isna method for DataFrame/Series detects missing values.
  • It returns a Boolean DataFrame/Series.
  • isnull is equivalent to isna.

type(np.NaN)

float

# All of the rows where the subject age is missing.
stops[stops['subject_age'].isna()]

	stop_id	stop_cause	service_area	subject_race	subject_sex	subject_age	timestamp	stop_date	stop_time	sd_resident	arrested	searched	obtained_consent	contraband_found	property_seized
606	1309398	Moving Violation	Unknown	H	M	NaN	2016-01-04 00:01:00	2016-01-04	0:01	N	N	N	NaN	NaN	NaN
607	1309404	Moving Violation	510	W	M	NaN	2016-01-04 00:01:00	2016-01-04	0:01	Y	N	N	NaN	NaN	NaN
653	1309400	Moving Violation	620	W	F	NaN	2016-01-04 13:00:00	2016-01-04	13:00	Y	N	N	NaN	NaN	NaN
658	1309399	Moving Violation	620	W	F	NaN	2016-01-04 13:55:00	2016-01-04	13:55	Y	N	N	NaN	NaN	NaN
816	1309424	Moving Violation	620	W	F	NaN	2016-01-05 08:30:00	2016-01-05	8:30	N	N	N	NaN	NaN	NaN
...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...
102989	1412614	Moving Violation	120	W	F	NaN	NaN	2016-08-30	24:40:00	Y	N	N	NaN	NaN	NaN
102990	1414071	Moving Violation	110	W	M	NaN	NaN	2016-08-31	0:91	N	N	N	NaN	NaN	NaN
102991	1414020	Moving Violation	930	O	F	NaN	NaN	2016-08-31	98:35:00	Y	N	N	NaN	NaN	NaN
103002	1420099	Moving Violation	610	H	M	NaN	NaN	2016-09-27	0.565972222	N	N	N	NaN	NaN	NaN
103005	1422234	Moving Violation	620	W	M	NaN	NaN	2016-10-05	0.417361111	N	N	N	NaN	NaN	NaN

2767 rows × 15 columns

# Proportion of values missing in the subject_age column.
stops['subject_age'].isna().mean()

0.026850782622196777

# Proportion of missing values in all columns.
stops.isna().mean()

stop_id             0.000000
stop_cause          0.000068
service_area        0.000000
subject_race        0.001271
subject_sex         0.001805
subject_age         0.026851
timestamp           0.001669
stop_date           0.000000
stop_time           0.000000
sd_resident         0.187888
arrested            0.180988
searched            0.191371
obtained_consent    0.953508
contraband_found    0.951781
property_seized     0.952218
dtype: float64

Dropping observations with null values

• The dropna method,
  • when used on a Series, returns a new Series with all null entries removed.
  • when used on a DataFrame, returns a new DataFrame where all rows with at least one null value are removed.
• Don't drop rows unless absolutely necessary!
  • Usually, there is still useful information in the other columns.

stops.dropna().head()

	stop_id	stop_cause	service_area	subject_race	subject_sex	subject_age	timestamp	stop_date	stop_time	sd_resident	arrested	searched	obtained_consent	contraband_found	property_seized
0	1308198	Equipment Violation	530	W	M	28	2016-01-01 00:06:00	2016-01-01	0:06	Y	N	N	N	N	N
8	1308979	Moving Violation	310	H	F	25	2016-01-01 01:03:00	2016-01-01	1:03	Y	N	Y	N	N	N
20	1308187	Equipment Violation	510	H	M	31	2016-01-01 08:15:00	2016-01-01	8:15	Y	N	Y	N	Y	Y
21	1308186	Moving Violation	710	H	F	48	2016-01-01 08:21:00	2016-01-01	8:21	N	N	N	N	N	N
39	1308193	Moving Violation	710	H	F	32	2016-01-01 11:09:00	2016-01-01	11:09	N	N	N	N	N	N

stops.dropna().shape

(4619, 15)

stops.shape

(103051, 15)

Dropping observations with null values

When used on a DataFrame:

• .dropna() drops rows containing at least one null value.
• .dropna(how='all') drops rows containing only null values.
• .dropna(axis=1) drops columns containing at least one null value.
• Other keyword arguments: thresh, subset.

nans = pd.DataFrame([['dog', 0, 1, np.NaN], ['cat', np.NaN, np.NaN, np.NaN], ['dog', 1, 2, 3]], columns='A B C D'.split())
nans

	A	B	C	D
0	dog	0.0	1.0	NaN
1	cat	NaN	NaN	NaN
2	dog	1.0	2.0	3.0

nans.dropna(how='any')

	A	B	C	D
2	dog	1.0	2.0	3.0

nans.dropna(how='all')

	A	B	C	D
0	dog	0.0	1.0	NaN
1	cat	NaN	NaN	NaN
2	dog	1.0	2.0	3.0

nans.dropna(subset=['B', 'C'])

	A	B	C	D
0	dog	0.0	1.0	NaN
2	dog	1.0	2.0	3.0

nans.dropna(axis=1)

	A
0	dog
1	cat
2	dog

Filling null values

As you've seen, the fillna method replaces all null values. Specifically:

• .fillna(val) fills null entries with the value val.
• .fillna(dict) fills null entries using a dictionary dict of column/row values.
• .fillna(method='bfill') and .fillna(method='ffill') fill null entries using neighboring non-null entries.

nans

	A	B	C	D
0	dog	0.0	1.0	NaN
1	cat	NaN	NaN	NaN
2	dog	1.0	2.0	3.0

# Filling all NaNs with the same value.
nans.fillna('zebra')

	A	B	C	D
0	dog	0.0	1.0	zebra
1	cat	zebra	zebra	zebra
2	dog	1.0	2.0	3.0

# Filling NaNs differently for each column.
nans.fillna({'B': 'f0', 'C': 'f1', 'D': 'f2'})

	A	B	C	D
0	dog	0.0	1.0	f2
1	cat	f0	f1	f2
2	dog	1.0	2.0	3.0

# Dictionary of column means.
# Note that most numerical methods ignore null values.
means = {c: nans[c].mean() for c in nans.columns[1:]}
means

{'B': 0.5, 'C': 1.5, 'D': 3.0}

# Filling NaNs with column means
nans.fillna(means)

	A	B	C	D
0	dog	0.0	1.0	3.0
1	cat	0.5	1.5	3.0
2	dog	1.0	2.0	3.0

# Another way of doing the same thing.
nans.iloc[:, 1:].apply(lambda x: x.fillna(x.mean()), axis=0)

	B	C	D
0	0.0	1.0	3.0
1	0.5	1.5	3.0
2	1.0	2.0	3.0

# bfill stands for "backfill".
nans.fillna(method='bfill')

	A	B	C	D
0	dog	0.0	1.0	3.0
1	cat	1.0	2.0	3.0
2	dog	1.0	2.0	3.0

# ffill stands for "forward fill".
nans.fillna(method='ffill')

	A	B	C	D
0	dog	0.0	1.0	NaN
1	cat	0.0	1.0	NaN
2	dog	1.0	2.0	3.0

Filling null values, so far

• Both in Project 1, and in our brief exploration today, we filled null values either with a constant (like 0), or by using other information in the same column as the null value.

• What if we could use values in other columns to inform us on how to fill null values in a particular column? For instance, what if we could fill missing values in the 'D' column differently for rows where:
  • The 'A' column contains 'dog'.
  • The 'A' column contains 'cat'.

• To do so, we'd need to know whether the missingness of a particular column (like 'D') looks like it depends on the value of 'A'.

• To establish this relationship, we'd need to perform a permutation test, which is a form of hypothesis test.

Hypothesis testing

Hypothesis testing

• There are many competing ways of viewing how the world works.
  • That is, competing models for how the data were generated.
• How do we decide which models are unlikely to be true, and which are reasonable?

• We'll start with an intuitive exploration, and cover more details next class.

Example: Coin flipping

• Suppose we find a coin on the ground. We flip it 100 times, and see 59 heads and 41 tails.

• We want to answer the question "does this observation (59 heads, 41 tails) look like it came from a fair coin?"

• More precisely, we want to decide between two possible hypotheses.
  • Null Hypothesis: The coin is fair.
  • Alternative Hypothesis: The coin is biased in favor of heads.

• To decide, we need to know how rare it is to see 59 heads and 41 tails, or a result that's even more biased in favor of heads, when flipping a fair coin 100 times.

Test statistics

To decide, we need to know how rare it is to see 59 heads and 41 tails, or a result that's even more biased in favor of heads, when flipping a fair coin 100 times.

• To make this notion precise, we need to summarize our observation (59 heads and 41 tails) into a single number, called a test statistic, which will help us decide between the two hypotheses.

• Then, we can find the distribution of the test statistic, under the null hypothesis, which will give us a range of typical values for the test statistic under the assumption that the null hypothesis is true.

• Question: What are some possible test statistics in this scenario?

For the alternative hypothesis "the coin was biased towards heads", we could use:

• $N_H$ (number of heads).
• $\frac{N_H}{100}$ (proportion of heads).
• $N_H - 50$ (difference from expected number of heads).

For simplicity, we'll start with $N_H$.

• If a value of this test statistic is large, it means there are many heads in 100 flips. This would suggest that the coin flipped is biased in favor of heads (alternative hypothesis).

• If a value of this test statistic is small, it means there are few heads in 100 flips. This would suggest that the coin flipped is not biased in favor of heads (null hypothesis).

Generating the null distribution

• Now that we've chosen a test statistic, we need to generate the distribution of the test statistic under the assumption the null hypothesis is true, i.e. the null distribution.

• This distribution will give us, for instance:
  • The probability of seeing 4 heads in 100 flips of a fair coin.
  • The probability of seeing at most 46 heads in 100 flips of a fair coin.
  • The probability of seeing at least 59 heads in 100 flips of a fair coin.

Generating the null distribution, using math

The number of heads in 100 flips of a fair coin follows the $\text{Binomial(100, 0.5)}$ distribution, in which

$$P(\text{# heads} = k) = {100 \choose k} (0.5)^k{(1-0.5)^{100-k}} = {100 \choose k} 0.5^{100}$$

from scipy.special import comb

def p_k_heads(k):
    return comb(100, k) * (0.5) ** 100

The probability that we see at least 59 heads is then:

sum([p_k_heads(k) for k in range(59, 101)])

0.04431304005703377

Let's look at this distribution visually.

plot_df = pd.DataFrame().assign(k = range(101))
plot_df['p_k'] = p_k_heads(plot_df['k'])
plot_df['color'] = plot_df['k'].apply(lambda k: 'orange' if k >= 59 else 'blue')

fig = plot_df.plot(kind='bar', x='k', y='p_k', color='color', width=1000)
fig.add_annotation(text='This red area is called the p-value!', x=77, y=0.008, showarrow=False)

Making a decision

We saw that, in 100 flips of a fair coin, $P(\text{# heads} \geq 59)$ is only ~4.4%.

• This is quite low – it suggests that our observed result is quite unlikely under the null.

• As such, we will reject the null hypothesis – our observation is not consistent with the hypothesis that the coin is fair.

• The null still may be true – it's possible that the coin we flipped was fair, and we just happened to see a rare result. For the same reason, we also cannot "accept" the alternative.

• This probability – the probability of seeing a result at least as extreme as the observed, under the null hypothesis – is called the p-value.
  • If the p-value is below a pre-defined cutoff (often 5%), we reject the null.
  • Otherwise, we fail to reject the null.

Fun fact

• One researcher found that coin flips aren't 50/50, but rather are closer to 51/49, biased towards whichever side started facing up.
• Read this for more details.

Summary, next time

Summary

• Data cleaning is the process of transforming data so that it is an accurate representation of the data generating process.
• Unfaithful data is data that is not representative of the data generating process. When working with messy data, we must look for:
  • Missing values (i.e. "null" values).
  • Incorrect values.
• Useful methods to be aware of: fillna, isna/isnull, dropna.
• Hypothesis testing allows us to make confident conclusions regarding a data generating process, given some observed data.

Next time

• Hypothesis testing via simulation, as you saw in DSC 10.
• More examples and test statistics (e.g. TVD).