In [1]:
import matplotlib
from datascience import *
%matplotlib inline
import matplotlib.pyplot as plots
import numpy as np
plots.style.use('fivethirtyeight')

Standard Units¶

In [2]:
exams = Table.read_table('exams_fa18.csv')
exams.show(5)
Midterm Final
91 88
89.5 84
78 71.5
87 88
72 72.5

... (1237 rows omitted)

In [3]:
exams.hist(overlay=False, bins=np.arange(0,101,5))
In [4]:
def standard_units(x):
    """Convert array of values to standard units"""
    return (x - np.average(x)) / np.std(x)
In [5]:
midterm_su = standard_units(exams.column('Midterm'))
exams = exams.with_column('Midterm in Standard Units', midterm_su)

final_su = standard_units(exams.column('Final'))
exams = exams.with_column('Final in Standard Units', final_su)

exams.show(10)
Midterm Final Midterm in Standard Units Final in Standard Units
91 88 1.16278 0.717241
89.5 84 1.03032 0.390935
78 71.5 0.0147889 -0.628768
87 88 0.809552 0.717241
72 72.5 -0.515053 -0.547191
71.5 61 -0.559206 -1.48532
82 92.5 0.368017 1.08433
84.5 87 0.588784 0.635664
85 92 0.632938 1.04355
78 71 0.0147889 -0.669556

... (1232 rows omitted)

In [6]:
exams.select(
    'Midterm in Standard Units', 'Final in Standard Units'
).hist(overlay=False, bins=np.arange(-4,2,0.1))

The SD and Bell Shaped Curves¶

In [7]:
births = Table.read_table('baby.csv')
In [8]:
births.hist('Maternal Height', bins = np.arange(56.5, 72.6, 1))
In [9]:
heights = births.column('Maternal Height')
np.mean(heights), np.std(heights)
Out[9]:
(64.049403747870528, 2.5250254409674375)
In [10]:
# 1 SD below & above the mean
np.mean(heights) - np.std(heights), np.mean(heights) + np.std(heights)
Out[10]:
(61.524378306903088, 66.574429188837968)

Central Limit Theorem¶

In [11]:
united = Table.read_table('united.csv')
united
Out[11]:
Date Flight Number Destination Delay
6/1/15 73 HNL 257
6/1/15 217 EWR 28
6/1/15 237 STL -3
6/1/15 250 SAN 0
6/1/15 267 PHL 64
6/1/15 273 SEA -6
6/1/15 278 SEA -8
6/1/15 292 EWR 12
6/1/15 300 HNL 20
6/1/15 317 IND -10

... (13815 rows omitted)

In [12]:
united_bins = np.arange(-20, 300, 10)
united.hist('Delay', bins=united_bins)
In [13]:
delays = united.column('Delay')
delay_mean = np.mean(delays)
delay_sd = np.std(delays)
delay_mean, delay_sd
Out[13]:
(16.658155515370705, 39.480199851609314)
In [14]:
percentile(50, delays)
Out[14]:
2
In [15]:
def one_sample_mean(sample_size):
    """ 
    Takes a sample from the population of flights 
    and computes its mean
    """
    sampled_flights = united.sample(sample_size)
    return np.mean(sampled_flights.column('Delay'))
In [16]:
one_sample_mean(100)
Out[16]:
20.780000000000001
In [17]:
def ten_thousand_sample_means(sample_size):
    means = make_array()
    for i in np.arange(10000):
        mean = one_sample_mean(sample_size)
        means = np.append(means, mean)
    return means
In [18]:
sample_means_100 = ten_thousand_sample_means(100)
In [19]:
sample_means_100
Out[19]:
array([ 13.26,  13.2 ,  20.8 , ...,  11.17,  19.01,  16.86])
In [20]:
len(sample_means_100)
Out[20]:
10000
In [21]:
Table().with_column(
    'Mean of 100 flight delays', sample_means_100).hist(bins=20)

print('Population Average:', delay_mean)

Population Average: 16.6581555154
In [22]:
sample_means_400 = ten_thousand_sample_means(400)
Table().with_column(
    'Mean of 400 flight delays', sample_means_400).hist(bins=20)

print('Population Average:', delay_mean)

Population Average: 16.6581555154
In [ ]: